From e4cf7b7910ec3ec2f85f24a5ffa78e1ccf828a57 Mon Sep 17 00:00:00 2001 From: Marcel Date: Sat, 8 Jan 2022 23:53:49 +0100 Subject: [PATCH] exam2 + remaining files of the semester --- .gitignore | 3 +- Exam1/Exam1_Solution.ipynb | 2405 +++ Exam1/Exam2021_1_Feedback.jl | 22 + Exam2.zip | Bin 0 -> 405170 bytes Exam2/Exam2.ipynb | 1147 ++ Exam2/Exam2_FakeResults.pdf | Bin 0 -> 405136 bytes Exam2/jlFiles/printmat.jl | 246 + Problemsets/.DS_Store | Bin 6148 -> 6148 bytes Problemsets/Data/FFmFactorsPs.csv | 389 + Problemsets/Data/Options_prices_US_Canada.csv | 13953 ++++++++++++++++ Problemsets/Data/PS08_Data.xlsx | Bin 0 -> 548781 bytes Problemsets/PS07_Optimisation.ipynb | 420 + Problemsets/PS08_Volatility.ipynb | 579 + Problemsets/PS09a_PyCall.ipynb | 271 + Problemsets/PS09b_RCall.ipynb | 284 + Problemsets/PS09c_WarAndPeace.ipynb | 132 + Problemsets/PS10_DataFrames.ipynb | 210 + Problemsets/PS10b_OptionDeltas.ipynb | 229 + .../Solutions/PS01_Basics_Solution.ipynb | 1688 ++ .../Solutions/PS02_Stats_Solution.ipynb | 1742 ++ Problemsets/Solutions/PS03_CLT_Solution.ipynb | 5438 ++++++ .../PS04_YieldToMaturity_Solution.ipynb | 522 + .../PS05_TradingStrategy_Solution.ipynb | 1839 ++ .../PS07_Optimisation_Solution.ipynb | 1193 ++ .../Solutions/PS08_Volatility_Solution.ipynb | 6294 +++++++ .../Solutions/PS09a_PyCall_Solution.ipynb | 369 + .../Solutions/PS09b_RCall_Solution.ipynb | 363 + .../PS09c_WarAndPeace_Solution.ipynb | 389 + .../Solutions/PS10_DataFrames_Solution.ipynb | 559 + .../PS10b_OptionDeltas_Solution.ipynb | 645 + Problemsets/jlFiles/OlsNW.jl | 112 + Problemsets/jlFiles/OptionsCalculations.jl | 104 + 32 files changed, 41546 insertions(+), 1 deletion(-) create mode 100644 Exam1/Exam1_Solution.ipynb create mode 100644 Exam1/Exam2021_1_Feedback.jl create mode 100644 Exam2.zip create mode 100644 Exam2/Exam2.ipynb create mode 100644 Exam2/Exam2_FakeResults.pdf create mode 100644 Exam2/jlFiles/printmat.jl create mode 100644 Problemsets/Data/FFmFactorsPs.csv create mode 100644 Problemsets/Data/Options_prices_US_Canada.csv create mode 100644 Problemsets/Data/PS08_Data.xlsx create mode 100644 Problemsets/PS07_Optimisation.ipynb create mode 100644 Problemsets/PS08_Volatility.ipynb create mode 100644 Problemsets/PS09a_PyCall.ipynb create mode 100644 Problemsets/PS09b_RCall.ipynb create mode 100644 Problemsets/PS09c_WarAndPeace.ipynb create mode 100644 Problemsets/PS10_DataFrames.ipynb create mode 100644 Problemsets/PS10b_OptionDeltas.ipynb create mode 100644 Problemsets/Solutions/PS01_Basics_Solution.ipynb create mode 100644 Problemsets/Solutions/PS02_Stats_Solution.ipynb create mode 100644 Problemsets/Solutions/PS03_CLT_Solution.ipynb create mode 100644 Problemsets/Solutions/PS04_YieldToMaturity_Solution.ipynb create mode 100644 Problemsets/Solutions/PS05_TradingStrategy_Solution.ipynb create mode 100644 Problemsets/Solutions/PS07_Optimisation_Solution.ipynb create mode 100644 Problemsets/Solutions/PS08_Volatility_Solution.ipynb create mode 100644 Problemsets/Solutions/PS09a_PyCall_Solution.ipynb create mode 100644 Problemsets/Solutions/PS09b_RCall_Solution.ipynb create mode 100644 Problemsets/Solutions/PS09c_WarAndPeace_Solution.ipynb create mode 100644 Problemsets/Solutions/PS10_DataFrames_Solution.ipynb create mode 100644 Problemsets/Solutions/PS10b_OptionDeltas_Solution.ipynb create mode 100644 Problemsets/jlFiles/OlsNW.jl create mode 100644 Problemsets/jlFiles/OptionsCalculations.jl diff --git a/.gitignore b/.gitignore index 2170e45..aba41e0 100644 --- a/.gitignore +++ b/.gitignore @@ -1,2 +1,3 @@ /.DS_Store -/.ipynb_checkpoints \ No newline at end of file +/.ipynb_checkpoints +.DS_Store \ No newline at end of file diff --git a/Exam1/Exam1_Solution.ipynb b/Exam1/Exam1_Solution.ipynb new file mode 100644 index 0000000..5452850 --- /dev/null +++ b/Exam1/Exam1_Solution.ipynb @@ -0,0 +1,2405 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "printyellow (generic function with 1 method)" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "using Printf, Dates, Statistics, DelimitedFiles, StatsBase\n", + "\n", + "include(\"jlFiles/printmat.jl\")" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "using Plots\n", + "\n", + "gr(size=(480,320))\n", + "default(fmt = :svg)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Load Data \n", + "\n", + "Returns from FX investments (for a US investor) in percent are in `Data_Returns.csv` (extract the returns into a matrix `R`) and log forward premia in percent are in `Data_Forwardpremia.csv` (extract the forward premia into a matrix `fp`). Convert the first columns to Julia dates.\n", + "\n", + "Test if the dates in the two files are the same.\n", + "\n", + "The currency names/abbreviations are\n", + "`[\"AUD\",\"CAD\",\"EUR\",\"JPY\",\"NZD\",\"NOK\",\"SEK\",\"CHF\",\"GBP\"]` " + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(300, 9)" + ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "CurrNames = [\"AUD\",\"CAD\",\"EUR\",\"JPY\",\"NZD\",\"NOK\",\"SEK\",\"CHF\",\"GBP\"]\n", + "\n", + "x = readdlm(\"Data_Returns.csv\",',',skipstart=1) #return data\n", + "dN = Date.(x[:,1]) #covert to Julia date, eg. 2001-12-31\n", + "R = Float64.(x[:,2:end])\n", + "\n", + "x = readdlm(\"Data_Forwardpremia.csv\",',',skipstart=2) #forward premia, skip 2 rows\n", + "dN2 = Date.(x[:,1]) \n", + "fp = Float64.(x[:,2:end])\n", + "\n", + "(T,n) = size(R) #number of data points, number of currencies" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Same dates? true\n" + ] + } + ], + "source": [ + "println(\"Same dates? \",dN == dN2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Plot the forward premia\n", + "\n", + "against time. Create a figure with 3x3 subplots. In the plot show the forward premia multiplied by 12 so they can be interpreted as the (annualised) interest rate differential (foreign minus US). \n", + "\n", + "Make sure to \n", + "1. have tick marks on the x-axis at `[Date(2000),Date(2010),Date(2020)]`\n", + "\n", + "2. limit the y-axis to `(-6,6)`\n", + "\n", + "3. Put the currency name/abbreviation in the title of each subfigure." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "xTicksLoc = [Date(2000),Date(2010),Date(2020)]\n", + "xTicksLab = Dates.format.(xTicksLoc,\"Y\")\n", + "\n", + "p1 = plot( dN,fp*12,\n", + " layout = (3,3),\n", + " legend = false,\n", + " size = (800,600),\n", + " linecolor = :blue,\n", + " xticks = (xTicksLoc,xTicksLab),\n", + " ylims = (-6,6),\n", + " title = reshape(CurrNames,1,:),\n", + " titlefont = font(10) )\n", + "display(p1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Plot cumulated log returns\n", + "\n", + "in a 3x3 figure (similar to before). The cumulated log returns for a given currency are $r_1,r_1+r_2,r_1+r_2+r_3,...$ etc where $r_t$ is the log return in period $t$.\n", + "\n", + "Make sure to\n", + "1. use the same tick marks on the x-axis as before\n", + "2. limit the y-axis to (-1,1)\n", + "\n", + "Hints:\n", + "1. log returns are $r= \\ln(1+R/100)$ since the returns in `R` are in percent. In Julia, use `log()` for the natural logarithm.\n", + "2. cumulate using `cumsum()` or by a loop so you get a $T \\times 9$ matrix." + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "r = log.(1.0.+R/100)\n", + "P = cumsum(r,dims=1)\n", + "\n", + "p1 = plot( dN,P,\n", + " layout = (3,3),\n", + " legend = false,\n", + " size = (800,600),\n", + " linecolor = :blue,\n", + " xticks = (xTicksLoc,xTicksLab),\n", + " ylims = (-1,1),\n", + " title = reshape(CurrNames,1,:),\n", + " titlefont = font(10) )\n", + "display(p1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Implement a carry trade strategy\n", + "\n", + "\n", + "1. Find the 4 currencies with the highest forward premia (interest rate differential) in $t-1$ and give each a portfolio weight `w[t,i]=1/4`. These are the investment currencies.\n", + "\n", + "2. Find the 4 currencies with the lowest forward premia in $t-1$ and give each a portfolio weight `w[t,i]=-1/4`. These are the funding currencies.\n", + "\n", + "3. The portfolio return in t is `w[t,:]'*R[t,:]`. For all periods that lacks data, set the portfolio return to 0.\n", + "\n", + "4. Create log returns, cumulate and plot (as above).\n", + "\n", + "5. In the plot, add a comparison with an equally weighted portfolio (1/9 in each currency). " + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [], + "source": [ + "m = 4 #number of long/short positions \n", + "\n", + "(R_CT,w_CT) = (zeros(T),zeros(T,n))\n", + "for t = 2:T #loop over periods, save portfolio returns\n", + " #local v,wt #local/global is needed in script\n", + " v = sortperm(fp[t-1,:])\n", + " w = zeros(n)\n", + " w[v[1:m]] .= -1/m #low interest rate currencies\n", + " w[v[end-m+1:end]] .= 1/m #high interest rate currencies \n", + " R_CT[t] = w'R[t,:]\n", + " w_CT[t,:] = w \n", + "end\n", + "\n", + "R_EW = vec(mean(R,dims=2)); #equally weighted portfolio" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "P_CT = cumsum(log.(1.0.+R_CT/100))\n", + "P_EW = cumsum(log.(1.0.+R_EW/100));\n", + "\n", + "p1 = plot( dN,[P_CT P_EW],\n", + " label = [\"CT\" \"EW\"],\n", + " size = (800,600),\n", + " linecolor = [:red :blue],\n", + " xticks = (xTicksLoc,xTicksLab),\n", + " ylims = (-1,1),\n", + " title = \"Comparison of performance\")\n", + "display(p1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Report some return statistics\n", + "\n", + "1. For each of the two portfolio returns, report the average returns, standard deviation and their ratio (the \"Sharpe ratio\").\n", + "\n", + "2. For each currency, report the frequency of periods that the currency is in the \"high\" carry trade portfolio." + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " CT EW\n", + "mean 0.283 0.018\n", + "std 2.038 2.255\n", + "SR 0.139 0.008\n", + "\n", + "and in case you want annualised numbers...\n", + " CT EW\n", + "mean 3.401 0.221\n", + "std 7.061 7.813\n", + "SR 0.482 0.028\n", + "\n" + ] + } + ], + "source": [ + "μ = mean([R_CT R_EW],dims=1)\n", + "σ = std([R_CT R_EW],dims=1)\n", + "SR = μ./σ\n", + "\n", + "printmat([μ;σ;SR],colNames=[\"CT\",\"EW\"],rowNames=[\"mean\",\"std\",\"SR\"])\n", + "\n", + "println(\"and in case you want annualised numbers...\")\n", + "(μ_a,σ_a,SR_a) = (μ*12,σ*sqrt(12),SR*sqrt(12))\n", + "printmat([μ_a;σ_a;SR_a],colNames=[\"CT\",\"EW\"],rowNames=[\"mean\",\"std\",\"SR\"])" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "...and for log returns, in %\n", + " CT EW\n", + "mean 0.262 -0.007\n", + "std 2.046 2.256\n", + "SR 0.128 -0.003\n", + "\n", + "and in case you want annualised numbers...\n", + " CT EW\n", + "mean 3.147 -0.083\n", + "std 7.089 7.816\n", + "SR 0.444 -0.011\n", + "\n" + ] + } + ], + "source": [ + "println(\"...and for log returns, in %\")\n", + "\n", + "μ = mean(log.(1.0.+[R_CT R_EW]/100),dims=1)*100\n", + "σ = std(log.(1.0.+[R_CT R_EW]/100),dims=1)*100\n", + "SR = μ./σ\n", + "\n", + "printmat([μ;σ;SR],colNames=[\"CT\",\"EW\"],rowNames=[\"mean\",\"std\",\"SR\"])\n", + "\n", + "println(\"and in case you want annualised numbers...\")\n", + "(μ_a,σ_a,SR_a) = (μ*12,σ*sqrt(12),SR*sqrt(12))\n", + "printmat([μ_a;σ_a;SR_a],colNames=[\"CT\",\"EW\"],rowNames=[\"mean\",\"std\",\"SR\"])" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "AUD 0.983\n", + "CAD 0.467\n", + "EUR 0.023\n", + "JPY 0.000\n", + "NZD 0.957\n", + "NOK 0.710\n", + "SEK 0.317\n", + "CHF 0.000\n", + "GBP 0.530\n", + "\n" + ] + } + ], + "source": [ + "FreqInHi = mean(0 .< w_CT,dims=1) #frequence of times the weight is positive\n", + "\n", + "printmat(vec(FreqInHi),rowNames=CurrNames)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "@webio": { + "lastCommId": null, + "lastKernelId": null + }, + "anaconda-cloud": {}, + "kernelspec": { + "display_name": "Julia 1.6.3", + "language": "julia", + "name": "julia-1.6" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.6.3" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Exam1/Exam2021_1_Feedback.jl b/Exam1/Exam2021_1_Feedback.jl new file mode 100644 index 0000000..dc318e8 --- /dev/null +++ b/Exam1/Exam2021_1_Feedback.jl @@ -0,0 +1,22 @@ + +using Printf +include("jlFiles/printmat.jl") + +println("Three ways of doing the same thing") + +fp = [1 2; + 1.1 2.1] + +x1 = fp*10 #easiest and fastest +printmat(x1) + +x2 = map(i->fp[:,i]*10,1:2) #cumbersome +x2 = hcat(x2...) +printmat(x2) + +x3 = zeros(size(fp)) #even more cumbersome +for i=1:size(fp,1),j=1:size(fp,2) + x3[i,j] = fp[i,j]*10 +end +printmat(x3) + diff --git a/Exam2.zip b/Exam2.zip new file mode 100644 index 0000000000000000000000000000000000000000..f5a44fc12c704b90cf862abb77a72a36de00a693 GIT binary patch literal 405170 zcmZs?V{j%w*DdD8;& z-nF~BYqx?l7&toMKMf?|r^~4sHz|UpzelO4e$Tie?}K~H~#Ru>k<2|G}XDrL3>)Z0BhDKVbg>`oDl(rpx{wVni0}b3njw zE>b-yguCpVMM&N`0__l#RDKHz0web#qIf4Ph!si zs;Mf{PX3;MXrF)D|NZ>Mej26qMe_S|MUM{g1_pkJF&a%o<;rM<6@G|Q_|F&eiNx}6 zLqDdf=)%#G;#C6Ax~3;N6Z^&rgS9PQ>*U6xVJp{-Tl z=b3AX%n2FF2@v}A0@UZv>@uKI2>AE$GknKj6w=H2 zX7hPa&NE}_zKblc|+pU4{YA`59i1fqdVsUa1upO-RPH 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unchanged)." + ] + }, + { + "cell_type": "code", + "execution_count": 97, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Average across the simulations: 1.989\n", + "\n", + "Std across the samples (with ρ=0.75) and in theory:\n", + "simulations theory\n", + " 0.534 0.206\n", + "\n" + ] + } + ], + "source": [ + "M = 10_000\n", + "\n", + "(T,ρ2,σ,μ) = (500,0.75,3,2)\n", + "\n", + "μi2 = fill(NaN, M)\n", + "σi2 = fill(NaN, M)\n", + "\n", + "# Monte Carlo simulation\n", + "for i = 1:M\n", + " y = SimAR1(T, ρ2, σ, μ)\n", + " μi2[i] = mean(y)\n", + " σi2[i] = std(y)\n", + "end\n", + "\n", + "# Theoretical results\n", + "y2 = SimAR1(10_000,ρ2,σ,μ)\n", + "σy2 = std(y2)\n", + "s2 = σy2/sqrt(T)\n", + "\n", + "printmat(\"Average across the simulations:\", mean(μi2))\n", + "println(\"Std across the samples (with ρ=0.75) and in theory:\")\n", + "printmat([\"simulations\", std(μi2)], [\"theory\", s2])" + ] + }, + { + "cell_type": "code", + "execution_count": 98, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "You decide to test the hypothesis that $\\mu=2$. Your decision rule is \n", + "\n", + "- reject the hypothesis if $|(\\mu_i-2)/s|>1.645$ with $s=\\sigma_y/\\sqrt{T}$\n", + "\n", + "With this decision rule, you are clearly assuming that the theoretical result (definition of $s$) is correct.\n", + "\n", + "Estimate both $\\mu_i$ and $\\sigma_y$ from each sample.\n", + "\n", + "In what fraction of the $M$ simulation do you reject your hypothesis when $\\rho=0$ and when $\\rho=0.75$? For the other parameters, use `(T,σ,μ) = (500,3,2)` (same as before)." + ] + }, + { + "cell_type": "code", + "execution_count": 99, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Frequency of rejections:\n", + " with ρ=0 with ρ=0.75 \n", + " 0.098 0.536\n", + "\n" + ] + } + ], + "source": [ + "(T,σ,μ) = (500,3,2)\n", + "\n", + "rejection1 = 0\n", + "rejection2 = 0\n", + "\n", + "# Count how many times we reject the hypothesis\n", + "for i = 1:M\n", + " # rejections for ρ = 0\n", + " s1 = σi1[i]/sqrt(T)\n", + " if abs((μi1[i] - 2)/s1) > 1.645\n", + " rejection1 += 1\n", + " end\n", + "\n", + " # rejections for ρ = 0.75\n", + " s2 = σi2[i]/sqrt(T)\n", + " if abs((μi2[i] - 2)/s2) > 1.645\n", + " rejection2 += 1\n", + " end\n", + "end\n", + "\n", + "println(\"Frequency of rejections:\")\n", + "printmat([\"with ρ=0 \", rejection1 / (M)], [\"with ρ=0.75 \", rejection2 / (M)])" + ] + } + ], + "metadata": { + "@webio": { + "lastCommId": null, 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Sales Constraints" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Load Packages and Utility Functions" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "printyellow (generic function with 1 method)" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "using Printf, LinearAlgebra, Optim\n", + "\n", + "include(\"jlFiles/printmat.jl\")" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "using Plots\n", + "\n", + "#pyplot(size=(600,400)) #use pyplot or gr\n", + "gr(size=(480,320))\n", + "default(fmt = :svg)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Traditional MV Calculations \n", + "\n", + "(no constraints)" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "3" + ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "μ = [11.5, 9.5, 6]/100 #expected returns\n", + "Σ = [166 34 58; #covariance matrix\n", + " 34 64 4;\n", + " 58 4 100]/100^2\n", + "\n", + "assetNames = [\"A\",\"B\",\"C\"];\n", + "n = length(μ)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## A Function for Calculating the (traditional) Mean-Variance Frontier\n" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "MVCalc" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "\"\"\"\n", + " MVCalc(μstar,μ,Σ)\n", + "\n", + "Calculate the std and weights of a portfolio (with mean return μstar) on MVF of risky assets.\n", + "\n", + "# Remark\n", + "- Only (λ,δ) and thus (w,stdRp) depend on μstar. We could therefore speed up the computations a bit\n", + "by doing the loop over different μstar values inside the function (and thus not recalculate Σ_1,a,b,c).\n", + "\"\"\"\n", + "function MVCalc(μstar,μ,Σ)\n", + " n = length(μ)\n", + " Σ_1 = inv(Σ)\n", + " a = μ'Σ_1*μ\n", + " b = μ'Σ_1*ones(n)\n", + " c = ones(n)'Σ_1*ones(n)\n", + " λ = (c*μstar - b)/(a*c-b^2)\n", + " δ = (a-b*μstar)/(a*c-b^2)\n", + " w = Σ_1 *(μ*λ.+δ)\n", + " StdRp = sqrt(w'Σ*w)\n", + " return StdRp,w\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "μstar_range = range(0.04,0.15,length=201)\n", + "L = length(μstar_range)\n", + "\n", + "StdRp = fill(NaN,L)\n", + "for i = 1:L\n", + " StdRp[i] = MVCalc(μstar_range[i],μ,Σ)[1]\n", + "end\n", + "\n", + "p1 = plot( StdRp*100,μstar_range*100,\n", + " linecolor = :red,\n", + " linewidth = 2,\n", + " label = \"MVF\",\n", + " legend = :topleft,\n", + " xlim = (0,15),\n", + " ylim = (0,15),\n", + " title = \"MVF (no portfolio constraints)\",\n", + " xlabel = \"Std(Rp), %\",\n", + " ylabel = \"ERp, %\" )\n", + "scatter!(sqrt.(diag(Σ))*100,μ*100,color=:red,label=\"assets\")\n", + "display(p1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# The Maximization Problem\n", + "\n", + "The investor maximizes:\n", + "\n", + "$\\text{E}R_{p}-\\frac{k}{2} \\text{Var}(R_{p})$,\n", + "\n", + "where $\\text{E}R_{p}= w'\\mu$ and $\\text{Var}(R_{p}=w'\\Sigma w$\n", + "\n", + "subject to $\\Sigma_{i=1}^n w_{i} = 1$.\n", + "\n", + "We can trace out the mean-variance frontier by solving this problem for different values of $k$.\n", + "\n", + "To impose the restriction, optimise over `v` and let `w=[v;1-sum(v)]`" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "objfun1 (generic function with 1 method)" + ] + }, + "execution_count": 6, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "function objfun1(v,k,μ,Σ)\n", + " w = [v;1-sum(v)]\n", + " ERp = w'μ\n", + " VarRp = w'Σ*w\n", + " StdRp = sqrt(VarRp)\n", + " U = ERp - k/2*VarRp\n", + " loss = -U #minimise this\n", + " return loss, w, ERp, StdRp\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "A 0.313\n", + "B 0.695\n", + "C -0.008\n", + "\n", + "σₚ 0.079\n", + "μₚ 0.102\n", + "\n" + ] + } + ], + "source": [ + "k = 10\n", + "Sol = optimize(v -> objfun1(v,k,μ,Σ)[1],zeros(n-1))\n", + "\n", + "v = Optim.minimizer(Sol)\n", + "(_,w,ERp,StdRp) = objfun1(v,k,μ,Σ)\n", + "\n", + "printmat(w,rowNames=assetNames)\n", + "printmat([StdRp,ERp],rowNames=[\"σₚ\",\"μₚ\"])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 1\n", + "\n", + "Redo the optimisation for the following range of $k$ values:\n", + "`k = [range(0.1,1.9,length=30);range(1,30,length=25)]`\n", + "\n", + "Then plot the resulting \"efficient frontier\" (similar to the MV plot above). Is is the same as the plot above?" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 2\n", + "\n", + "Change the optimisation problem to `U = ERp + cost - k/2*VarRp` where `cost` is (sum of the *negative* portfolio weights)*0.05. This is a (crude/dramatic) attaempt to illustrate the impact of costs of short selling ($w_i<0$) assets. Plot the new \"effecient frontier\" together with the old one." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 3\n", + "\n", + "Compare the portfolio weights (with/without the extra `cost`)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "@webio": { + "lastCommId": null, + "lastKernelId": null + }, + "anaconda-cloud": {}, + "kernelspec": { + "display_name": "Julia 1.6.3", + "language": "julia", + "name": "julia-1.6" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.6.3" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Problemsets/PS08_Volatility.ipynb b/Problemsets/PS08_Volatility.ipynb new file mode 100644 index 0000000..25c98b9 --- /dev/null +++ b/Problemsets/PS08_Volatility.ipynb @@ -0,0 +1,579 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Load Packages and Extra Functions" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "printyellow (generic function with 1 method)" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "using Printf, Dates, Statistics, Distributions, XLSX\n", + "\n", + "include(\"jlFiles/printmat.jl\")" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "using Plots\n", + "\n", + "gr(size=(480,320))\n", + "default(fmt = :svg)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Loading Daily S&P 500 Data\n", + "\n", + "The data file contains some missing values indicated by the value -999.99. Do not use those observations when the S&P level is missing. \n", + "\n", + "1. Calculate daily returns (treat missing days as if they do not exist).\n", + "\n", + "2. Create a vector of dates (from the first column in the data set)." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Number of days in the sample: 17864\n" + ] + } + ], + "source": [ + "xx = XLSX.readxlsx(\"Data/PS08_Data.xlsx\") #reads a sheet from the xlsx file\n", + "x = xx[\"SP500RfPs!A2:B17870\"] #pick out some cells\n", + "\n", + "...do the rest" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 1\n", + "\n", + "1. Show a normalised histogram of the returns. Use the bins `-5:0.25:5`\n", + "2. Add a curve for the density of the best fitting $N(\\mu,\\sigma)$ distribution." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(μ,σ) = (mean(R),std(R))\n", + "\n", + "xGrid = -5:0.25:5\n", + "pdfX = pdf.(Normal(μ,σ),xGrid) #\"Distributions\" wants σ, not σ^2\n", + "\n", + "do the rest..." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 2\n", + "\n", + "1. Calculate the quantile 0.01 in the data and according to the fitted normal distribution. Change the sign and call it \"99% Value at Risk\". Hint: `quantile(R,0.01)`and `quantile(Normal(μ,σ),0.01)`\n", + "\n", + "2. Calculate the expected return conditional on the return being less than the 5th quantile, both in data and according to the normal distribution. Change sign and call it \"expected shortfall\". Hint: `mean(TruncatedNormal(μ,σ,-Inf,quantile001))`\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 3\n", + "\n", + "Construct an simple estimate of $\\sigma_t^2$ as a backward looking exponential moving average\n", + "\n", + "$\\sigma_t^2 = \\lambda \\sigma_{t-1}^2 + (1-\\lambda) (R_{t-1} -\\mu_{t-1})^2$,\n", + "where $\\mu_{t}=\\lambda \\mu_{t-1} + (1-\\lambda) R_{t-1}$ \n", + "\n", + "Use $\\lambda=0.94$\n", + "\n", + "Plot the standard deviation in a time series plot." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 4\n", + "\n", + "Calculate 99% VaR and ES according to time-varying normal distributions `Normal(μ[t],sqrt(σ²[t]))` and plot the results (as time series plots)." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "@webio": { + "lastCommId": null, + "lastKernelId": null + }, + "anaconda-cloud": {}, + "kernelspec": { + "display_name": "Julia 1.6.3", + "language": "julia", + "name": "julia-1.6" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.6.3" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Problemsets/PS09a_PyCall.ipynb b/Problemsets/PS09a_PyCall.ipynb new file mode 100644 index 0000000..945fc52 --- /dev/null +++ b/Problemsets/PS09a_PyCall.ipynb @@ -0,0 +1,271 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Python\n", + "\n", + "using the [PyCall.jl](https://github.com/JuliaPy/PyCall.jl) package." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "CovNWFn" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "using Printf, DelimitedFiles, LinearAlgebra, Statistics\n", + "\n", + "include(\"jlFiles/printmat.jl\")\n", + "include(\"jlFiles/OlsNW.jl\") #functions for OLS" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Sample size: (388,)\n" + ] + } + ], + "source": [ + "x = readdlm(\"Data/FFmFactorsPs.csv\",',',skipstart=1)\n", + "\n", + " #yearmonth, market, small minus big, high minus low\n", + "(ym,Rme,RSMB,RHML) = (x[:,1],x[:,2]/100,x[:,3]/100,x[:,4]/100) \n", + "x = nothing\n", + "\n", + "printlnPs(\"Sample size:\",size(Rme))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Do OLS (in Julia)\n", + "\n", + "use the function sin the file OlsNW.jl to do OLS. Report point estimates and standard errors." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\u001b[34m\u001b[1mOLS Results (assuming iid residuals):\u001b[22m\u001b[39m\n", + "\n", + " b std_iid\n", + "c 0.007 0.002\n", + "SMB 0.217 0.073\n", + "HML -0.429 0.074\n", + "\n" + ] + } + ], + "source": [ + "Y = Rme\n", + "T = size(Y,1)\n", + "X = [ones(T) RSMB RHML]\n", + "\n", + "(b,u,Yhat,V,R2) = OlsGMFn(Y,X)\n", + "std_iid = sqrt.(diag(V))\n", + "\n", + "printblue(\"OLS Results (assuming iid residuals):\\n\")\n", + "xNames = [\"c\",\"SMB\",\"HML\"]\n", + "printmat([b std_iid],colNames=[\"b\",\"std_iid\"],rowNames=xNames)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Getting Started with PyCall" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [], + "source": [ + "using PyCall\n", + "sm = pyimport(\"statsmodels.api\"); #activate this package" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "PyObject \n", + "\"\"\"\n", + " OLS Regression Results \n", + "==============================================================================\n", + "Dep. Variable: y R-squared: 0.134\n", + "Model: OLS Adj. R-squared: 0.130\n", + "Method: Least Squares F-statistic: 29.85\n", + "Date: Mon, 08 Nov 2021 Prob (F-statistic): 8.88e-13\n", + "Time: 16:17:13 Log-Likelihood: 672.28\n", + "No. Observations: 388 AIC: -1339.\n", + "Df Residuals: 385 BIC: -1327.\n", + "Df Model: 2 \n", + "Covariance Type: nonrobust \n", + "==============================================================================\n", + " coef std err t P>|t| [0.025 0.975]\n", + "------------------------------------------------------------------------------\n", + "const 0.0070 0.002 3.167 0.002 0.003 0.011\n", + "x1 0.2170 0.074 2.949 0.003 0.072 0.362\n", + "x2 -0.4291 0.074 -5.821 0.000 -0.574 -0.284\n", + "==============================================================================\n", + "Omnibus: 58.863 Durbin-Watson: 1.849\n", + "Prob(Omnibus): 0.000 Jarque-Bera (JB): 146.539\n", + "Skew: -0.749 Prob(JB): 1.51e-32\n", + "Kurtosis: 5.612 Cond. No. 38.8\n", + "==============================================================================\n", + "\n", + "Notes:\n", + "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n", + "\"\"\"\n" + ] + } + ], + "source": [ + "resultsP = sm.OLS(Y, X).fit() #can use Python functions directly\n", + "\n", + "println(resultsP.summary())" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[:HC0_se, :HC1_se, :HC2_se, :HC3_se, :_HCCM, :__class__, :__delattr__, :__dict__, :__dir__, :__doc__, :__eq__, :__format__, :__ge__, :__getattribute__, :__gt__, :__hash__, :__init__, :__init_subclass__, :__le__, :__lt__, :__module__, :__ne__, :__new__, :__reduce__, :__reduce_ex__, :__repr__, :__setattr__, :__sizeof__, :__str__, :__subclasshook__, :__weakref__, :_abat_diagonal, :_cache, :_data_attr, :_data_in_cache, :_get_robustcov_results, :_is_nested, :_use_t, :_wexog_singular_values, :aic, :bic, :bse, :centered_tss, :compare_f_test, :compare_lm_test, :compare_lr_test, :condition_number, :conf_int, :conf_int_el, :cov_HC0, :cov_HC1, :cov_HC2, :cov_HC3, :cov_kwds, :cov_params, :cov_type, :df_model, :df_resid, :diagn, :eigenvals, :el_test, :ess, :f_pvalue, :f_test, :fittedvalues, :fvalue, :get_influence, :get_prediction, :get_robustcov_results, :info_criteria, :initialize, :k_constant, :llf, :load, :model, :mse_model, :mse_resid, :mse_total, :nobs, :normalized_cov_params, :outlier_test, :params, :predict, :pvalues, :remove_data, :resid, :resid_pearson, :rsquared, :rsquared_adj, :save, :scale, :ssr, :summary, :summary2, :t_test, :t_test_pairwise, :tvalues, :uncentered_tss, :use_t, :wald_test, :wald_test_terms, :wresid]\n" + ] + } + ], + "source": [ + "println(keys(resultsP)) #print all keys (field names)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 1\n", + "\n", + "Print the Julia and Python estimates (of the coefficients) in a table so we can compare directly." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 2\n", + "\n", + "Print the smallest and largest values of the difference between the residuals according to Julia and those according to Python." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# OLS (in Julia) with Robust Standard Errors\n", + "\n", + "Use standard errors that are robust to heteroskedastcity and autocorrelation (2 lags)." + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\u001b[34m\u001b[1mOLS Results (robust std):\u001b[22m\u001b[39m\n", + "\n", + " b std_nw\n", + "c 0.007 0.002\n", + "SMB 0.217 0.129\n", + "HML -0.429 0.118\n", + "\n" + ] + } + ], + "source": [ + "(b,u,Yhat,V,R2) = OlsNWFn(Y,X,2)\n", + "std_nw = sqrt.(diag(V))\n", + "\n", + "printblue(\"OLS Results (robust std):\\n\")\n", + "xNames = [\"c\",\"SMB\",\"HML\"]\n", + "printmat([b std_nw],colNames=[\"b\",\"std_nw\"],rowNames=xNames)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 3 \n", + "\n", + "Now redo the Python estimation with the same sort of robust standard errors. Hint: `resultsP.get_robustcov_results()`" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "@webio": { + "lastCommId": null, + "lastKernelId": null + }, + "kernelspec": { + "display_name": "Julia 1.6.3", + "language": "julia", + "name": "julia-1.6" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.6.3" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Problemsets/PS09b_RCall.ipynb b/Problemsets/PS09b_RCall.ipynb new file mode 100644 index 0000000..ef5536b --- /dev/null +++ b/Problemsets/PS09b_RCall.ipynb @@ -0,0 +1,284 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# R\n", + "\n", + "using the [RCall.jl](https://juliainterop.github.io/RCall.jl/stable/) package." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "CovNWFn" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "using Printf, DelimitedFiles, LinearAlgebra, Statistics\n", + "\n", + "include(\"jlFiles/printmat.jl\")\n", + "include(\"jlFiles/OlsNW.jl\") #functions for OLS" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Sample size: (388,)\n" + ] + } + ], + "source": [ + "x = readdlm(\"Data/FFmFactorsPs.csv\",',',skipstart=1)\n", + "\n", + " #yearmonth, market, small minus big, high minus low\n", + "(ym,Rme,RSMB,RHML) = (x[:,1],x[:,2]/100,x[:,3]/100,x[:,4]/100) \n", + "x = nothing\n", + "\n", + "printlnPs(\"Sample size:\",size(Rme))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Do OLS (in Julia)\n", + "\n", + "use the function sin the file OlsNW.jl to do OLS. Report point estimates and standard errors." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\u001b[34m\u001b[1mOLS Results (assuming iid residuals):\u001b[22m\u001b[39m\n", + "\n", + " b std_iid\n", + "c 0.007 0.002\n", + "SMB 0.217 0.073\n", + "HML -0.429 0.074\n", + "\n" + ] + } + ], + "source": [ + "Y = Rme\n", + "T = size(Y,1)\n", + "X = [ones(T) RSMB RHML]\n", + "\n", + "(b,u,Yhat,V,R2) = OlsGMFn(Y,X)\n", + "std_iid = sqrt.(diag(V))\n", + "\n", + "printblue(\"OLS Results (assuming iid residuals):\\n\")\n", + "xNames = [\"c\",\"SMB\",\"HML\"]\n", + "printmat([b std_iid],colNames=[\"b\",\"std_iid\"],rowNames=xNames)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Getting Started with RCall" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [], + "source": [ + "using RCall" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "RObject{VecSxp}\n", + "\n", + "Call:\n", + "lm(formula = Y ~ X + 0)\n", + "\n", + "Residuals:\n", + " Min 1Q Median 3Q Max \n", + "-0.20224 -0.02477 0.00335 0.02663 0.11840 \n", + "\n", + "Coefficients:\n", + " Estimate Std. Error t value Pr(>|t|) \n", + "X1 0.006983 0.002205 3.167 0.00166 ** \n", + "X2 0.216968 0.073565 2.949 0.00338 ** \n", + "X3 -0.429088 0.073710 -5.821 1.23e-08 ***\n", + "---\n", + "Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n", + "\n", + "Residual standard error: 0.04295 on 385 degrees of freedom\n", + "Multiple R-squared: 0.1488,\tAdjusted R-squared: 0.1422 \n", + "F-statistic: 22.44 on 3 and 385 DF, p-value: 2.07e-13\n", + "\n", + "\n" + ] + }, + { + "data": { + "text/plain": [ + "RObject{VecSxp}\n", + "\n", + "Call:\n", + "lm(formula = Y ~ X + 0)\n", + "\n", + "Coefficients:\n", + " X1 X2 X3 \n", + " 0.006983 0.216968 -0.429088 \n", + "\n" + ] + }, + "execution_count": 5, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "@rput X Y\n", + "\n", + "ResultsR = reval(\"summary(mod <- lm(Y ~ X+0))\") #print summary of regression\n", + "println(ResultsR)\n", + "\n", + "resultsR = reval(\"mod <- lm(Y ~ X+0)\") #get all output" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[:coefficients, :residuals, :effects, :rank, Symbol(\"fitted.values\"), :assign, :qr, Symbol(\"df.residual\"), :xlevels, :call, :terms, :model]\n" + ] + } + ], + "source": [ + "println(names(resultsR)) #print all keys (field names)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 1\n", + "\n", + "Print the Julia and Python estimates (of the coefficients) in a table so we can compare directly." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 2\n", + "\n", + "Print the smallest and largest values of the difference between the residuals according to Julia and those according to Python." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# OLS (in Julia) with Robust Standard Errors\n", + "\n", + "Use standard errors that are robust to heteroskedastcity and autocorrelation (2 lags)." + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\u001b[34m\u001b[1mOLS Results (robust std):\u001b[22m\u001b[39m\n", + "\n", + " b std_nw\n", + "c 0.007 0.002\n", + "SMB 0.217 0.129\n", + "HML -0.429 0.118\n", + "\n" + ] + } + ], + "source": [ + "(b,u,Yhat,V,R2) = OlsNWFn(Y,X,2)\n", + "std_nw = sqrt.(diag(V))\n", + "\n", + "printblue(\"OLS Results (robust std):\\n\")\n", + "xNames = [\"c\",\"SMB\",\"HML\"]\n", + "printmat([b std_nw],colNames=[\"b\",\"std_nw\"],rowNames=xNames)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 3 \n", + "\n", + "Now redo the R estimation with the same sort of robust standard errors. Hint: the `NeweyWest` in the `sandwich`package." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "@webio": { + "lastCommId": null, + "lastKernelId": null + }, + "kernelspec": { + "display_name": "Julia 1.6.3", + "language": "julia", + "name": "julia-1.6" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.6.3" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Problemsets/PS09c_WarAndPeace.ipynb b/Problemsets/PS09c_WarAndPeace.ipynb new file mode 100644 index 0000000..f9cfec7 --- /dev/null +++ b/Problemsets/PS09c_WarAndPeace.ipynb @@ -0,0 +1,132 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [], + "source": [ + "using Downloads" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Download a Book from Internet\n", + "\n", + "and read it into a string in Julia. Then report the number of letters etc (see below)" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "tags": [] + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "File to download: http://www.gutenberg.org/files/2600/2600-0.txt\n", + "\n", + "check the subfolder Results\n", + "\n" + ] + } + ], + "source": [ + "if !isdir(\"Results\")\n", + " error(\"create the subfolder Results before running this program\")\n", + "end\n", + "\n", + "http = \"http://www.gutenberg.org/files/2600/2600-0.txt\"\n", + "\n", + "println(\"File to download: \",http)\n", + "Downloads.download(http,\"Results/WarAndPeace.txt\")\n", + "\n", + "println(\"\\ncheck the subfolder Results\\n\")" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "tags": [] + }, + "outputs": [], + "source": [ + "fh = open(\"Results/WarAndPeace.txt\")\n", + "str = read(fh,String) \n", + "close(fh)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 1\n", + "\n", + "1. Count the number of letters and the unique letters in `str`. Hint: `length() and `unique()`\n", + "\n", + "2. Count the number of word and lines. Hint: `split(str)` and `split(str,\"\\n\")`\n", + "\n", + "3. Count the number of unique words." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 2\n", + "\n", + "1. How often is Borodinó mentioned. Hint: `occursin.(,words)`\n", + "\n", + "2. Print all lines that contain the word Borodinó. Hint: `occursin(,line)`" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 3\n", + "\n", + "1. Change Borodinó everywhere to Berëzina and then count the occurances" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "@webio": { + "lastCommId": null, + "lastKernelId": null + }, + "anaconda-cloud": {}, + "kernel_info": { + "name": "julia-1.2" + }, + "kernelspec": { + "display_name": "Julia 1.6.3", + "language": "julia", + "name": "julia-1.6" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.6.3" + }, + "nteract": { + "version": "0.24.1" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Problemsets/PS10_DataFrames.ipynb b/Problemsets/PS10_DataFrames.ipynb new file mode 100644 index 0000000..3f85bc5 --- /dev/null +++ b/Problemsets/PS10_DataFrames.ipynb @@ -0,0 +1,210 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Load Packages" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "tags": [] + }, + "outputs": [ + { + "data": { + "text/plain": [ + "printyellow (generic function with 1 method)" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "using Printf, Dates, Statistics, CSV, DataFrames\n", + "include(\"jlFiles/printmat.jl\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Loading Some Data with CSV.jl" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The first 4 lines of Data/Options_prices_US_Canada.csv:\n", + "\n", + "symbol,exchange,date,adjusted close,option symbol,expiration,strike,call/put,style,ask,bid,volume,open interest,unadjusted\n", + "SPX,CBOE,03/30/17,2368.06,SPXW 170331C00300000,03/31/17,300,C,E,2073.9,2062.9,0,0,2368.927\n", + "SPX,CBOE,03/30/17,2368.06,SPXW 170331P00300000,03/31/17,300,P,E,0.1,0,0,0,2368.927\n", + "SPX,CBOE,03/30/17,2368.06,SPXW 170331C00400000,03/31/17,400,C,E,1974.1,1962.7,0,0,2368.927\n", + "\n" + ] + } + ], + "source": [ + "DataFile = \"Data/Options_prices_US_Canada.csv\"\n", + "\n", + "println(\"The first 4 lines of $(DataFile):\\n\")\n", + "txt = readlines(DataFile)\n", + "printmat(txt[1:4])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Use `normalizenames` to get names that can be used in Julia as variables names and specify the `dateformat` used in the csv file (to convert to proper Julia dates). The dates in the file are given as `03/30/17` which CSV/DataFrames interpret as 30 March year 17 (AD). We add `Dates.Year(2000)` to get year 2017." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "tags": [] + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\u001b[1m13952×10 DataFrame\u001b[0m\n", + "\u001b[1m Row \u001b[0m│\u001b[1m symbol \u001b[0m\u001b[1m date \u001b[0m\u001b[1m close \u001b[0m\u001b[1m expiration \u001b[0m\u001b[1m strike \u001b[0m\u001b[1m call_put \u001b[0m\u001b[1m ask \u001b[0m\u001b[1m\u001b[0m ⋯\n", + "\u001b[1m \u001b[0m│\u001b[90m String3 \u001b[0m\u001b[90m Date \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Date \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m String1 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m\u001b[0m ⋯\n", + "───────┼────────────────────────────────────────────────────────────────────────\n", + " 1 │ SPX 2017-03-30 2368.06 2017-03-31 300.0 C 2073.9 ⋯\n", + " 2 │ SPX 2017-03-30 2368.06 2017-03-31 300.0 P 0.1\n", + " 3 │ SPX 2017-03-30 2368.06 2017-03-31 400.0 C 1974.1\n", + " 4 │ SPX 2017-03-30 2368.06 2017-03-31 400.0 P 0.05\n", + " 5 │ SPX 2017-03-30 2368.06 2017-03-31 500.0 C 1874.1 ⋯\n", + " 6 │ SPX 2017-03-30 2368.06 2017-03-31 500.0 P 0.05\n", + " 7 │ SPX 2017-03-30 2368.06 2017-03-31 600.0 C 1774.1\n", + " 8 │ SPX 2017-03-30 2368.06 2017-03-31 600.0 P 0.05\n", + " 9 │ SPX 2017-03-30 2368.06 2017-03-31 700.0 C 1673.9 ⋯\n", + " 10 │ SPX 2017-03-30 2368.06 2017-03-31 700.0 P 0.05\n", + " 11 │ SPX 2017-03-30 2368.06 2017-03-31 750.0 C 1624.1\n", + " ⋮ │ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱\n", + " 13943 │ XIU 2017-03-30 23.06 2019-03-15 21.0 C 3.21\n", + " 13944 │ XIU 2017-03-30 23.06 2019-03-15 21.0 P 1.71 ⋯\n", + " 13945 │ XIU 2017-03-30 23.06 2019-03-15 22.0 C 2.53\n", + " 13946 │ XIU 2017-03-30 23.06 2019-03-15 22.0 P 2.03\n", + " 13947 │ XIU 2017-03-30 23.06 2019-03-15 23.0 C 1.97\n", + " 13948 │ XIU 2017-03-30 23.06 2019-03-15 23.0 P 2.51 ⋯\n", + " 13949 │ XIU 2017-03-30 23.06 2019-03-15 24.0 C 1.5\n", + " 13950 │ XIU 2017-03-30 23.06 2019-03-15 24.0 P 3.12\n", + " 13951 │ XIU 2017-03-30 23.06 2019-03-15 25.0 C 1.13\n", + " 13952 │ XIU 2017-03-30 23.06 2019-03-15 25.0 P 3.64 ⋯\n", + "\u001b[36m 3 columns and 13931 rows omitted\u001b[0m" + ] + } + ], + "source": [ + "df1 = CSV.read(DataFile,DataFrame,normalizenames=true,dateformat=\"mm/dd/yy\")\n", + "\n", + "df1.date .+= Dates.Year(2000) #03/30/17 to 03/30/2017\n", + "df1.expiration .+= Dates.Year(2000)\n", + "\n", + "select!(df1,Not([:exchange,:option_symbol,:style,:unadjusted])) #deleting some columns\n", + "rename!(df1,:adjusted_close => :close) #renaming a column\n", + "\n", + "show(df1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 1\n", + "\n", + "Create a new DataFrame that contains only the data for SPX and those option contracts that were traded (volume > 0). Hint: `df1[vv, :]` picks out the rows of the data frame for which `vv` is `true`. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 2\n", + "\n", + "Create a *group* for each expiration date. These groups can be referred to as `dataG2[key]`.\n", + "\n", + "Hints: `groupby()`" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 3\n", + "\n", + "Print the number of contracts (`nrow`) and the sum of the open interest `:open_interest=>sum` for each of the expiration dates.\n", + "\n", + "Hint: `combine()`" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 4 \n", + "Creating two new DataFrames: for expiration date 2017-04-21 and another for 2017-06-16.\n", + "\n", + "Hint: `dataG2[(expiration = Date(\"2017-04-21\"),)]`" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 5\n", + "\n", + "For the expiration date 2017-04-21, calculate the mid price as the average of the `.ask` and `.bid`. \n", + "\n", + "Plot the mid price as a function of the strike price `.strike` for put options. Add a curve another curve for the call options." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "@webio": { + "lastCommId": null, + "lastKernelId": null + }, + "anaconda-cloud": {}, + "kernelspec": { + "display_name": "Julia 1.7.0", + "language": "julia", + "name": "julia-1.7" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.7.0" + }, + "nteract": { + "version": "0.23.1" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Problemsets/PS10b_OptionDeltas.ipynb b/Problemsets/PS10b_OptionDeltas.ipynb new file mode 100644 index 0000000..36de03f --- /dev/null +++ b/Problemsets/PS10b_OptionDeltas.ipynb @@ -0,0 +1,229 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Load Packages and Extra Functions" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "printyellow (generic function with 1 method)" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "using Printf, Distributions, OffsetArrays, FiniteDiff\n", + "\n", + "include(\"jlFiles/OptionsCalculations.jl\")\n", + "include(\"jlFiles/printmat.jl\")" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "using Plots\n", + "#pyplot(size=(600,400))\n", + "gr(size=(480,320))\n", + "default(fmt = :svg)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# The Black-Scholes Model\n", + "\n", + "The next cell calculates call `C` and put prices `P` from the Black-Scholes formula.\n", + "\n", + "The key parameters are:\n", + "`(S,K,m,y,σ) = (current undelying price, strike price,time to expiration,interest rate, volatility)`" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "call price at K=42: 2.893 \n", + "\n", + "put price at K=42: 1.856 \n", + "\n" + ] + } + ], + "source": [ + "(S,K,m,y,σ) = (42,42,0.5,0.05,0.2)\n", + "\n", + "C = OptionBlackSPs(S,K,m,y,σ)\n", + "printlnPs(\"call price at K=$K: \",C,\"\\n\")\n", + "\n", + "P = OptionBlackSPs(S,K,m,y,σ,0,true)\n", + "printlnPs(\"put price at K=$K: \",P,\"\\n\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 1\n", + "\n", + "For a range of different prices of the underlying asset `S=30.0:60.0`, calculat the call and put prices and plot them (with `S` on the hoizontal axis)." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# The Binomial Option Pricing Model\n", + "\n", + "The next cell contains functions to implement a binomial option pricing model for European style (exercise at expiration only) and American style (exercise any day) options. We use the CRR (Cox-Ross-Rubinstein) parameterisation.\n", + "\n", + "The key parameters are the same as before, but also `n` which is the number of time steps used in the calculations (which defaults to 250)." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "BOPM_American (generic function with 3 methods)" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "\"\"\"\n", + "CRRparams(σ,m,n,y)\n", + "\n", + " BOPM parameters according to CRR\n", + "\"\"\"\n", + "function CRRparams(σ,m,y,n)\n", + " h = m/n #time step size (in years)\n", + " u = exp(σ*sqrt(h)) #up move\n", + " d = exp(-σ*sqrt(h)) #down move\n", + " p = (exp(y*h) - d)/(u-d) #rn prob of up move\n", + " return h,u,d,p\n", + "end \n", + "\n", + "function BOPM_European(S,K,m,y,σ,isPut=false,n=250)\n", + " (h,u,d,p) = CRRparams(σ,m,y,n)\n", + " STree = BuildSTree(S,n,u,d)\n", + " price = EuOptionPrice(STree,K,y,h,p,isPut)[0][]\n", + " return price\n", + "end\n", + "\n", + "function BOPM_American(S,K,m,y,σ,isPut=false,n=250)\n", + " (h,u,d,p) = CRRparams(σ,m,y,n)\n", + " STree = BuildSTree(S,n,u,d)\n", + " price = AmOptionPrice(STree,K,y,h,p,isPut)[1][0][1]\n", + " return price\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "call (Eu) 2.891\n", + "put (Eu) 1.854\n", + "call (Am) 2.891\n", + "put (Am) 1.954\n", + "\n", + "\u001b[31m\u001b[1mNotice that c_e=c_a but that p_a>=p_e\u001b[22m\u001b[39m\n" + ] + } + ], + "source": [ + "c_e = BOPM_European(S,K,m,y,σ,false) #call price, European style\n", + "p_e = BOPM_European(S,K,m,y,σ,true) #put price, European\n", + "\n", + "c_a = BOPM_American(S,K,m,y,σ,false) #call, American\n", + "p_a = BOPM_American(S,K,m,y,σ,true) #put, American\n", + "\n", + "printmat([c_e,p_e,c_a,p_a],rowNames=[\"call (Eu)\",\"put (Eu)\",\"call (Am)\",\"put (Am)\"])\n", + "\n", + "printred(\"Notice that c_e=c_a but that p_a>=p_e\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 2\n", + "\n", + "Redo the calculation of p_e and p_a for `S=30:60` and plot the prices." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Delta\n", + "\n", + "To hedge an option contract (eg. if we are short one option contract), we buy $\\Delta$ units of the underlying asset, where $\\Delta$ is the partial derivative of the option price with respect to the price of the underlying asset.\n", + "\n", + "Calculate $\\Delta$ for both the European and American puts, using the BOPM function. Do the calculation for each value in `S_range` and plot the results.\n", + "\n", + "To calculate the derivatives we could use, for instance, \n", + "`FiniteDiff.finite_difference_derivative(the function,an S value)`" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "@webio": { + "lastCommId": null, + "lastKernelId": null + }, + "anaconda-cloud": {}, + "kernelspec": { + "display_name": "Julia 1.6.3", + "language": "julia", + "name": "julia-1.6" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.6.3" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Problemsets/Solutions/PS01_Basics_Solution.ipynb b/Problemsets/Solutions/PS01_Basics_Solution.ipynb new file mode 100644 index 0000000..31199ec --- /dev/null +++ b/Problemsets/Solutions/PS01_Basics_Solution.ipynb @@ -0,0 +1,1688 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Basic Julia Tutorial" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Running/Installing Julia\n", + "\n", + "This is a notebook. To run in you need Julia and jupyter or nteract (see below).\n", + "\n", + "1. To run Julia without a local installation, use (for instance) [CoCalc](https://cocalc.com).\n", + "\n", + "2. To install Julia on your machine, [download and install it](https://julialang.org/downloads/). \n", + "\n", + "3. You may also want to use [VS Code with the Julia extension](https://github.com/julia-vscode/julia-vscode) IDE. See also the [julia-vscode.org](https://www.julia-vscode.org/) for more information.\n", + "\n", + "4. To run notebooks with either jupyter or nteract from your local installation, see [IJulia](https://github.com/JuliaLang/IJulia.jl) for instructions. See also below for some hints.\n", + "\n", + "5. As an alternative to jupyter/nteract you can also run notebooks in VS Code (see above).\n", + "\n", + "\n", + "\n", + "## More Details on Installing jupyter and nteract\n", + "\n", + "1. If you want to use nteract do as follows: install IJulia, but do *not* call on `notebook()` to avoid the automatic Python installation. Then install nteract.\n", + "\n", + "2. After installing IJulia and issuing the command `notebook()` a new local Python installation will be made - unless IJulia finds an existing installation.\n", + "\n", + "3. If you already have a Python installation, run `ENV[\"JUPYTER\"] = \"C:\\\\Miniconda3\\\\Scripts\\\\jupyter.exe\"` (change the path as needed) before you install IJulia. The best is perhaps to add this to your startup.jl file (~/.julia/config/startup.jl, for instance, C:\\Users\\yourusername\\.julia\\config\\startup.jl if you are on Windows). You can test whether it works by running ```run(`$(ENV[\"JUPYTER\"]) --version`)``` from the Julia REPL.\n", + "\n", + "4. On Windows, it helps to allow the Python installer to add python to the system path." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Documentation and Help\n", + "\n", + "1. Cheat sheet at [QuantEcon](https://cheatsheets.quantecon.org/julia-cheatsheet.html)\n", + "2. [Wiki book](https://en.wikibooks.org/wiki/Introducing_Julia)\n", + "3. [Short tutorials](https://techytok.com/from-zero-to-julia/) \n", + "4. [ThinkJulia](https://benlauwens.github.io/ThinkJulia.jl/latest/book.html) is a free on-line book\n", + "5. The [official Julia on-line manual](https://docs.julialang.org)\n", + "6. Discussion lists are found at\n", + " * https://discourse.julialang.org/\n", + " * https://stackoverflow.com/questions/tagged/julia-lang\n", + " * https://www.reddit.com/r/Julia/\n", + " * https://gitter.im/JuliaLang/julia\n", + "7. In Julia, do ```? cos``` to get help with the cos function " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# About Notebooks\n", + "\n", + "This cell is a \"Markdown\" cell. This is meant for comments and documentation, not computations.\n", + "\n", + "You can change a cell to \"Code\" or \"Markdown\" in the menu.\n", + "\n", + "Markdown cells can handle LaTeX. An example: $\\alpha = \\beta/2$. A Markdown cell can also contain some *formatting*, like lists of this kind\n", + "\n", + "1. To insert a new cell, use the menu. \n", + "\n", + "2. The next cell is \"Code\". You can run it. Text after a # sign is treated as a comment.\n", + "\n", + "3. The subsequent cell shows how to get help on a command." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "2" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "a = 2 #this is a comment\n", + " #run this cell by using the menu, or by Shift+Enter" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "search: \u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ms\u001b[22m \u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ms\u001b[22mh \u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ms\u001b[22md \u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ms\u001b[22mc \u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ms\u001b[22mpi a\u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ms\u001b[22m a\u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ms\u001b[22mh a\u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ms\u001b[22md sin\u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ms\u001b[22m sin\u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ms\u001b[22md sin\u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22m\u001b[0m\u001b[1ms\u001b[22mpi \u001b[0m\u001b[1mc\u001b[22m\u001b[0m\u001b[1mo\u001b[22mn\u001b[0m\u001b[1ms\u001b[22mt\n", + "\n" + ] + }, + { + "data": { + "text/latex": [ + "\\begin{verbatim}\n", + "cos(x)\n", + "\\end{verbatim}\n", + "Compute cosine of \\texttt{x}, where \\texttt{x} is in radians.\n", + "\n", + "\\rule{\\textwidth}{1pt}\n", + "\\begin{verbatim}\n", + "cos(A::AbstractMatrix)\n", + "\\end{verbatim}\n", + "Compute the matrix cosine of a square matrix \\texttt{A}.\n", + "\n", + "If \\texttt{A} is symmetric or Hermitian, its eigendecomposition (\\href{@ref}{\\texttt{eigen}}) is used to compute the cosine. Otherwise, the cosine is determined by calling \\href{@ref}{\\texttt{exp}}.\n", + "\n", + "\\section{Examples}\n", + "\\begin{verbatim}\n", + "julia> cos(fill(1.0, (2,2)))\n", + "2×2 Matrix{Float64}:\n", + " 0.291927 -0.708073\n", + " -0.708073 0.291927\n", + "\\end{verbatim}\n" + ], + "text/markdown": [ + "```\n", + "cos(x)\n", + "```\n", + "\n", + "Compute cosine of `x`, where `x` is in radians.\n", + "\n", + "---\n", + "\n", + "```\n", + "cos(A::AbstractMatrix)\n", + "```\n", + "\n", + "Compute the matrix cosine of a square matrix `A`.\n", + "\n", + "If `A` is symmetric or Hermitian, its eigendecomposition ([`eigen`](@ref)) is used to compute the cosine. Otherwise, the cosine is determined by calling [`exp`](@ref).\n", + "\n", + "# Examples\n", + "\n", + "```jldoctest\n", + "julia> cos(fill(1.0, (2,2)))\n", + "2×2 Matrix{Float64}:\n", + " 0.291927 -0.708073\n", + " -0.708073 0.291927\n", + "```\n" + ], + "text/plain": [ + "\u001b[36m cos(x)\u001b[39m\n", + "\n", + " Compute cosine of \u001b[36mx\u001b[39m, where \u001b[36mx\u001b[39m is in radians.\n", + "\n", + " ────────────────────────────────────────────────────────────────────────────\n", + "\n", + "\u001b[36m cos(A::AbstractMatrix)\u001b[39m\n", + "\n", + " Compute the matrix cosine of a square matrix \u001b[36mA\u001b[39m.\n", + "\n", + " If \u001b[36mA\u001b[39m is symmetric or Hermitian, its eigendecomposition (\u001b[36meigen\u001b[39m) is used to\n", + " compute the cosine. Otherwise, the cosine is determined by calling \u001b[36mexp\u001b[39m.\n", + "\n", + "\u001b[1m Examples\u001b[22m\n", + "\u001b[1m ≡≡≡≡≡≡≡≡≡≡\u001b[22m\n", + "\n", + "\u001b[36m julia> cos(fill(1.0, (2,2)))\u001b[39m\n", + "\u001b[36m 2×2 Matrix{Float64}:\u001b[39m\n", + "\u001b[36m 0.291927 -0.708073\u001b[39m\n", + "\u001b[36m -0.708073 0.291927\u001b[39m" + ] + }, + "execution_count": 2, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "? cos #to get help on the cos() function" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Load Packages and Extra Functions\n", + "\n", + "There are many packages for Julia, for instance, for plotting or statistical methods (see [JuliaHub](https://juliahub.com/) for a list). To install a package, you do either \n", + "\n", + "1. (works everywhere) run `import Pkg` and then `Pkg.add(\"Packagename\")`\n", + "\n", + "2. (works in the Julia console, REPL) enter the \"package manager mode\" by typing `]`, then run `add PackageName`. You leave the package manager mode by ctrl-c or backspace.\n", + "\n", + "Once a package is installed, you can use it by running\n", + "\n", + "```\n", + "using PackageName\n", + "```" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "printyellow (generic function with 1 method)" + ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "using Printf\n", + "\n", + "include(\"jlFiles/printmat.jl\") #just a function for prettier matrix printing" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [], + "source": [ + "using Plots #this loads the Plots package\n", + "\n", + "#pyplot(size=(600,400)) #choice of plotting backend\n", + "gr(size=(480,320))\n", + "default(fmt = :svg) #try :png or :svg" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Scalars and Matrices\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Create a Scalar and a Matrix" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "q is a scalar. To print, use println() or printlnPs()\n", + "1\n", + "\n", + "Q is a matrix. To print, use display() or printmat()\n", + " 1 2 3 \n", + " 4 5 6 \n", + "\n" + ] + } + ], + "source": [ + "q = 1 #create a scalar\n", + "Q = [ 1 2 3; #create 2x3 matrix\n", + " 4 5 6 ] \n", + "println(\"q is a scalar. To print, use println() or printlnPs()\")\n", + "println(q)\n", + "\n", + "println(\"\\nQ is a matrix. To print, use display() or printmat()\")\n", + "printmat(Q) #case sensitive (q and Q are different)\n", + " #the \\n adds a line break" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Picking Out Parts of a Matrix" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\n", + "element [1,2] of Q: 2\n", + "\n", + "columns 2 and 3 of Q: \n", + " 2 3 \n", + " 5 6 \n", + "\n", + "\n", + "line 1 of Q (as a vector): \n", + " 1 \n", + " 2 \n", + " 3 \n", + "\n" + ] + } + ], + "source": [ + "println(\"\\n\",\"element [1,2] of Q: \", #commands continue on\n", + " Q[1,2]) #the next line (until finished)\n", + "\n", + "println(\"\\ncolumns 2 and 3 of Q: \")\n", + "printmat(Q[:,2:3])\n", + "\n", + "println(\"\\nline 1 of Q (as a vector): \")\n", + "printmat(Q[1,:])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Task\n", + "Create a 2x2 matrix A\n", + "\n", + "```\n", + "10 10\n", + "21 22\n", + "```\n", + "\n", + "Print it\n", + "\n", + "Change the second row to `[1 2]`. Print again." + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " 10 10 \n", + " 21 22 \n", + "\n", + " 10 10 \n", + " 1 2 \n", + "\n" + ] + } + ], + "source": [ + "A = [10 10;\n", + " 21 22]\n", + "printmat(A)\n", + "A[2,:] = [1 2]\n", + "printmat(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Basic Linear Algebra\n", + "\n", + "The syntax for linear algebra is similar to the standard text book approach. For instance, \n", + "* `Q'Q` (or `Q'*Q`) multiplies the transpose ($Q'$) with the matrix ($Q$)\n", + "* `A*B` does matrix multiplication\n", + "* `100*Q` multiplies each element of the matrix ($Q$) by 100. (You can also do `100Q`.)\n", + "\n", + "However, to add a scalar to each element of a matrix, use `100 .+ Q`. Notice the dot." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "transpose of Q:\n", + " 1 4 \n", + " 2 5 \n", + " 3 6 \n", + "\n", + "Q'Q:\n", + " 17 22 27 \n", + " 22 29 36 \n", + " 27 36 45 \n", + "\n", + "scalar * matrix:\n", + " 100 200 300 \n", + " 400 500 600 \n", + "\n", + "scalar .+ matrix:\n", + " 101 102 103 \n", + " 104 105 106 \n", + "\n" + ] + } + ], + "source": [ + "println(\"transpose of Q:\")\n", + "printmat(Q')\n", + "\n", + "println(\"Q'Q:\")\n", + "printmat(Q'Q)\n", + "\n", + "println(\"scalar * matrix:\")\n", + "printmat(100*Q)\n", + "\n", + "println(\"scalar .+ matrix:\") #notice the dot\n", + "printmat(100 .+ Q)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Task\n", + "\n", + "Consider the system of linear equations\n", + "\n", + "$\n", + "Ax = q\n", + "$, \n", + "\n", + "where \n", + "$ \n", + "q = \\begin{bmatrix} 1 \\\\2 \\end{bmatrix}\n", + "$\n", + "\n", + "You could solve it as\n", + "$\n", + "x = A^{-1}q\n", + "$\n", + "\n", + "\n", + "Solve (numrically) for the vector $x$. Hint: `inv()` Then check that $Ax$ indeed equals $q$" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " x\n", + " -1.800\n", + " 1.900\n", + "\n", + " A*x q\n", + " 1.000 1.000\n", + " 2.000 2.000\n", + "\n" + ] + } + ], + "source": [ + "\n", + "\n", + "\n", + "q = [1,2]\n", + "x = inv(A)*q\n", + "printmat(x,colNames=[\"x\"])\n", + "\n", + "printmat([A*x q],colNames=[\"A*x\",\"q\"])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Creating a Sequence and a Vector" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\n", + "θ is a sequence: 1:10:21\n", + "\n", + "ρ is a vector: \n", + " 1 \n", + " 11 \n", + " 21 \n", + "\n" + ] + } + ], + "source": [ + "θ = 1:10:21 #a range, type \\theta[TAB] to get this symbol\n", + "println(\"\\n\",\"θ is a sequence: \",θ)\n", + "\n", + "ρ = collect(θ) #make the sequence into a vector, \\rho[TAB]\n", + "println(\"\\n\",\"ρ is a vector: \")\n", + "printmat(ρ)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Task\n", + "\n", + "Create a sequence $\\alpha$ which starts at 0 and ends at 1, taking 11 steps. (Hint: you can also use `range()`)" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " 0.000\n", + " 0.100\n", + " 0.200\n", + " 0.300\n", + " 0.400\n", + " 0.500\n", + " 0.600\n", + " 0.700\n", + " 0.800\n", + " 0.900\n", + " 1.000\n", + "\n" + ] + } + ], + "source": [ + "α = 0:0.1:1\n", + "#α = range(0,1,length=11)\n", + "printmat(α)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Comparing Things\n", + "\n", + "To see if the scalar z <= 0, do \n", + "```\n", + "vv = z <= 0\n", + "```\n", + "to get a single output (true or false).\n", + "\n", + "Instead, if x is an array, do \n", + "```\n", + "vv = x .<= 0 #notice the dot.\n", + "```\n", + "to get an array of outputs (same dimension as x)" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "x values: \n", + " -1.500\n", + " -1.000\n", + " -0.500\n", + " 0.000\n", + " 0.500\n", + "\n", + "true if x is in (-1,0]: \n", + " 0 \n", + " 0 \n", + " 1 \n", + " 1 \n", + " 0 \n", + "\n", + "x values that are in (-1,0]: \n", + " -0.500\n", + " 0.000\n", + "\n" + ] + } + ], + "source": [ + "x = [-1.5,-1.0,-0.5,0,0.5] #this is a vector\n", + "\n", + "println(\"x values: \")\n", + "printmat(x)\n", + "\n", + "vv = -1 .< x .<= 0 #true for x values (-1,0], vv is a vector\n", + "println(\"true if x is in (-1,0]: \")\n", + "printmat(vv)\n", + "\n", + "x2 = x[vv] #x values for which vv==true\n", + "println(\"x values that are in (-1,0]: \")\n", + "printmat(x2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Task\n", + "\n", + "Which elements of $\\alpha$ are (weakly) between 0.6 and 0.99?" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " vv[-1.5, -1.0, -0.5, 0.0, 0.5]\n", + " 0 0.000\n", + " 0 0.100\n", + " 0 0.200\n", + " 0 0.300\n", + " 0 0.400\n", + " 0 0.500\n", + " 1 0.600\n", + " 1 0.700\n", + " 1 0.800\n", + " 1 0.900\n", + " 0 1.000\n", + "\n" + ] + } + ], + "source": [ + "vv = 0.6 .<= α .<= 0.99\n", + "printmat(vv,α,colNames=[\"vv\",x])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Finding Things\n", + "\n", + "can be done by, for instance, `findfirst()`, `findall()` and `indexin()`. \n", + "\n", + "Instead, if you just need to check if the number `z` is in an array (or any collection) `x`, then use `in(z,x)`." + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "x values: \n", + " -1.500\n", + " -1.000\n", + " -0.500\n", + " 0.000\n", + " 0.500\n", + "\n", + "(first) index v in x such x[v]==0: 4\n", + "\n", + "all indices v in x such x[v]>=0: \n", + " 4 \n", + " 5 \n", + "\n", + "\n", + "indices in x so that x[v] equals the vector y=[-1.0, 0.0]: \n", + " 2 \n", + " 4 \n", + "\n", + "\n", + "testing if 0 is in x: true\n" + ] + } + ], + "source": [ + "x = [-1.5,-1.0,-0.5,0,0.5] #this is a vector\n", + "\n", + "println(\"x values: \")\n", + "printmat(x)\n", + "\n", + "v1 = findfirst(x.==0)\n", + "println(\"(first) index v in x such x[v]==0: \",v1)\n", + "\n", + "v2 = findall(x.>=0)\n", + "println(\"\\nall indices v in x such x[v]>=0: \")\n", + "printmat(v2)\n", + "\n", + "y = [-1.0,0]\n", + "v3 = indexin(y,x)\n", + "println(\"\\nindices in x so that x[v] equals the vector y=$y: \")\n", + "printmat(v3)\n", + "\n", + "v4 = in(0,x)\n", + "println(\"\\ntesting if 0 is in x: \",v4)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# if-else\n", + "\n", + "allows you to run different comman ddepending on a condition which you specify.\n", + "\n", + "(extra) There are also other (more compact) possibilities for `if-else` case\n", + "\n", + "1. `y = ifelse(z <= 2,z,2)` or \n", + "2. `z <= 2 ? y = z : y = 2`\n", + "(In this partcular case, `y=minimum(z,2)` also works.)\n", + "\n", + "You can also insert an `ifelse` command." + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "1.05\n" + ] + } + ], + "source": [ + "z = 1.05\n", + "\n", + "if z <= 2 #(a) if true, run the next command (y=z) and then jump to end\n", + " y = z\n", + "else #(b) if (a) is false, do this instead\n", + " y = 2\n", + "end\n", + "\n", + "#y = ifelse(z <= 2,z,2) #these two versions also work\n", + "#z <= 2 ? y = z : y = 2\n", + "\n", + "println(y)" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "1.05\n" + ] + } + ], + "source": [ + "if z < 1 #(a) if true, run the next command (y=1) and then jump to end\n", + " y = 1\n", + "elseif 1 <= z <= 2 #(b) if (a) is false, try this instead \n", + " y = z\n", + "else #(c) if also (b) is false, do this\n", + " y = 2\n", + "end\n", + "\n", + "println(y)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Task\n", + "\n", + "\n", + "Use `if-else` to calculate the tax paid by someone with an income of $X$. The tax schedule is progressive so the first 100 of the income pays 40% tax, while income above 100 has a marginal tax rate of 0.6 (that is, $X-100$ pays 60% if $X>100)$. What is the total tax paid and the average tax rate.\n", + "Report the results for an income of 150." + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " 70.000 0.467\n", + "\n" + ] + } + ], + "source": [ + "X = 150\n", + "\n", + "if X <= 100\n", + " tax = 0.4*X\n", + "else\n", + " tax = 0.4*100 + 0.6*(X-100)\n", + "end\n", + "\n", + "avgtax = tax/X\n", + "printmat(tax,avgtax)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Loops\n", + "\n", + "The are two types of loops: \"for loops\" and \"while loops\". \n", + "\n", + "The *for loop* is best when you know how many times you want to loop (for instance, over all $m$ rows in a matrix). \n", + "\n", + "The *while loop* is best when you want to keep looping until something happens, for instance, that $x_0$ and $x_1$ get really close.\n", + "\n", + "The default behaviour in *IJulia*, *inside functions* and at the *REPL* promt is that assignments of `x` inside the loop overwrites `x` defined before the loop. To get the same behavior in scripts, you need to add `global x` somewhere inside the loop.\n", + "\n", + "To make sure that the `y` calculated inside the loop does not affect `y` outside the loop, add `local y`.\n", + "\n", + "A variable (here `z2`) that does not exist before the loop is local to the loop." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### A Simple \"for loop\"\n", + "\n", + "The \"for loop\" in the next cell makes 3 iterations and changes a global $x$ variable." + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "i=3, x=3 and z2=-998\n", + "i=6, x=9 and z2=-998\n", + "i=9, x=18 and z2=-998\n", + "\n", + "After loop: x=18 and y=-999\n" + ] + } + ], + "source": [ + "x = 0\n", + "y = -999\n", + "for i = 3:3:9 #or `i in 3:3:9`, or `v=[3,6,9]; i in v`\n", + " #global x #only needed in script\n", + " local y #don't overwrite y outside loop\n", + " x = x + i #adding i to the \"old\" x\n", + " y = i\n", + " z2 = -998 #notice: z2 has not been used before \n", + " println(\"i=$i, x=$x and z2=$z2\") #$x prints the value of x\n", + "end\n", + "\n", + "println(\"\\nAfter loop: x=$x and y=$y\")\n", + "#println(z2) #does not work: z2 is local to the loop" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Task\n", + "\n", + "You have the following financial data\n", + "\n", + "```\n", + "P = [100;108;109] #price series, after dividends\n", + "D = [0;2;0] #dividends\n", + "```\n", + "\n", + "For each $t$, calculate the returns\n", + "\n", + "$\n", + "R_t = (P_t+D_t)/P_{t-1} - 1\n", + "$,\n", + "\n", + "by using a loop." + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " Return\n", + "period 2 0.100\n", + "period 3 0.009\n", + "\n" + ] + } + ], + "source": [ + "P = [100;108;109] #price series, after dividends\n", + "D = [0;2;0] #dividends\n", + "\n", + "R = zeros(length(P)) #where to store the results\n", + "for t = 2:length(P) #P[2] is the 2nd element of P \n", + " R[t] = (P[t] + D[t])/P[t-1] - 1\n", + "end\n", + "R = R[2:end]\n", + "\n", + "printmat(R,colNames=[\"Return\"],rowNames=string.(\"period \",2:3))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### A Double \"for loop\"\n", + "\n", + "An example of a nested for for loop\n", + "\n", + "```\n", + "for j = 1:n, i = 1:m\n", + " #do something\n", + "end\n", + "```\n", + "\n", + "If you prefer, could also write a longer version to do the same thing\n", + "```\n", + "for j = 1:n \n", + " for i = 1:m\n", + " #do something\n", + " end \n", + "end\n", + "```\n", + "\n", + "The next cell uses a double loop to fill a matrix. " + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "new x matrix: \n", + "\n", + " 11 12 13 \n", + " 21 22 23 \n", + " 31 32 33 \n", + " 41 42 43 \n", + "\n" + ] + } + ], + "source": [ + "(m,n) = (4,3) #same as m=4;n=3\n", + "x = fill(-999,(m,n)) #to put results in, initialized as -999\n", + "for i = 1:m, j = 1:n\n", + " x[i,j] = 10*i + j\n", + "end\n", + "\n", + "println(\"new x matrix: \\n\")\n", + "printmat(x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## A Simple \"while loop\"\n", + "\n", + "The \"while loop\" in the next cell iterates until two variables ($x_0$ and $x_1$) get close.\n", + "\n", + "The background to the example is that we want to solve a function $f(x)=x^2$ for the $x$ value that makes $f(x)=2$. The Newton-Raphson algorithm starts with a value $x_0$ and updates it to\n", + "$\n", + "x_1 = x_0 + (2-f(x_0))/f'(x_0)\n", + "$\n", + "where $f'(x_0)$ is the derivative of $f()$ evaluated at $x_0$. The algorithm iterates until $x_0$ and $x_1$ are close. Clearly, we are trying to find the square root of 2." + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Solving x^2 = 2 with Newton-Raphson:\n", + "\n", + " 10 is changed to 5.100\n", + " 5.100 is changed to 2.746\n", + " 2.746 is changed to 1.737\n", + " 1.737 is changed to 1.444\n", + " 1.444 is changed to 1.415\n", + " 1.415 is changed to 1.414\n", + "\n", + "The result should be close to 1.414\n" + ] + } + ], + "source": [ + "println(\"Solving x^2 = 2 with Newton-Raphson:\\n\")\n", + "\n", + "x₀ = Inf #x\\_0[TAB]\n", + "x₁ = 10\n", + "\n", + "while abs(x₁-x₀) > 0.001 #keep going until they get similar\n", + " #global x₀, x₁ #only needed in script\n", + " local y, dy #don't overwrite any y,dy outside loop\n", + " x₀ = x₁ #initial guess is taken from old guess\n", + " y = x₀^2 #value of function\n", + " dy = 2*x₀ #derivative of function\n", + " x₁ = x₀ + (2 - y)/dy #updating the guess, Newton-Raphson\n", + " printlnPs(x₀,\" is changed to \",x₁)\n", + "end\n", + "\n", + "printlnPs(\"\\nThe result should be close to \",sqrt(2))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Task\n", + "\n", + "Start at `x=1` and create a while loop that multiplies the previous $x$ by 1.1.(That is, you earn a 10% return each year.) Continue while `x<2`. Print the iterations (iteration number, `x`). So, how many years did it take to double the investment?" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " 1 1.100\n", + " 2 1.210\n", + " 3 1.331\n", + " 4 1.464\n", + " 5 1.611\n", + " 6 1.772\n", + " 7 1.949\n", + " 8 2.144\n" + ] + } + ], + "source": [ + "x = 1\n", + "iter = 0\n", + "while x < 2\n", + " x = x*1.1\n", + " iter = iter + 1 \n", + " printlnPs(iter,x)\n", + "end" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# A Simple Function\n", + "\n", + "The next cell defines a new function, `fn1()`. It takes a scalar input (`x`) and returns a scalar output (`y`).\n", + "\n", + "If you instead use a vector as the input, then the computation fails. (The reason is that you cannot do x^2 on a vector. You could on a square matrix, though.)\n", + "\n", + "However, using the \"dot\" syntax\n", + "```\n", + "y = fn1.(x)\n", + "```\n", + "gives an array as output where element `y[i,j] = fn1(x[i,j])`." + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "fn1 (generic function with 1 method)" + ] + }, + "execution_count": 23, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "function fn1(x) #define a new function \n", + " y = (x-1.1)^2 - 0.5\n", + " return y\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "result from fn1(1.5): -0.340\n", + "\n", + "result from fn1.(x): \n", + " -0.490\n", + " -0.340\n", + "\n" + ] + } + ], + "source": [ + "y = fn1(1.5)\n", + "printlnPs(\"result from fn1(1.5): \",y)\n", + "\n", + "x = [1;1.5]\n", + "#y = fn1(x) #would give an error \n", + "y = fn1.(x) #calling on the function, dot. to do for each element in x\n", + "printlnPs(\"\\nresult from fn1.(x): \")\n", + "printmat(y)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Task\n", + "\n", + "Write a function to calculate \n", + "$B =\\exp{(-my)}$, \n", + "where $B$ is the price of a bond and $m$ is the time to maturity (measured in years) and $y$ is the (continously compounded) interest rate.\n", + "\n", + "If $y=0.103$ and $m=0.5$, what is $B$?" + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " 0.950\n" + ] + } + ], + "source": [ + "BondPrice(y,m) = exp(-m*y) #B as fn of y\n", + "\n", + "printlnPs(BondPrice(0.103,0.5))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# A First Plot\n", + "\n", + "With the Plots package you create a a simple plot like this:\n", + "\n", + "1. Plot two curve by using the `plot([x1 x2],[y1 y2])` command\n", + "2. Configure curves by `linecolor =`, `linestyle =` etc\n", + "2. Add titles and labels by `title = `, `xlabel = `, etc\n", + "\n", + "Notice: the *first plot is slow*." + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "x = -3:6/99:6 \n", + "\n", + "p1 = plot( [x x],[fn1.(x) cos.(x)],\n", + " linecolor = [:red :blue],\n", + " linestyle = [:solid :dash],\n", + " linewidth = [2 1],\n", + " label = [\"fn1()\" \"cos\"],\n", + " title = \"My Results\",\n", + " xlabel = \"x\",\n", + " ylabel = \"my output value\" )\n", + "display(p1) #not needed in notebook, but useful in script" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Task\n", + "\n", + "Consider different interest rates in the interval (-0.05,0.1). Plot the prices of 3-month and 10-year bonds as functions of the interest rate." + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "execution_count": 27, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "y = -0.05:0.001:0.10\n", + "\n", + "B3m = BondPrice.(y,3/12)\n", + "B10y = BondPrice.(y,10)\n", + "\n", + "plot([y y],[B3m B10y],label = [\"3m\" \"10y\"],xlabel=\"interest rate\",ylabel=\"Bond price\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Types: Integers, Floats, Bools and Others\n", + "\n", + "Julia has many different types of variables: signed integers (like 2 or -5), floating point numbers (2.0 and -5.1), bools (false/true), bitarrays (similar to bools, but with more efficient use of memory), strings (\"hello\"), Dates (2017-04-23) and many more types." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Integers and Floats" + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Finding the type of a, b, A and B:\n", + "Int64 Float64 Vector{Int64} Vector{Float64}\n" + ] + } + ], + "source": [ + "a = 2 #integer, Int (Int64 on most machines)\n", + "b = 2.0 #floating point, (Float64 on most machines)\n", + "A = [1,2]\n", + "B = [1.0,2.0]\n", + "\n", + "println(\"Finding the type of a, b, A and B:\")\n", + "println(typeof(a),\" \",typeof(b),\" \",typeof(A),\" \",typeof(B))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Bools and BitArrays\n", + "\n", + "Bools are \"true\" or \"false\". BitArrays are (more memory efficient) versions of this." + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Finding the type of c and C:\n", + "Bool BitVector\n" + ] + } + ], + "source": [ + "c = 2 > 1.1\n", + "C = A .> 1.5 #A is an array, so C is too\n", + "\n", + "println(\"Finding the type of c and C:\")\n", + "println(typeof(c),\" \",typeof(C))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Calculations with Mixed Types\n", + "\n", + "A calculation like \"integer\" + \"float\" works and the type of the result will be a float (the more flexible type). Similarly, \"bool\" + \"integer\" will give an integer. These promotion rules make it easy to have mixed types in calculations, and also provide a simple way of converting a variable from one type to another. (There are also an explicit convert() function that might be quicker.)" + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "3.0\n", + "3\n" + ] + } + ], + "source": [ + "println(1+2.0) #integer + Float\n", + "println((1>0) + 2) #bool + integer" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "@webio": { + "lastCommId": null, + "lastKernelId": null + }, + "anaconda-cloud": {}, + "kernelspec": { + "display_name": "Julia 1.6.3", + "language": "julia", + "name": "julia-1.6" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.6.3" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Problemsets/Solutions/PS02_Stats_Solution.ipynb b/Problemsets/Solutions/PS02_Stats_Solution.ipynb new file mode 100644 index 0000000..4752b12 --- /dev/null +++ b/Problemsets/Solutions/PS02_Stats_Solution.ipynb @@ -0,0 +1,1742 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Load Packages and Extra Functions" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "printyellow (generic function with 1 method)" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "using Statistics, Printf, Dates, DelimitedFiles, Distributions, LinearAlgebra\n", + "\n", + "include(\"jlFiles/printmat.jl\")" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "using Plots\n", + "\n", + "#pyplot(size=(600,400))\n", + "gr(size=(480,320))\n", + "default(fmt = :svg)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Load Data from a csv File" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\n", + "first four lines of x:\n", + "1970-01-01 -5.480 -4.875 -8.100\n", + "1970-02-01 1.465 7.546 5.130\n", + "1970-03-01 -7.232 1.061 -1.060\n", + "1970-04-01 -24.488 -8.481 -11.000\n", + "\n" + ] + } + ], + "source": [ + "x = readdlm(\"Data/Portfolios_SGLV.csv\",',',skipstart=1) #reading the csv file\n", + " #skip 1st line\n", + "println(\"\\nfirst four lines of x:\")\n", + "printmat(x[1:4,:])\n", + "\n", + "dN = Date.(x[:,1]) #creating variables\n", + "Re = convert.(Float64,x[:,2:3]) #small growth (SG), large value (LV)\n", + "Rme = convert.(Float64,x[:,end]); #market" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Task 1: Means and Standard Deviations\n", + "\n", + "Estimate and print the means and standard deviations of the three series (in `Re` and `Rme`)." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " SG LV Market\n", + "mean 0.218 0.677 0.593\n", + "std 8.079 5.772 4.595\n", + "\n" + ] + } + ], + "source": [ + "μ = mean([Re Rme],dims=1) \n", + "σ = std([Re Rme],dims=1) \n", + "\n", + "printmat([μ;σ],colNames=[\"SG\",\"LV\",\"Market\"],rowNames=[\"mean\",\"std\"])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Task 2: Correlations\n", + "\n", + "Estimate and print the correlation matrix." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " 1.000 0.563 0.804\n", + " 0.563 1.000 0.774\n", + " 0.804 0.774 1.000\n", + "\n" + ] + } + ], + "source": [ + "printmat(cor([Re Rme]))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Task 3: OLS\n", + "\n", + "Code up a function to do OLS. It should take `y` and `x` as inputs and return the slope coefficients, the standard errors and R2 as output. Notice that you can calculate the standard erors as follows\n", + "1. `cov_b = inv(x'x)*var(u)` where `u` are the residuals\n", + "2. `std_b = sqrt.(diag(cov_b))`\n", + "\n", + "Hint: cell 3 (or so) in [OLS notebook on Paul Söderlind's Github](https://github.com/PaulSoderlind/FinancialEconometrics/blob/master/Ch02_OLS1.ipynb)\n", + "\n", + "Then, regress each of return `Re[:,1]` and `Re[:,2]` on the market `Rme` and a constant. Report the coefficients and standard errors." + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "OLS (generic function with 1 method)" + ] + }, + "execution_count": 6, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "function OLS(y,x) \n", + " b = x\\y\n", + " u = y - x*b\n", + " R2 = 1 - var(u)/var(y)\n", + " cov_b = inv(x'x)*var(u)\n", + " std_b = sqrt.(diag(cov_b))\n", + " return b, std_b, R2\n", + "end " + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Small growth\n", + " coef StdErr\n", + "c -0.620 0.196\n", + "Rme 1.414 0.042\n", + "\n", + "Large value\n", + " coef StdErr\n", + "c 0.100 0.149\n", + "Rme 0.972 0.032\n", + "\n" + ] + } + ], + "source": [ + "T = size(Re,1) #number of time periods\n", + "\n", + "c = ones(T) #a vector with ones\n", + "x = [c Rme] #x is a Tx2 matrix\n", + "b1 = x\\Re[:,1] \n", + "b2 = x\\Re[:,2] \n", + "(b1,std_b1,) = OLS(Re[:,1],x)\n", + "(b2,std_b2,) = OLS(Re[:,2],x)\n", + "\n", + "println(\"Small growth\")\n", + "printmat([b1 std_b1],colNames=[\"coef\",\"StdErr\"],rowNames=[\"c\",\"Rme\"])\n", + "println(\"Large value\")\n", + "printmat([b2 std_b2],colNames=[\"coef\",\"StdErr\"],rowNames=[\"c\",\"Rme\"])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Task 4: Scatter Plot\n", + "\n", + "Make a scatter plot of `Rme` (on the horizontal axis) and `SG`. Do the same for `Rme` and `LV`. Add a 45 degree line and also the regression line from OLS." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", 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"plot!([-40;40],[-40;40],color=:black,linewidth=0.5) \n", + "plot!(xGrid,yLine,color=:red,linestyle=:dash)\n", + "display(p1)" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + 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stocks, %\",\n", + " guidefont = font(8) )\n", + "plot!([-40;40],[-40;40],color=:black,linewidth=0.5)\n", + "plot!(xGrid,yLine,color=:red,linestyle=:dash)\n", + "display(p1)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "@webio": { + "lastCommId": null, + "lastKernelId": null + }, + "anaconda-cloud": {}, + "kernelspec": { + "display_name": "Julia 1.6.3", + "language": "julia", + "name": "julia-1.6" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.6.3" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Problemsets/Solutions/PS03_CLT_Solution.ipynb b/Problemsets/Solutions/PS03_CLT_Solution.ipynb new file mode 100644 index 0000000..3331682 --- /dev/null +++ b/Problemsets/Solutions/PS03_CLT_Solution.ipynb @@ -0,0 +1,5438 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "printyellow (generic function with 1 method)" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "using Printf, Statistics, Random, Distributions\n", + "include(\"jlFiles/printmat.jl\")" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "using Plots\n", + "\n", + "gr(size=(480,320))\n", + "default(fmt = :svg)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 1 (done)\n", + "\n", + "Generate 10000 values of $x=z-2$ where $z$ has a `Chisq(2)` distribution. Show a histogram (with bins from at least -3 to 6) and comment on whether the distribution looks normal. Add the pdf of $x$. \n", + "\n", + "Notice that the (population) mean and standard deviation of a `Chisq(n)` variable are $\\mu=n$ and $\\sigma=\\sqrt{2n}$." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "estimated mean and std: 0.018 2.027\n", + "\n", + "\u001b[31m\u001b[1mstrongly non-normal\u001b[22m\u001b[39m\n" + ] + } + ], + "source": [ + "Random.seed!(123) #so the same random numbers are drawn \n", + " #each time you run this cell\n", + "x = rand(Chisq(2),10_000) .- 2\n", + "\n", + "printlnPs(\"estimated mean and std: \",mean(x),\" \",std(x))\n", + "\n", + "zGrid = range(0,8,length=101)\n", + "\n", + "p1 = histogram( x,bins = -3:0.1:6,\n", + " normalized = true, \n", + " legend=false,\n", + " title = \"Histogram of x\",\n", + " xlabel = \"x\" )\n", + "plot!(zGrid.-2,pdf(Chisq(2),zGrid),linewidth=2) #pdf(x) value equals pdf(z) value, where x=z-2\n", + "display(p1)\n", + "\n", + "printred(\"\\nstrongly non-normal\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 2: Law of Large Numbers\n", + "\n", + "\n", + "Use the $x$ series above and calculate the average of the first $s$ observations. Do this for $s=1$, $s=2$, ..., $s=2000$. Plot the averages (on the vertical axis) against $s$ (on the horizontal axis).\n", + "\n", + "Hint: `cumsum()` or a loop over `s`." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "\n", + "avg_s = cumsum(x)./(1:10_000)\n", + "\n", + "plot( 1:2000,avg_s[1:2000],\n", + " title = \"sample average over observation 1 to s\",\n", + " xlabel = \"s\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 3\n", + "\n", + "(a) Code up a function `SimulateAvg(T,NSim)` which calculates `NSim` different sample averages from `NSim` different samples of length `T` where the data is drawn from $x=z-2$ where $z$ has a `Chisq(2)` distribution. \n", + "\n", + "(b) Simulate 10000 samples of length 5, and estimate the average of x in each sample. Plot a histogram over the averages, using the bins `-3:0.1:3`. Repeat for sample lengths of 25, 100 and 1000." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "SimulateAvg (generic function with 1 method)" + ] + }, + "execution_count": 5, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "function SimulateAvg(T,NSim)\n", + " avg = fill(NaN,NSim)\n", + " for i = 1:NSim\n", + " x_i = rand(Chisq(2),T) .- 2\n", + " avg[i] = mean(x_i)\n", + " end\n", + " return avg\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [], + "source": [ + "NSim = 10_000\n", + "\n", + "avg_5 = SimulateAvg(5,NSim)\n", + "avg_25 = SimulateAvg(25,NSim)\n", + "avg_100 = SimulateAvg(100,NSim)\n", + "avg_1000 = SimulateAvg(1000,NSim);" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + 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= histogram( avg_5,bins = bins,\n", + " normalized = true,legend=false,\n", + " title = \"Histogram of average, T=5\" )\n", + "p2 = histogram( avg_25,bins = bins,\n", + " normalized = true,legend=false,\n", + " title = \"Histogram of average, T=25\" )\n", + "p3 = histogram( avg_100,bins = bins,\n", + " normalized = true,legend=false,\n", + " title = \"Histogram of average, T=100\" )\n", + "p4 = histogram( avg_1000,bins = bins,\n", + " normalized = true,legend=false,\n", + " title = \"Histogram of average, T=1000\" )\n", + "plot(p1,p2,p3,p4,layout = (2,2),size=(800,600))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 4: Central Limit Theorem\n", + "\n", + "Multiply each averages (calculated above) with the square root of the sample size (5, 25, 100, or 1000) and divide by $\\sigma$ (to get $\\sqrt{T} \\bar{x} /\\sigma$) and plot histograms again. In each subplot, add the pdf of a N$(0,1)$ variable. Recall that $\\sigma=2$.\n", + "\n", + "Notice $\\sqrt{T} \\bar{x}$ should converge to a normally distributed variable (with a zero mean since $x$ has). Dividing by $\\sigma$ should make it a standard normal." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + 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+ "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "execution_count": 8, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "σ = 2\n", + "pdfN01 = pdf(Normal(0,1),bins)\n", + "\n", + "p1 = histogram( sqrt(5)*avg_5/σ,bins = bins,\n", + " normalized = true,legend=false,\n", + " title = \"Histogram of √T average/σ, T=5\" )\n", + "plot!(bins,pdfN01,linewidth=2)\n", + "p2 = histogram( sqrt(25)*avg_25/σ,bins = bins,\n", + " normalized = true,legend=false,\n", + " title = \"Histogram of √T average/σ, T=25\" )\n", + "plot!(bins,pdfN01,linewidth=2)\n", + "p3 = histogram( sqrt(100)*avg_100/σ,bins = bins,\n", + " normalized = true,legend=false,\n", + " title = \"Histogram of √T average/σ, T=100\" )\n", + "plot!(bins,pdfN01,linewidth=2)\n", + "p4 = histogram( sqrt(1000)*avg_1000/σ,bins = bins,\n", + " normalized = true,legend=false,\n", + " title = \"Histogram of √T average/σ, T=1000\" )\n", + "plot!(bins,pdfN01,linewidth=2)\n", + "plot(p1,p2,p3,p4,layout = (2,2),size=(800,600))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "@webio": { + "lastCommId": null, + "lastKernelId": null + }, + "kernelspec": { + "display_name": "Julia 1.6.3", + "language": "julia", + "name": "julia-1.6" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.6.3" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Problemsets/Solutions/PS04_YieldToMaturity_Solution.ipynb b/Problemsets/Solutions/PS04_YieldToMaturity_Solution.ipynb new file mode 100644 index 0000000..9ffc037 --- /dev/null +++ b/Problemsets/Solutions/PS04_YieldToMaturity_Solution.ipynb @@ -0,0 +1,522 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "printyellow (generic function with 1 method)" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "using Printf, Roots\n", + "\n", + "include(\"jlFiles/printmat.jl\")" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "using Plots\n", + "\n", + "#pyplot(size=(600,400))\n", + "gr(size=(480,320))\n", + "default(fmt = :svg)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Yield to Maturity" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The yield to maturity (ytm) is the $\\theta$ that solves\n", + "\n", + "$P = \\sum_{k=1}^{K} \\frac{cf_{k}}{(1+\\theta) ^{m_{k}}}$,\n", + "\n", + "where $cf_k$ is the cash flow from the bond (portfolio) $m_k$ periods ahead and $P$ is the current price of the bond (portfolio).\n", + "\n", + "We typically have to find $\\theta$ by a numerical method." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 1\n", + "\n", + "Start by coding a function that calculates a price of the bond as a function of the cash flows ($cf$, a vector), times to the cash flows ($m$, also a vector) and the discount rate ($\\theta$, a single number).\n", + "\n", + "Assume that the bond portfolio pays 0.2 each year for 10 years (1 year from now, 2 years from now,...). (It pays no face value. This is an annuity.) \n", + "\n", + "(1) What is a fair price if the discount rate $\\theta$ is 0.05?\n", + "(2) Plot the fair price as a function of $\\theta$ (use θ = -0.02:0.005:0.1)" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "PV (generic function with 1 method)" + ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "function PV(θ,cf,m) #cf is a vector of all cash flows at times m[1],m[2],...\n", + " cdisc = cf./((1.0.+θ).^m) \n", + " P = sum(cdisc) \n", + " return P\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Price at θ=0.05: 1.544\n" + ] + }, + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "cf = 0.2*ones(10)\n", + "m = 1:10\n", + "θ = -0.02:0.005:0.1\n", + "\n", + "printlnPs(\"Price at θ=0.05: \",PV(0.05,cf,m))\n", + "\n", + "n = length(θ)\n", + "Pfair = fill(NaN,n)\n", + "for i = 1:n\n", + " Pfair[i] = PV(θ[i],cf,m)\n", + "end \n", + "#Pfair = PV.(θ,Ref(cf),Ref(m)) #this works too\n", + "\n", + "plot( θ,Pfair,\n", + " title = \"Present value of cash flow\",\n", + " xlabel = \"θ\",\n", + " legend = false )" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 2\n", + "\n", + "Assume the price this bond (portfolio) is 1.54. Solve for the ytm. Compare with the previous figure. Repeat with a bond price of 1.71.\n", + "\n", + "Hint: the `find_zero()`command of the `Roots.jl` package." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The price and yield to maturity is \n", + " P θ\n", + " 1.540 0.051\n", + " 1.710 0.030\n", + "\n" + ] + } + ], + "source": [ + "P1 = 1.54\n", + "P2 = 1.71\n", + "\n", + "ytm1 = find_zero(θ->PV(θ,cf,m)-P1,(-0.1,0.1)) #solving for ytm\n", + "ytm2 = find_zero(θ->PV(θ,cf,m)-P2,(-0.1,0.1)) \n", + "\n", + "printlnPs(\"The price and yield to maturity is \")\n", + "printmat([P1,P2],[ytm1,ytm2],colNames=[\"P\",\"θ\"])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 3a\n", + "\n", + "Now interpret `cf` as the cash flow to a company where `cf[1]` also incorporates the initial investment and `m[1]=0`. (Alternatively, let `P` represent the initial investment.)\n", + "\n", + "The cash flow process is as follows. Find the IRR (internal rate of return).\n", + "\n", + "Hint: `findzero(fn,0)` would start searching at 0." + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [], + "source": [ + "cf = [-150,100,0,130]\n", + "m = [0,1,2,3];" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.2349421128831387" + ] + }, + "execution_count": 7, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "IRR = find_zero(θ->PV(θ,cf,m),0)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 3b\n", + "\n", + "Change the cash flow as in the cell below. Find the IRR. Could there be several solutions?\n", + "\n", + "(Hint: `find_zeros()`)" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [], + "source": [ + "cf = [-140,320,0,-190];" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "2-element Vector{Float64}:\n", + " 0.046501435936330826\n", + " 0.9160439083362444" + ] + }, + "execution_count": 9, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "find_zeros(θ->PV(θ,cf,m),-0.5,1.5)" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "execution_count": 10, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "θ = -0.1:0.05:1.1\n", + "\n", + "n = length(θ)\n", + "Pfair = fill(NaN,n)\n", + "for i = 1:n\n", + " Pfair[i] = PV(θ[i],cf,m)\n", + "end \n", + "\n", + "plot( θ,Pfair,\n", + " title = \"NPV\",\n", + " xlabel = \"θ\",\n", + " legend = false )" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "@webio": { + "lastCommId": null, + "lastKernelId": null + }, + "anaconda-cloud": {}, + "kernelspec": { + "display_name": "Julia 1.6.3", + "language": "julia", + "name": "julia-1.6" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.6.3" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Problemsets/Solutions/PS05_TradingStrategy_Solution.ipynb b/Problemsets/Solutions/PS05_TradingStrategy_Solution.ipynb new file mode 100644 index 0000000..6421e99 --- /dev/null +++ b/Problemsets/Solutions/PS05_TradingStrategy_Solution.ipynb @@ -0,0 +1,1839 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "printyellow (generic function with 1 method)" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "using Printf, Dates, Statistics, DelimitedFiles, StatsBase\n", + "\n", + "include(\"jlFiles/printmat.jl\")" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "using Plots\n", + "\n", + "gr(size=(480,320))\n", + "default(fmt = :svg)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Load Data" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(10340, 25)" + ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "x = readdlm(\"Data/25_Portfolios_5x5_Daily.CSV\",',',skipstart=1) #daily return data\n", + "ym = round.(Int,x[:,1]) #yearmonthday, like 20071231\n", + "\n", + "dN = Date.(string.(ym),\"yyyymmdd\") #covert to Julia date, eg. 2001-12-31\n", + "\n", + "\n", + "vv = Date(1980,1,1) .<= dN .<= Date(2020,12,31) #pick out the correct sample\n", + "ym = ym[vv]\n", + "R = x[vv,2:end] #returns\n", + "\n", + "(T,n) = size(R) #number of data points, number of assets" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 1: Two Simple Strategies\n", + "\n", + "`R_1`: go long each of asset 1-24 (each with the weight 1/24) and short asset 25\n", + "\n", + "`R_2`: go long asset 1 and short asset 25\n", + "\n", + "The returns of these portfolios are easy to calculate without having to explicitly construct the portfolio weights, but it still a good preparation for later to do the explicit calculations as follows:\n", + "\n", + "1. Construct the vector of portfolio weights `w`\n", + "2. The portfolio return in `t` is `w'*R[t,:]`.\n", + "\n", + "Also, do not be afraid of loops: they are quick.\n", + "\n", + "Show means and standard deviations of the two strategies. Annualize the mean by `*252` and the standard deviation by `*sqrt(252)`.\n", + "\n", + "Plot histograms with bins that are 0.25 wide. (Don't annualize anything in the histograms.)" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [], + "source": [ + "w1 = [ones(24)/24;-1] #static\n", + "w2 = [1;zeros(23);-1]\n", + "\n", + "(R_1,R_2) = (fill(NaN,T),fill(NaN,T))\n", + "for t = 1:T \n", + " R_1[t] = w1'*R[t,:]\n", + " R_2[t] = w2'*R[t,:]\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " R_1 R_2\n", + "mean -0.348 -8.665\n", + "std 13.860 18.996\n", + "SR -0.025 -0.456\n", + "\n" + ] + } + ], + "source": [ + "R_all = [R_1 R_2]\n", + "\n", + "μ = mean(R_all,dims=1)*252\n", + "σ = std(R_all,dims=1)*sqrt(252)\n", + "SR = μ./σ\n", + "\n", + "printmat([μ;σ;SR],colNames=[\"R_1\",\"R_2\"],rowNames=[\"mean\",\"std\",\"SR\"])" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + 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trading strategy.\n", + "\n", + "1. Find the 3 least volatile assets over `t-22:t-1` and give each a portfolio weight `w[t,i]=1/3`. \n", + "\n", + "2. Find the 3 most volatile assets over `t-22:t-1` and give each a portfolio weight `w[t,i]=-1/3`. \n", + "\n", + "3. The portfolio return in `t` is `w[t,:]'*R[t,:]`.\n", + "\n", + "4. Compare the average and std (annualized) with the previous portfolios, over periods `23:T`\n", + "\n", + "Hint: `v = sortperm(x)` gives indices such that `v[1:2]` are the indices of the lowest 2 elements in x. Try `sortperm([12,11,13])` to see." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " 2 \n", + " 1 \n", + "\n" + ] + } + ], + "source": [ + "v = sortperm([12,11,13])\n", + "\n", + "printmat(v[1:2])" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [], + "source": [ + "σ_moving = fill(NaN,T,n)\n", + "for t = 22:T\n", + " σ_moving[t,:] = std(R[t-21:t,:],dims=1)\n", + "end " + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [], + "source": [ + "m = 3 #number of long/short positions \n", + "\n", + "R_3 = fill(NaN,T)\n", + "for t = 23:T #loop over periods, save portfolio returns\n", + " #local s,w #local/global is needed in script\n", + " s = sortperm(σ_moving[t-1,:])\n", + " w = zeros(n)\n", + " w[s[1:m]] .= 1/m\n", + " w[s[end-m+1:end]] .= -1/m\n", + " R_3[t] = w'R[t,:]\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " R_1 R_2 Vol sort\n", + "avg return -0.281 -8.739 3.628\n", + "std 13.867 19.002 14.867\n", + "SR -0.020 -0.460 0.244\n", + "\n" + ] + } + ], + "source": [ + "R_all = [R_1[23:end] R_2[23:end] R_3[23:end]]\n", + "\n", + "μ = mean(R_all,dims=1)*252\n", + "σ = std(R_all,dims=1)*sqrt(252)\n", + "SR = μ./σ\n", + "\n", + "printmat([μ;σ;SR],colNames=[\"R_1\",\"R_2\",\"Vol sort\"],rowNames=[\"avg return\",\"std\",\"SR\"])" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "@webio": { + "lastCommId": null, + "lastKernelId": null + }, + "anaconda-cloud": {}, + "kernelspec": { + "display_name": "Julia 1.6.3", + "language": "julia", + "name": "julia-1.6" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.6.3" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Problemsets/Solutions/PS07_Optimisation_Solution.ipynb b/Problemsets/Solutions/PS07_Optimisation_Solution.ipynb new file mode 100644 index 0000000..b89ffb5 --- /dev/null +++ b/Problemsets/Solutions/PS07_Optimisation_Solution.ipynb @@ -0,0 +1,1193 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Mean Variance Frontier with Short Sales Constraints" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Load Packages and Utility Functions" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "printyellow (generic function with 1 method)" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "using Printf, LinearAlgebra, Optim\n", + "\n", + "include(\"jlFiles/printmat.jl\")" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "using Plots\n", + "\n", + "#pyplot(size=(600,400)) #use pyplot or gr\n", + "gr(size=(480,320))\n", + "default(fmt = :svg)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Traditional MV Calculations \n", + "\n", + "(no constraints)" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "3" + ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "μ = [11.5, 9.5, 6]/100 #expected returns\n", + "Σ = [166 34 58; #covariance matrix\n", + " 34 64 4;\n", + " 58 4 100]/100^2\n", + "\n", + "assetNames = [\"A\",\"B\",\"C\"];\n", + "n = length(μ)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## A Function for Calculating the (traditional) Mean-Variance Frontier\n" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "MVCalc" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "\"\"\"\n", + " MVCalc(μstar,μ,Σ)\n", + "\n", + "Calculate the std and weights of a portfolio (with mean return μstar) on MVF of risky assets.\n", + "\n", + "# Remark\n", + "- Only (λ,δ) and thus (w,stdRp) depend on μstar. We could therefore speed up the computations a bit\n", + "by doing the loop over different μstar values inside the function (and thus not recalculate Σ_1,a,b,c).\n", + "\"\"\"\n", + "function MVCalc(μstar,μ,Σ)\n", + " n = length(μ)\n", + " Σ_1 = inv(Σ)\n", + " a = μ'Σ_1*μ\n", + " b = μ'Σ_1*ones(n)\n", + " c = ones(n)'Σ_1*ones(n)\n", + " λ = (c*μstar - b)/(a*c-b^2)\n", + " δ = (a-b*μstar)/(a*c-b^2)\n", + " w = Σ_1 *(μ*λ.+δ)\n", + " StdRp = sqrt(w'Σ*w)\n", + " return StdRp,w\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "μstar_range = range(0.04,0.15,length=201)\n", + "L = length(μstar_range)\n", + "\n", + "StdRp = fill(NaN,L)\n", + "for i = 1:L\n", + " StdRp[i] = MVCalc(μstar_range[i],μ,Σ)[1]\n", + "end\n", + "\n", + "p1 = plot( StdRp*100,μstar_range*100,\n", + " linecolor = :red,\n", + " linewidth = 2,\n", + " label = \"MVF\",\n", + " legend = :topleft,\n", + " xlim = (0,15),\n", + " ylim = (0,15),\n", + " title = \"MVF (no portfolio constraints)\",\n", + " xlabel = \"Std(Rp), %\",\n", + " ylabel = \"ERp, %\" )\n", + "scatter!(sqrt.(diag(Σ))*100,μ*100,color=:red,label=\"assets\")\n", + "display(p1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# The Maximization Problem\n", + "\n", + "The investor maximizes:\n", + "\n", + "$\\text{E}R_{p}-\\frac{k}{2} \\text{Var}(R_{p})$,\n", + "\n", + "where $\\text{E}R_{p}= w'\\mu$ and $\\text{Var}(R_{p}=w'\\Sigma w$\n", + "\n", + "subject to $\\Sigma_{i=1}^n w_{i} = 1$.\n", + "\n", + "We can trace out the mean-variance frontier by solving this problem for different values of $k$.\n", + "\n", + "To impose the restriction, optimise over `v` and let `w=[v;1-sum(v)]`" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "objfun1 (generic function with 1 method)" + ] + }, + "execution_count": 6, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "function objfun1(v,k,μ,Σ)\n", + " w = [v;1-sum(v)]\n", + " ERp = w'μ\n", + " VarRp = w'Σ*w\n", + " StdRp = sqrt(VarRp)\n", + " U = ERp - k/2*VarRp\n", + " loss = -U #minimise this\n", + " return loss, w, ERp, StdRp\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "A 0.313\n", + "B 0.695\n", + "C -0.008\n", + "\n", + "σₚ 0.079\n", + "μₚ 0.102\n", + "\n" + ] + } + ], + "source": [ + "k = 10\n", + "Sol = optimize(v -> objfun1(v,k,μ,Σ)[1],zeros(n-1))\n", + "\n", + "v = Optim.minimizer(Sol)\n", + "(_,w,ERp,StdRp) = objfun1(v,k,μ,Σ)\n", + "\n", + "printmat(w,rowNames=assetNames)\n", + "printmat([StdRp,ERp],rowNames=[\"σₚ\",\"μₚ\"])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 1\n", + "\n", + "Redo the optimisation for the following range of $k$ values:\n", + "`k = [range(0.1,0.9,length=30);range(1,30,length=25)]` EDIT\n", + "\n", + "Then plot the resulting \"efficient frontier\" (similar to the MV plot above). Is is the same as the plot above?" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [], + "source": [ + "k = [range(0.1,0.9,length=30);range(1,30,length=25)]\n", + "L = length(k)\n", + "\n", + "σμM = fill(NaN,L,2) #to fill with calculation results\n", + "wM = fill(NaN,L,n)\n", + "for i = 1:L\n", + " Sol = optimize(v -> objfun1(v,k[i],μ,Σ)[1],zeros(n-1))\n", + " v = Optim.minimizer(Sol)\n", + " (_,w,ERp,StdRp) = objfun1(v,k[i],μ,Σ)\n", + " σμM[i,:] = [StdRp,ERp]\n", + " wM[i,:] = w\n", + "end\n", + "\n", + "#printmat(k,σμM)" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "p2 = plot( σμM[:,1]*100,σμM[:,2]*100,\n", + " linecolor = :blue,\n", + " linewidth = 2,\n", + " label = \"MVF\",\n", + " legend = :topleft,\n", + " xlim = (0,15),\n", + " ylim = (0,15),\n", + " title = \"MVF (no portfolio constraints)\",\n", + " xlabel = \"Std(Rp), %\",\n", + " ylabel = \"ERp, %\" )\n", + "scatter!(sqrt.(diag(Σ))*100,μ*100,color=:red,label=\"assets\")\n", + "display(p2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 2\n", + "\n", + "Change the optimisation problem to `U = ERp + cost - k/2*VarRp` where `cost` is (sum of the *negative* portfolio weights)*0.05. This is a (crude/dramatic) attempt to illustrate the impact of costs of short selling ($w_i<0$) assets. Plot the new \"effecient frontier\" together with the old one." + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "objfun2 (generic function with 1 method)" + ] + }, + "execution_count": 10, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "function objfun2(v,k,μ,Σ)\n", + " w = [v;1-sum(v)]\n", + " ERp = w'μ\n", + " VarRp = w'Σ*w\n", + " StdRp = sqrt(VarRp)\n", + " cost = sum(((w.<0).*w))*0.05 #<0,short positions reduce the return\n", + " U = ERp + cost - k/2*VarRp\n", + " loss = -U #minimise this\n", + " return loss, w, ERp, StdRp\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [], + "source": [ + "σμMb = fill(NaN,L,2) #to fill with calculation results\n", + "wMb = fill(NaN,L,n)\n", + "for i = 1:L\n", + " Sol = optimize(v -> objfun2(v,k[i],μ,Σ)[1],zeros(n-1))\n", + " v = Optim.minimizer(Sol)\n", + " (_,w,ERp,StdRp) = objfun2(v,k[i],μ,Σ)\n", + " σμMb[i,:] = [StdRp,ERp]\n", + " wMb[i,:] = w\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "p3 = plot( [σμM[:,1] σμMb[:,1]]*100,[σμM[:,2] σμMb[:,2]]*100,\n", + " linecolor = [:blue :black],\n", + " linewidth = 2,\n", + " label = [\"MVF (trad)\" \"MVF (with cost)\"],\n", + " legend = :topleft,\n", + " xlim = (0,15),\n", + " ylim = (0,15),\n", + " title = \"MVF\",\n", + " xlabel = \"Std(Rp), %\",\n", + " ylabel = \"ERp, %\" )\n", + "scatter!(sqrt.(diag(Σ))*100,μ*100,color=:red,label=\"assets\")\n", + "display(p3)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 3\n", + "\n", + "Compare the portfolio weights (with/without the extra `cost`)" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " A B C A B C\n", + "0.1 33.248 7.705 -39.954 3.617 -0.000 -2.617\n", + "0.13 26.055 6.174 -31.229 2.892 0.000 -1.892\n", + "0.16 21.419 5.188 -25.607 2.429 0.000 -1.429\n", + "0.18 18.183 4.499 -21.682 2.102 0.000 -1.102\n", + "0.21 15.798 3.989 -18.788 1.866 -0.000 -0.866\n", + "0.24 13.962 3.600 -16.562 1.681 0.000 -0.681\n", + "0.27 12.509 3.292 -14.800 1.534 0.000 -0.534\n", + "0.29 11.330 3.042 -13.371 1.417 0.000 -0.417\n", + "0.32 10.354 2.831 -12.185 1.320 0.000 -0.320\n", + "0.35 9.531 2.659 -11.191 1.240 0.000 -0.240\n", + "0.38 8.831 2.508 -10.339 1.166 0.000 -0.166\n", + "0.4 8.226 2.381 -9.607 1.105 -0.000 -0.105\n", + "0.43 7.698 2.267 -8.966 1.055 0.000 -0.055\n", + "0.46 7.234 2.168 -8.402 1.005 0.000 -0.005\n", + "0.49 6.824 2.080 -7.904 1.000 0.000 -0.000\n", + "0.51 6.456 2.003 -7.458 1.000 0.000 -0.000\n", + "0.54 6.126 1.933 -7.059 1.000 0.000 -0.000\n", + "0.57 5.827 1.870 -6.697 1.000 0.000 -0.000\n", + "0.6 5.557 1.811 -6.368 1.000 0.000 -0.000\n", + "0.62 5.311 1.759 -6.069 1.000 0.000 -0.000\n", + "0.65 5.085 1.711 -5.796 1.000 -0.000 -0.000\n", + "0.68 4.878 1.668 -5.546 1.000 0.000 -0.000\n", + "0.71 4.687 1.626 -5.313 1.000 0.000 -0.000\n", + "0.73 4.509 1.589 -5.098 1.000 0.000 -0.000\n", + "0.76 4.347 1.554 -4.901 1.000 -0.000 -0.000\n", + "0.79 4.193 1.522 -4.715 1.000 0.000 -0.000\n", + "0.82 4.051 1.492 -4.543 1.000 0.000 -0.000\n", + "0.84 3.918 1.463 -4.381 1.000 0.000 -0.000\n", + "0.87 3.793 1.436 -4.229 1.000 0.000 -0.000\n", + "0.9 3.677 1.411 -4.087 1.000 0.000 -0.000\n", + "1.0 3.307 1.333 -3.640 1.000 0.000 -0.000\n", + "2.21 1.487 0.945 -1.432 0.744 0.256 -0.000\n", + "3.42 0.954 0.831 -0.785 0.547 0.453 -0.000\n", + "4.62 0.699 0.777 -0.476 0.452 0.548 0.000\n", + "5.83 0.550 0.746 -0.296 0.397 0.603 0.000\n", + "7.04 0.452 0.725 -0.177 0.361 0.639 0.000\n", + "8.25 0.383 0.711 -0.094 0.335 0.665 0.000\n", + "9.46 0.332 0.699 -0.031 0.316 0.684 0.000\n", + "10.67 0.292 0.691 0.017 0.292 0.691 0.017\n", + "11.88 0.261 0.684 0.055 0.260 0.684 0.055\n", + "13.08 0.234 0.679 0.087 0.234 0.679 0.087\n", + "14.29 0.213 0.674 0.113 0.213 0.674 0.113\n", + "15.5 0.195 0.670 0.135 0.195 0.670 0.135\n", + "16.71 0.179 0.667 0.154 0.179 0.667 0.154\n", + "17.92 0.166 0.664 0.170 0.166 0.664 0.170\n", + "19.12 0.154 0.661 0.184 0.154 0.661 0.184\n", + "20.33 0.144 0.659 0.197 0.144 0.659 0.197\n", + "21.54 0.134 0.657 0.208 0.134 0.657 0.208\n", + "22.75 0.126 0.656 0.218 0.126 0.656 0.218\n", + "23.96 0.119 0.654 0.227 0.119 0.654 0.227\n", + "25.17 0.113 0.653 0.235 0.113 0.653 0.235\n", + "26.38 0.107 0.651 0.242 0.107 0.651 0.242\n", + "27.58 0.101 0.650 0.249 0.101 0.650 0.249\n", + "28.79 0.096 0.649 0.255 0.096 0.649 0.255\n", + "30.0 0.091 0.648 0.261 0.091 0.648 0.261\n", + "\n" + ] + } + ], + "source": [ + "printmat(wM,wMb,colNames=[assetNames;assetNames],rowNames=string.(round.(k,digits=2)))" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "execution_count": 14, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "plot_array = [] #an alternative way to create subplots\n", + "for i = 1:n\n", + " push!(plot_array,\n", + " plot(k,[wM[:,i] wMb[:,i]],linecolor = [:blue :black],linestyle = [:solid :dash],title = assetNames[i],\n", + " xlabel=\"k\",label=[\"traditional\" \"with cost\"],size = (700,450),ylims=(-3,3)) )\n", + "end\n", + "plot(plot_array...) # note the \"...\" " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "@webio": { + "lastCommId": null, + "lastKernelId": null + }, + "anaconda-cloud": {}, + "kernelspec": { + "display_name": "Julia 1.6.4", + "language": "julia", + "name": "julia-1.6" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.6.4" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Problemsets/Solutions/PS08_Volatility_Solution.ipynb b/Problemsets/Solutions/PS08_Volatility_Solution.ipynb new file mode 100644 index 0000000..36b3c89 --- /dev/null +++ b/Problemsets/Solutions/PS08_Volatility_Solution.ipynb @@ -0,0 +1,6294 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Load Packages and Extra Functions" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "printyellow (generic function with 1 method)" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "using Printf, Dates, Statistics, Distributions, XLSX\n", + "\n", + "include(\"jlFiles/printmat.jl\")" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "using Plots\n", + "\n", + "gr(size=(480,320))\n", + "default(fmt = :svg)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Loading Daily S&P 500 Data\n", + "\n", + "The data file contains some missing values indicated by the value -999.99. Do not use those observations when the S&P level is missing. \n", + "\n", + "1. Calculate daily returns (treat missing days as if they do not exist).\n", + "\n", + "2. Create a vector of dates (from the first column in the data set)." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Number of days in the sample: 17864\n" + ] + } + ], + "source": [ + "xx = XLSX.readxlsx(\"Data/PS08_Data.xlsx\") #reads a sheet from the xlsx file\n", + "x = xx[\"SP500RfPs!A2:B17870\"] #pick out some cells\n", + "\n", + "vv = x[:,2] .!= -999.99 #obs when SP500 is not -999.99\n", + "x = x[vv,:] #only rows with data for x\n", + "\n", + "R = (x[2:end,2]./x[1:end-1,2] .- 1) * 100 #returns, %, will make it Float64 \n", + "dN_SP = Date.(x[2:end,1]) #extract correct sample, convert to Dates\n", + "\n", + "T = length(R)\n", + "println(\"Number of days in the sample: $T\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 1\n", + "\n", + "1. Show a normalised histogram of the returns. Use the bins `-5:0.25:5`\n", + "2. Add a curve for the density of the best fitting $N(\\mu,\\sigma)$ distribution." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(μ,σ) = (mean(R),std(R))\n", + "\n", + "xGrid = -5:0.25:5\n", + "pdfX = pdf.(Normal(μ,σ),xGrid) #Distributions.jl wants σ, not σ^2\n", + "\n", + "histogram( R,bins = -5:0.25:5,\n", + " normalized = true, #normalized to have area=1\n", + " label = \"histogram\",\n", + " legend = :left,\n", + " title = \"Histogram: daily equity returns\",\n", + " titlefont = font(10),\n", + " xlabel = \"Return, %\",\n", + " guidefont = font(8) )\n", + "plot!(xGrid,pdfX,linewidth=3,label=\"fitted N()\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 2\n", + "\n", + "1. Calculate quantile 0.01 in the data and according to the fitted normal distribution. Change the sign and call it \"99% Value at Risk\". Hint: `quantile(R,0.01)`and `quantile(Normal(μ,σ),0.01)`\n", + "\n", + "2. Calculate the expected return conditional on the return being less than the 5th quantile, both in data and according to the normal distribution. Change sign and call it \"expected shortfall\". Hint: `mean(TruncatedNormal(μ,σ,-Inf,quantile001))`\n" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " empirical N()\n", + "VaR 2.672 2.262\n", + "ES 3.973 2.597\n", + "\n" + ] + } + ], + "source": [ + "q_d = quantile(R,0.01)\n", + "q_n = quantile(Normal(μ,σ),0.01)\n", + "\n", + "vv = R .< q_d\n", + "ER_d = mean(R[vv])\n", + "ER_n = mean(TruncatedNormal(μ,σ,-Inf,q_n))\n", + "\n", + "xx = [q_d q_n;ER_d ER_n]\n", + "printmat(-xx,colNames=[\"empirical\",\"N()\"],rowNames = [\"VaR\",\"ES\"] )" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 3\n", + "\n", + "Construct an simple estimate of $\\sigma_t^2$ as a backward looking exponential moving average\n", + "\n", + "$\\sigma_t^2 = \\lambda \\sigma_{t-1}^2 + (1-\\lambda) (R_{t-1} -\\mu_{t-1})^2$,\n", + "where $\\mu_{t}=\\lambda \\mu_{t-1} + (1-\\lambda) R_{t-1}$ \n", + "\n", + "Use $\\lambda=0.94$\n", + "\n", + "Plot the standard deviation in a time series plot." + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [], + "source": [ + "λ = 0.94\n", + "σ² = zeros(T) #vector, time-varying variance\n", + "μ = zeros(T)\n", + "for t = 2:T\n", + " μ[t] = λ*μ[t-1] + (1-λ)*R[t-1]\n", + " σ²[t] = λ*σ²[t-1] + (1-λ)*(R[t-1]-μ[t-1])^2 #RiskMetrics approach\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "xTicksLoc = Date(1950):Year(10):Date(2020)\n", + "xTicksLab = Dates.format.(xTicksLoc,\"Y\") #see Dates.format for more options\n", + "\n", + "p1 = plot( dN_SP,sqrt.(σ²),\n", + " linecolor = :blue,\n", + " #ylim = (-1,35),\n", + " legend = false,\n", + " xticks = (xTicksLoc,xTicksLab),\n", + " title = \"Dynamic std\",\n", + " ylabel= \"%\" )\n", + "hline!([σ],linecolor=:black,line=(:dash,1))\n", + "display(p1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 4\n", + "\n", + "Calculate 99% VaR and ES according to time-varying normal distributions `Normal(μ[t],sqrt(σ²[t]))` and plot the results (as time series plots)." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [], + "source": [ + "(VaR,ES) = (zeros(T),zeros(T))\n", + "\n", + "for t = 2:T\n", + " q_n = quantile(Normal(μ[t],sqrt(σ²[t])),0.01)\n", + " ER_n = mean(TruncatedNormal(μ[t],sqrt(σ²[t]),-Inf,q_n))\n", + " VaR[t] = -q_n\n", + " ES[t] = -ER_n \n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "\n", + "p1 = plot( dN_SP,[VaR ES],\n", + " linecolor = [:blue :red],\n", + " #ylim = (-1,35),\n", + " legend = false,\n", + " xticks = (xTicksLoc,xTicksLab),\n", + " title = \"Dynamic VaR\",\n", + " ylabel= \"%\" )\n", + "display(p1)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "@webio": { + "lastCommId": null, + "lastKernelId": null + }, + "anaconda-cloud": {}, + "kernelspec": { + "display_name": "Julia 1.7.0", + "language": "julia", + "name": "julia-1.7" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.7.0" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Problemsets/Solutions/PS09a_PyCall_Solution.ipynb b/Problemsets/Solutions/PS09a_PyCall_Solution.ipynb new file mode 100644 index 0000000..277591f --- /dev/null +++ b/Problemsets/Solutions/PS09a_PyCall_Solution.ipynb @@ -0,0 +1,369 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Python\n", + "\n", + "using the [PyCall.jl](https://github.com/JuliaPy/PyCall.jl) package." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "CovNWFn" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "using Printf, DelimitedFiles, LinearAlgebra, Statistics\n", + "\n", + "include(\"jlFiles/printmat.jl\")\n", + "include(\"jlFiles/OlsNW.jl\") #functions for OLS" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Sample size: (388,)\n" + ] + } + ], + "source": [ + "x = readdlm(\"Data/FFmFactorsPs.csv\",',',skipstart=1)\n", + "\n", + " #yearmonth, market, small minus big, high minus low\n", + "(ym,Rme,RSMB,RHML) = (x[:,1],x[:,2]/100,x[:,3]/100,x[:,4]/100) \n", + "x = nothing\n", + "\n", + "printlnPs(\"Sample size:\",size(Rme))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Do OLS (in Julia)\n", + "\n", + "use the function sin the file OlsNW.jl to do OLS. Report point estimates and standard errors." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\u001b[34m\u001b[1mOLS Results (assuming iid residuals):\u001b[22m\u001b[39m\n", + "\n", + " b std_iid\n", + "c 0.007 0.002\n", + "SMB 0.217 0.073\n", + "HML -0.429 0.074\n", + "\n" + ] + } + ], + "source": [ + "Y = Rme\n", + "T = size(Y,1)\n", + "X = [ones(T) RSMB RHML]\n", + "\n", + "(b,u,Yhat,V,R2) = OlsGMFn(Y,X)\n", + "std_iid = sqrt.(diag(V))\n", + "\n", + "printblue(\"OLS Results (assuming iid residuals):\\n\")\n", + "xNames = [\"c\",\"SMB\",\"HML\"]\n", + "printmat([b std_iid],colNames=[\"b\",\"std_iid\"],rowNames=xNames)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Getting Started with PyCall" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [], + "source": [ + "using PyCall\n", + "sm = pyimport(\"statsmodels.api\"); #activate this package" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "PyObject \n", + "\"\"\"\n", + " OLS Regression Results \n", + "==============================================================================\n", + "Dep. Variable: y R-squared: 0.134\n", + "Model: OLS Adj. R-squared: 0.130\n", + "Method: Least Squares F-statistic: 29.85\n", + "Date: Thu, 09 Dec 2021 Prob (F-statistic): 8.88e-13\n", + "Time: 10:00:28 Log-Likelihood: 672.28\n", + "No. Observations: 388 AIC: -1339.\n", + "Df Residuals: 385 BIC: -1327.\n", + "Df Model: 2 \n", + "Covariance Type: nonrobust \n", + "==============================================================================\n", + " coef std err t P>|t| [0.025 0.975]\n", + "------------------------------------------------------------------------------\n", + "const 0.0070 0.002 3.167 0.002 0.003 0.011\n", + "x1 0.2170 0.074 2.949 0.003 0.072 0.362\n", + "x2 -0.4291 0.074 -5.821 0.000 -0.574 -0.284\n", + "==============================================================================\n", + "Omnibus: 58.863 Durbin-Watson: 1.849\n", + "Prob(Omnibus): 0.000 Jarque-Bera (JB): 146.539\n", + "Skew: -0.749 Prob(JB): 1.51e-32\n", + "Kurtosis: 5.612 Cond. No. 38.8\n", + "==============================================================================\n", + "\n", + "Notes:\n", + "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n", + "\"\"\"\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "┌ Warning: `vendor()` is deprecated, use `BLAS.get_config()` and inspect the output instead\n", + "│ caller = npyinitialize() at numpy.jl:67\n", + "└ @ PyCall C:\\Users\\psoderlind\\.julia\\packages\\PyCall\\3fwVL\\src\\numpy.jl:67\n" + ] + } + ], + "source": [ + "resultsP = sm.OLS(Y, X).fit() #can use Python functions directly\n", + "\n", + "println(resultsP.summary())" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[:HC0_se, :HC1_se, :HC2_se, :HC3_se, :_HCCM, :__class__, :__delattr__, :__dict__, :__dir__, :__doc__, :__eq__, :__format__, :__ge__, :__getattribute__, :__gt__, :__hash__, :__init__, :__init_subclass__, :__le__, :__lt__, :__module__, :__ne__, :__new__, :__reduce__, :__reduce_ex__, :__repr__, :__setattr__, :__sizeof__, :__str__, :__subclasshook__, :__weakref__, :_abat_diagonal, :_cache, :_data_attr, :_data_in_cache, :_get_robustcov_results, :_is_nested, :_use_t, :_wexog_singular_values, :aic, :bic, :bse, :centered_tss, :compare_f_test, :compare_lm_test, :compare_lr_test, :condition_number, :conf_int, :conf_int_el, :cov_HC0, :cov_HC1, :cov_HC2, :cov_HC3, :cov_kwds, :cov_params, :cov_type, :df_model, :df_resid, :diagn, :eigenvals, :el_test, :ess, :f_pvalue, :f_test, :fittedvalues, :fvalue, :get_influence, :get_prediction, :get_robustcov_results, :info_criteria, :initialize, :k_constant, :llf, :load, :model, :mse_model, :mse_resid, :mse_total, :nobs, :normalized_cov_params, :outlier_test, :params, :predict, :pvalues, :remove_data, :resid, :resid_pearson, :rsquared, :rsquared_adj, :save, :scale, :ssr, :summary, :summary2, :t_test, :t_test_pairwise, :tvalues, :uncentered_tss, :use_t, :wald_test, :wald_test_terms, :wresid]\n" + ] + } + ], + "source": [ + "println(keys(resultsP)) #print all keys (field names)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 1\n", + "\n", + "Print the Julia and Python estimates (of the coefficients) in a table so we can compare directly." + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Comparing the estimates in Julia and Python\n", + " 0.007 0.007\n", + " 0.217 0.217\n", + " -0.429 -0.429\n", + "\n" + ] + } + ], + "source": [ + "b_P = resultsP.params #the numerical results are now a Julia vector\n", + "\n", + "println(\"Comparing the estimates in Julia and Python\")\n", + "printmat([b b_P])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 2\n", + "\n", + "Print the smallest and largest values of the difference between the residuals according to Julia and those according to Python." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "(-2.7755575615628914e-17, 4.163336342344337e-17)\n", + "\n" + ] + } + ], + "source": [ + "printmat(extrema(resultsP.resid - u))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# OLS (in Julia) with Robust Standard Errors\n", + "\n", + "Use standard errors that are robust to heteroskedastcity and autocorrelation (2 lags)." + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\u001b[34m\u001b[1mOLS Results (robust std):\u001b[22m\u001b[39m\n", + "\n", + " b std_nw\n", + "c 0.007 0.002\n", + "SMB 0.217 0.129\n", + "HML -0.429 0.118\n", + "\n" + ] + } + ], + "source": [ + "(b,u,Yhat,V,R2) = OlsNWFn(Y,X,2)\n", + "std_nw = sqrt.(diag(V))\n", + "\n", + "printblue(\"OLS Results (robust std):\\n\")\n", + "xNames = [\"c\",\"SMB\",\"HML\"]\n", + "printmat([b std_nw],colNames=[\"b\",\"std_nw\"],rowNames=xNames)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 3 \n", + "\n", + "Now redo the Python estimation with the same sort of robust standard errors. Hint: `resultsP.get_robustcov_results()`" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "PyObject \n", + "\"\"\"\n", + " OLS Regression Results \n", + "==============================================================================\n", + "Dep. Variable: y R-squared: 0.134\n", + "Model: OLS Adj. R-squared: 0.130\n", + "Method: Least Squares F-statistic: 11.87\n", + "Date: Thu, 09 Dec 2021 Prob (F-statistic): 9.94e-06\n", + "Time: 10:00:32 Log-Likelihood: 672.28\n", + "No. Observations: 388 AIC: -1339.\n", + "Df Residuals: 385 BIC: -1327.\n", + "Df Model: 2 \n", + "Covariance Type: HAC \n", + "==============================================================================\n", + " coef std err t P>|t| [0.025 0.975]\n", + "------------------------------------------------------------------------------\n", + "const 0.0070 0.002 2.850 0.005 0.002 0.012\n", + "x1 0.2170 0.129 1.688 0.092 -0.036 0.470\n", + "x2 -0.4291 0.118 -3.649 0.000 -0.660 -0.198\n", + "==============================================================================\n", + "Omnibus: 58.863 Durbin-Watson: 1.849\n", + "Prob(Omnibus): 0.000 Jarque-Bera (JB): 146.539\n", + "Skew: -0.749 Prob(JB): 1.51e-32\n", + "Kurtosis: 5.612 Cond. No. 38.8\n", + "==============================================================================\n", + "\n", + "Notes:\n", + "[1] Standard Errors are heteroscedasticity and autocorrelation robust (HAC) using 2 lags and without small sample correction\n", + "\"\"\"\n" + ] + } + ], + "source": [ + "resultsP2 = resultsP.get_robustcov_results(cov_type=\"HAC\",maxlags=2)\n", + "\n", + "println(resultsP2.summary())" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "@webio": { + "lastCommId": null, + "lastKernelId": null + }, + "kernelspec": { + "display_name": "Julia 1.7.0", + "language": "julia", + "name": "julia-1.7" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.7.0" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Problemsets/Solutions/PS09b_RCall_Solution.ipynb b/Problemsets/Solutions/PS09b_RCall_Solution.ipynb new file mode 100644 index 0000000..ba9030c --- /dev/null +++ b/Problemsets/Solutions/PS09b_RCall_Solution.ipynb @@ -0,0 +1,363 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# R\n", + "\n", + "using the [RCall.jl](https://juliainterop.github.io/RCall.jl/stable/) package." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "CovNWFn" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "using Printf, DelimitedFiles, LinearAlgebra, Statistics\n", + "\n", + "include(\"jlFiles/printmat.jl\")\n", + "include(\"jlFiles/OlsNW.jl\") #functions for OLS" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Sample size: (388,)\n" + ] + } + ], + "source": [ + "x = readdlm(\"Data/FFmFactorsPs.csv\",',',skipstart=1)\n", + "\n", + " #yearmonth, market, small minus big, high minus low\n", + "(ym,Rme,RSMB,RHML) = (x[:,1],x[:,2]/100,x[:,3]/100,x[:,4]/100) \n", + "x = nothing\n", + "\n", + "printlnPs(\"Sample size:\",size(Rme))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Do OLS (in Julia)\n", + "\n", + "use the function sin the file OlsNW.jl to do OLS. Report point estimates and standard errors." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\u001b[34m\u001b[1mOLS Results (assuming iid residuals):\u001b[22m\u001b[39m\n", + "\n", + " b std_iid\n", + "c 0.007 0.002\n", + "SMB 0.217 0.073\n", + "HML -0.429 0.074\n", + "\n" + ] + } + ], + "source": [ + "Y = Rme\n", + "T = size(Y,1)\n", + "X = [ones(T) RSMB RHML]\n", + "\n", + "(b,u,Yhat,V,R2) = OlsGMFn(Y,X)\n", + "std_iid = sqrt.(diag(V))\n", + "\n", + "printblue(\"OLS Results (assuming iid residuals):\\n\")\n", + "xNames = [\"c\",\"SMB\",\"HML\"]\n", + "printmat([b std_iid],colNames=[\"b\",\"std_iid\"],rowNames=xNames)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Getting Started with RCall" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [], + "source": [ + "using RCall" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "RObject{VecSxp}\n", + "\n", + "Call:\n", + "lm(formula = Y ~ X + 0)\n", + "\n", + "Residuals:\n", + " Min 1Q Median 3Q Max \n", + "-0.20224 -0.02477 0.00335 0.02663 0.11840 \n", + "\n", + "Coefficients:\n", + " Estimate Std. Error t value Pr(>|t|) \n", + "X1 0.006983 0.002205 3.167 0.00166 ** \n", + "X2 0.216968 0.073565 2.949 0.00338 ** \n", + "X3 -0.429088 0.073710 -5.821 1.23e-08 ***\n", + "---\n", + "Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n", + "\n", + "Residual standard error: 0.04295 on 385 degrees of freedom\n", + "Multiple R-squared: 0.1488,\tAdjusted R-squared: 0.1422 \n", + "F-statistic: 22.44 on 3 and 385 DF, p-value: 2.07e-13\n", + "\n", + "\n" + ] + }, + { + "data": { + "text/plain": [ + "RObject{VecSxp}\n", + "\n", + "Call:\n", + "lm(formula = Y ~ X + 0)\n", + "\n", + "Coefficients:\n", + " X1 X2 X3 \n", + " 0.006983 0.216968 -0.429088 \n", + "\n" + ] + }, + "execution_count": 5, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "@rput X Y\n", + "\n", + "ResultsR = reval(\"summary(mod <- lm(Y ~ X+0))\") #print summary of regression\n", + "println(ResultsR)\n", + "\n", + "resultsR = reval(\"mod <- lm(Y ~ X+0)\") #get all output" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[:coefficients, :residuals, :effects, :rank, Symbol(\"fitted.values\"), :assign, :qr, Symbol(\"df.residual\"), :xlevels, :call, :terms, :model]\n" + ] + } + ], + "source": [ + "println(names(resultsR)) #print all keys (field names)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 1\n", + "\n", + "Print the Julia and Python estimates (of the coefficients) in a table so we can compare directly." + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Comparing the estimates in Julia and R\n", + " 0.007 0.007\n", + " 0.217 0.217\n", + " -0.429 -0.429\n", + "\n" + ] + } + ], + "source": [ + "b_R = rcopy(resultsR[:coefficients]) #the numerical results are now a Julia vector\n", + "\n", + "println(\"Comparing the estimates in Julia and R\")\n", + "printmat([b b_R])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 2\n", + "\n", + "Print the smallest and largest values of the difference between the residuals according to Julia and those according to Python." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "(-1.942890293094024e-16, 6.453171330633722e-16)\n", + "\n" + ] + } + ], + "source": [ + "resid_R = rcopy(resultsR[:residuals]) \n", + "\n", + "printmat(extrema(resid_R - u))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# OLS (in Julia) with Robust Standard Errors\n", + "\n", + "Use standard errors that are robust to heteroskedastcity and autocorrelation (2 lags)." + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\u001b[34m\u001b[1mOLS Results (robust std):\u001b[22m\u001b[39m\n", + "\n", + " b std_nw\n", + "c 0.007 0.002\n", + "SMB 0.217 0.129\n", + "HML -0.429 0.118\n", + "\n" + ] + } + ], + "source": [ + "(b,u,Yhat,V,R2) = OlsNWFn(Y,X,2)\n", + "std_nw = sqrt.(diag(V))\n", + "\n", + "printblue(\"OLS Results (robust std):\\n\")\n", + "xNames = [\"c\",\"SMB\",\"HML\"]\n", + "printmat([b std_nw],colNames=[\"b\",\"std_nw\"],rowNames=xNames)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 3 \n", + "\n", + "Now redo the R estimation with the same sort of robust standard errors. Hint: the `NeweyWest` in the `sandwich`package." + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [], + "source": [ + "@rlibrary sandwich" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " bstd_nw (R)\n", + "c 0.007 0.002\n", + "SMB 0.217 0.129\n", + "HML -0.429 0.118\n", + "\n" + ] + } + ], + "source": [ + "reval(\"mod <- lm(Y ~ X+0)\")\n", + "\n", + "VCV_nwR = reval(NeweyWest(resultsR,lag=2,prewhite=0))\n", + "std_nwR = sqrt.(diag(rcopy(VCV_nwR)))\n", + "\n", + "printmat([b std_nwR],colNames=[\"b\",\"std_nw (R)\"],rowNames=xNames)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "@webio": { + "lastCommId": null, + "lastKernelId": null + }, + "kernelspec": { + "display_name": "Julia 1.7.0", + "language": "julia", + "name": "julia-1.7" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.7.0" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Problemsets/Solutions/PS09c_WarAndPeace_Solution.ipynb b/Problemsets/Solutions/PS09c_WarAndPeace_Solution.ipynb new file mode 100644 index 0000000..735522b --- /dev/null +++ b/Problemsets/Solutions/PS09c_WarAndPeace_Solution.ipynb @@ -0,0 +1,389 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [], + "source": [ + "using Downloads" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Download a Book from Internet\n", + "\n", + "and read it into a string in Julia. Then report the number of letters etc (see below)" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "tags": [] + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "File to download: http://www.gutenberg.org/files/2600/2600-0.txt\n", + "\n", + "check the subfolder Results\n", + "\n" + ] + } + ], + "source": [ + "if !isdir(\"Results\")\n", + " error(\"create the subfolder Results before running this program\")\n", + "end\n", + "\n", + "http = \"http://www.gutenberg.org/files/2600/2600-0.txt\"\n", + "\n", + "println(\"File to download: \",http)\n", + "Downloads.download(http,\"Results/WarAndPeace.txt\")\n", + "\n", + "println(\"\\ncheck the subfolder Results\\n\")" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "tags": [] + }, + "outputs": [], + "source": [ + "fh = open(\"Results/WarAndPeace.txt\")\n", + "str = read(fh,String) \n", + "close(fh)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 1\n", + "\n", + "1. Count the number of letters and the unique letters in `str`. Hint: `length() and `unique()`\n", + "\n", + "2. Count the number of word and lines. Hint: `split(str)` and `split(str,\"\\n\")`\n", + "\n", + "3. Count the number of unique words." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "number of letters and unique letters in W&P: 3293555 113\n", + "\n", + "unique letters: ['\\ufeff', 'T', 'h', 'e', ' ', 'P', 'r', 'o', 'j', 'c', 't', 'G', 'u', 'n', 'b', 'g', 'B', 'k', 'f', 'W', 'a', 'd', ',', 'y', 'L', 'l', 's', '\\r', '\\n', 'i', 'w', 'U', 'S', 'm', 'p', 'v', '.', 'Y', '-', 'I', ':', 'A', 'M', 'R', 'D', '2', '0', '1', '[', '#', '6', ']', 'J', '9', 'E', 'C', 'F', '8', 'V', '*', 'O', 'H', 'N', 'K', '/', '5', 'X', '7', '3', '“', '’', '—', '‘', '!', '?', '”', 'á', 'é', 'ë', 'í', ';', 'x', '(', ')', 'z', 'q', 'À', 'ó', 'ú', 'è', 'î', 'ô', 'à', 'ç', 'Q', 'â', 'ê', 'ï', 'Z', '4', 'ý', 'ö', 'ä', 'ü', 'Á', 'œ', 'É', '=', 'æ', '\"', '%', '\\'', '$']\n", + "\n", + "ASCII letters: ['T', 'h', 'e', ' ', 'P', 'r', 'o', 'j', 'c', 't', 'G', 'u', 'n', 'b', 'g', 'B', 'k', 'f', 'W', 'a', 'd', ',', 'y', 'L', 'l', 's', '\\r', '\\n', 'i', 'w', 'U', 'S', 'm', 'p', 'v', '.', 'Y', '-', 'I', ':', 'A', 'M', 'R', 'D', '2', '0', '1', '[', '#', '6', ']', 'J', '9', 'E', 'C', 'F', '8', 'V', '*', 'O', 'H', 'N', 'K', '/', '5', 'X', '7', '3', '!', '?', ';', 'x', '(', ')', 'z', 'q', 'Q', 'Z', '4', '=', '\"', '%', '\\'', '$']\n" + ] + } + ], + "source": [ + "n = length(str)\n", + "letters = unique(str)\n", + "println(\"number of letters and unique letters in W&P: \", n,\" \",length(letters))\n", + "\n", + "println(\"\\nunique letters: \",letters)\n", + "\n", + "vv = isascii.(letters)\n", + "println(\"\\nASCII letters: \",letters[vv])" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "tags": [] + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "number of words in W&P, including pre-amble, etc: 566334\n", + "number of lines in W&P, including pre-amble, etc: 66033\n" + ] + } + ], + "source": [ + "words = split(str)\n", + "println(\"number of words in W&P, including pre-amble, etc: \",length(words))\n", + "\n", + "lines = split(str,\"\\n\")\n", + "println(\"number of lines in W&P, including pre-amble, etc: \",length(lines))" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": { + "tags": [] + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "How many unique words are there in the file?\n", + "Number of unique words: 41971\n" + ] + } + ], + "source": [ + "println(\"How many unique words are there in the file?\")\n", + "UniqueWords = unique(words)\n", + "\n", + "println(\"Number of unique words: \",length(UniqueWords))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 2\n", + "\n", + "1. How often is Borodinó mentioned. Hint: `occursin.(,words)`\n", + "\n", + "2. Print all lines that contain the word Borodinó. Hint: `occursin(,line)`" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": { + "tags": [] + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "How often is Borodinó mentioned?\n", + "108\n", + "108\n" + ] + } + ], + "source": [ + "println(\"How often is Borodinó mentioned?\") \n", + "\n", + "println(sum(occursin.(\"Borodinó\",words))) #\"Borodinó\" or \"Borodinó. or similarly\n", + "\n", + "println(sum(z->occursin(\"Borodinó\",z),words)) #quicker approach" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "print the line numbers and the lines that contain the word Borodinó:\n", + "\n", + "38971 they reached Borodinó, seventy miles from Moscow. From Vyázma Napoleon\n", + "41375 the twenty-sixth the battle of Borodinó itself took place.\n", + "41377 Why and how were the battles of Shevárdino and Borodinó given and\n", + "41378 accepted? Why was the battle of Borodinó fought? There was not the least\n", + "41399 Before the battle of Borodinó our strength in proportion to the French\n", + "41416 In giving and accepting battle at Borodinó, Kutúzov acted involuntarily\n", + "41427 On the other question, how the battle of Borodinó and the preceding\n", + "41434 position at Borodinó.\n", + "41438 to it, from Borodinó to Utítsa, at the very place where the battle was\n", + "41445 the field of Borodinó.\n", + "41451 during the retreat passed many positions better than Borodinó. They did\n", + "41457 and that the position at Borodinó (the one where the battle was fought),\n", + "41463 Borodinó to the left of, and at a right angle to, the highroad (that\n", + "41481 later, when reports on the battle of Borodinó were written at leisure,\n", + "41486 the battle of Borodinó was fought by us on an entrenched position\n", + "41493 village of Nóvoe, and the center at Borodinó at the confluence of the\n", + "41496 To anyone who looks at the field of Borodinó without thinking of how\n", + "41503 to Borodinó (he could not have seen that position because it did not\n", + "41514 Borodinó—a plain no more advantageous as a position than any other plain\n", + "41531 and chief action of the battle of Borodinó was already lost on the\n", + "41551 distinct from the main course of the battle.) So the battle of Borodinó\n", + "41554 army and people) it has been described. The battle of Borodinó was not\n", + "41557 Shevárdino Redoubt, the Russians fought the battle of Borodinó on an\n", + "41749 hundred paces in front of the knoll and below it. This was Borodinó.\n", + "41780 “Borodinó,” the other corrected him.\n", + "41814 entrenchments. There, you see? There’s our center, at Borodinó, just\n", + "41855 A church procession was coming up the hill from Borodinó. First along\n", + "42128 Borodinó and thence turned to the left, passing an enormous number of\n", + "42133 him than any other spot on the plain of Borodinó.\n", + "42643 On August 25, the eve of the battle of Borodinó, M. de Beausset, prefect\n", + "42949 The fourth order was: The vice-King will occupy the village (Borodinó)\n", + "42957 given him, he was to advance from the left through Borodinó to the\n", + "42962 not be executed. After passing through Borodinó the vice-King was driven\n", + "42982 Many historians say that the French did not win the battle of Borodinó\n", + "42993 battle of Borodinó, and if this or that other arrangement depended on\n", + "43015 men at Borodinó was not due to Napoleon’s will, though he ordered the\n", + "43022 At the battle of Borodinó Napoleon shot at no one and killed no one.\n", + "43026 The French soldiers went to kill and be killed at the battle of Borodinó\n", + "43065 than previous ones because the battle of Borodinó was the first Napoleon\n", + "43078 Napoleon at the battle of Borodinó fulfilled his office as\n", + "43286 Pierre most of all was the view of the battlefield itself, of Borodinó\n", + "43289 Above the Kolochá, in Borodinó and on both sides of it, especially to\n", + "43297 riverbanks and in Borodinó. A white church could be seen through the\n", + "43298 mist, and here and there the roofs of huts in Borodinó as well as dense\n", + "43301 the whole space. Just as in the mist-enveloped hollow near Borodinó, so\n", + "43386 bridge across the Kolochá between Górki and Borodinó, which the French\n", + "43387 (having occupied Borodinó) were attacking in the first phase of the\n", + "43819 The chief action of the battle of Borodinó was fought within the seven\n", + "43820 thousand feet between Borodinó and Bagratión’s flèches. Beyond that\n", + "43825 battlefield. On the field between Borodinó and the flèches, beside the\n", + "43834 troops advanced on Borodinó from their left.\n", + "43838 to Borodinó, so that Napoleon could not see what was happening there,\n", + "43886 Borodinó had been occupied and the bridge over the Kolochá was in the\n", + "43890 as soon in fact as the adjutant had left Borodinó—the bridge had been\n", + "44222 times repulsed. In the center the French had not got beyond Borodinó,\n", + "44301 of the field of Borodinó.\n", + "44821 hundreds of years the peasants of Borodinó, Górki, Shevárdino, and\n", + "44903 Russians at Borodinó. The French invaders, like an infuriated animal\n", + "44909 wound it had received at Borodinó. The direct consequence of the battle\n", + "44910 of Borodinó was Napoleon’s senseless flight from Moscow, his retreat\n", + "44913 which at Borodinó for the first time the hand of an opponent of stronger\n", + "45069 from Smolénsk to Borodinó. The French army pushed on to Moscow, its\n", + "45078 consolidated. At Borodinó a collision took place. Neither army was\n", + "45097 Russian army were convinced that the battle of Borodinó was a victory.\n", + "45185 the twenty-sixth at Borodinó, and each day and hour and minute of the\n", + "45186 retreat from Borodinó to Filí.\n", + "45449 After the battle of Borodinó the abandonment and burning of Moscow was\n", + "45491 that could happen. They went away even before the battle of Borodinó and\n", + "45873 Borodinó.\n", + "45881 Toward the end of the battle of Borodinó, Pierre, having run down\n", + "46434 and commotion. Every day thousands of men wounded at Borodinó were\n", + "46442 Some said there had been another battle after Borodinó at which the\n", + "47598 to the second of September, that is from the battle of Borodinó to the\n", + "48315 ever since the battle of Borodinó, for all the generals who came to\n", + "48353 if after the battle of Borodinó, when the surrender of Moscow became\n", + "49092 particularly of the battle of Borodinó and of that vague sense of his\n", + "50087 ambulance station on the field of Borodinó. His feverish state and the\n", + "50105 Borodinó. They were accompanied by a doctor, Prince Andrew’s valet, his\n", + "50835 battle of Borodinó, there was a soiree, the chief feature of which was\n", + "51280 A few days before the battle of Borodinó, Nicholas received the\n", + "51740 The dreadful news of the battle of Borodinó, of our losses in killed and\n", + "51747 When he received the news of the battle of Borodinó and the abandonment\n", + "53596 The historians consider that, next to the battle of Borodinó and the\n", + "53703 whole campaign and by the battle of Borodinó, the Russian army—when\n", + "53711 Borodinó had been a victory, he alone—who as commander in chief might\n", + "53715 The beast wounded at Borodinó was lying where the fleeing hunter had\n", + "54244 or deliberately deceive themselves. No battle—Tarútino, Borodinó, or\n", + "54858 of Borodinó. He had sought it in philanthropy, in Freemasonry, in the\n", + "55314 day long. At the battle of Borodinó, when Bagratión was killed and nine\n", + "55319 And the quiet little Dokhtúrov rode thither, and Borodinó became the\n", + "55543 The undecided question as to whether the wound inflicted at Borodinó was\n", + "55658 That army could not recover anywhere. Since the battle of Borodinó\n", + "55805 The Battle of Borodinó, with the occupation of Moscow that followed it\n", + "55840 history: to say that the field of battle at Borodinó remained in the\n", + "55844 After the French victory at Borodinó there was no general engagement nor\n", + "55855 The period of the campaign of 1812 from the battle of Borodinó to the\n", + "55895 retreats after battles, the blow dealt at Borodinó and the renewed\n", + "57692 done at Mozháysk after the battle of Borodinó.\n", + "58038 French had given battle at Borodinó, did not achieve its purpose when it\n", + "58063 enemy in full strength at Borodinó—defeated at Krásnoe and the Berëzina\n", + "58772 activity in 1812, never once swerving by word or deed from Borodinó to\n", + "58819 Beginning with the battle of Borodinó, from which time his disagreement\n", + "58820 with those about him began, he alone said that the battle of Borodinó\n", + "58845 this enemy of decisive action, gave battle at Borodinó, investing the\n", + "58848 contradiction to everyone else, declared till his death that Borodinó\n", + "59762 Borodinó for more than a month had recently died in the Rostóvs’ house\n", + "60187 one at Borodinó.\n", + "61413 the cold in his head at Borodinó to the sparks which set Moscow on\n" + ] + } + ], + "source": [ + "println(\"print the line numbers and the lines that contain the word Borodinó:\\n\")\n", + "for (i,line) in enumerate(lines)\n", + " if occursin(\"Borodinó\",line)\n", + " println(i,\" \",line)\n", + " end\n", + "end" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 3\n", + "\n", + "1. Change Borodinó everywhere to Berëzina and then count the occurances" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "tags": [] + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Change Borodinó everywhere to Berëzina and then count the occurances\n", + "125\n", + "125\n" + ] + } + ], + "source": [ + "println(\"Change Borodinó everywhere to Berëzina and then count the occurances\")\n", + "str2 = replace(str,\"Borodinó\"=>\"Berëzina\");\n", + "words2 = split(str2)\n", + "println(sum(occursin.(\"Berëzina\",words2)))\n", + "\n", + "println(sum(z->occursin(\"Berëzina\",z),words2)) #quicker approach" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "@webio": { + "lastCommId": null, + "lastKernelId": null + }, + "anaconda-cloud": {}, + "kernel_info": { + "name": "julia-1.2" + }, + "kernelspec": { + "display_name": "Julia 1.7.0", + "language": "julia", + "name": "julia-1.7" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.7.0" + }, + "nteract": { + "version": "0.24.1" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Problemsets/Solutions/PS10_DataFrames_Solution.ipynb b/Problemsets/Solutions/PS10_DataFrames_Solution.ipynb new file mode 100644 index 0000000..6278251 --- /dev/null +++ b/Problemsets/Solutions/PS10_DataFrames_Solution.ipynb @@ -0,0 +1,559 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Load Packages" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "tags": [] + }, + "outputs": [ + { + "data": { + "text/plain": [ + "printyellow (generic function with 1 method)" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "using Printf, Dates, Statistics, CSV, DataFrames\n", + "include(\"jlFiles/printmat.jl\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Loading Some Data with CSV.jl" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The first 4 lines of Data/Options_prices_US_Canada.csv:\n", + "\n", + "symbol,exchange,date,adjusted close,option symbol,expiration,strike,call/put,style,ask,bid,volume,open interest,unadjusted\n", + "SPX,CBOE,03/30/17,2368.06,SPXW 170331C00300000,03/31/17,300,C,E,2073.9,2062.9,0,0,2368.927\n", + "SPX,CBOE,03/30/17,2368.06,SPXW 170331P00300000,03/31/17,300,P,E,0.1,0,0,0,2368.927\n", + "SPX,CBOE,03/30/17,2368.06,SPXW 170331C00400000,03/31/17,400,C,E,1974.1,1962.7,0,0,2368.927\n", + "\n" + ] + } + ], + "source": [ + "DataFile = \"Data/Options_prices_US_Canada.csv\"\n", + "\n", + "println(\"The first 4 lines of $(DataFile):\\n\")\n", + "txt = readlines(DataFile)\n", + "printmat(txt[1:4])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Use `normalizenames` to get names that can be used in Julia as variables names and specify the `dateformat` used in the csv file (to convert to proper Julia dates). The dates in the file are given as `03/30/17` which CSV/DataFrames interpret as 30 March year 17 (AD). We add `Dates.Year(2000)` to get year 2017." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "tags": [] + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\u001b[1m13952×10 DataFrame\u001b[0m\n", + "\u001b[1m Row \u001b[0m│\u001b[1m symbol \u001b[0m\u001b[1m date \u001b[0m\u001b[1m close \u001b[0m\u001b[1m expiration \u001b[0m\u001b[1m strike \u001b[0m\u001b[1m call_put \u001b[0m\u001b[1m ask \u001b[0m\u001b[1m\u001b[0m ⋯\n", + "\u001b[1m \u001b[0m│\u001b[90m String3 \u001b[0m\u001b[90m Date \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Date \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m String1 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m\u001b[0m ⋯\n", + "───────┼────────────────────────────────────────────────────────────────────────\n", + " 1 │ SPX 2017-03-30 2368.06 2017-03-31 300.0 C 2073.9 ⋯\n", + " 2 │ SPX 2017-03-30 2368.06 2017-03-31 300.0 P 0.1\n", + " 3 │ SPX 2017-03-30 2368.06 2017-03-31 400.0 C 1974.1\n", + " 4 │ SPX 2017-03-30 2368.06 2017-03-31 400.0 P 0.05\n", + " 5 │ SPX 2017-03-30 2368.06 2017-03-31 500.0 C 1874.1 ⋯\n", + " 6 │ SPX 2017-03-30 2368.06 2017-03-31 500.0 P 0.05\n", + " 7 │ SPX 2017-03-30 2368.06 2017-03-31 600.0 C 1774.1\n", + " 8 │ SPX 2017-03-30 2368.06 2017-03-31 600.0 P 0.05\n", + " 9 │ SPX 2017-03-30 2368.06 2017-03-31 700.0 C 1673.9 ⋯\n", + " 10 │ SPX 2017-03-30 2368.06 2017-03-31 700.0 P 0.05\n", + " 11 │ SPX 2017-03-30 2368.06 2017-03-31 750.0 C 1624.1\n", + " ⋮ │ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱\n", + " 13943 │ XIU 2017-03-30 23.06 2019-03-15 21.0 C 3.21\n", + " 13944 │ XIU 2017-03-30 23.06 2019-03-15 21.0 P 1.71 ⋯\n", + " 13945 │ XIU 2017-03-30 23.06 2019-03-15 22.0 C 2.53\n", + " 13946 │ XIU 2017-03-30 23.06 2019-03-15 22.0 P 2.03\n", + " 13947 │ XIU 2017-03-30 23.06 2019-03-15 23.0 C 1.97\n", + " 13948 │ XIU 2017-03-30 23.06 2019-03-15 23.0 P 2.51 ⋯\n", + " 13949 │ XIU 2017-03-30 23.06 2019-03-15 24.0 C 1.5\n", + " 13950 │ XIU 2017-03-30 23.06 2019-03-15 24.0 P 3.12\n", + " 13951 │ XIU 2017-03-30 23.06 2019-03-15 25.0 C 1.13\n", + " 13952 │ XIU 2017-03-30 23.06 2019-03-15 25.0 P 3.64 ⋯\n", + "\u001b[36m 3 columns and 13931 rows omitted\u001b[0m" + ] + } + ], + "source": [ + "df1 = CSV.read(DataFile,DataFrame,normalizenames=true,dateformat=\"mm/dd/yy\")\n", + "\n", + "df1.date .+= Dates.Year(2000) #03/30/17 to 03/30/2017\n", + "df1.expiration .+= Dates.Year(2000)\n", + "\n", + "select!(df1,Not([:exchange,:option_symbol,:style,:unadjusted])) #deleting some columns\n", + "rename!(df1,:adjusted_close => :close) #renaming a column\n", + "\n", + "show(df1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 1\n", + "\n", + "Create a new DataFrame that contains only the data for SPX and those option contracts that were traded (volume > 0). Hint: `df1[vv, :]` picks out the rows of the data frame for which `vv` is `true`. " + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\u001b[1m2359×10 DataFrame\u001b[0m\n", + "\u001b[1m Row \u001b[0m│\u001b[1m symbol \u001b[0m\u001b[1m date \u001b[0m\u001b[1m close \u001b[0m\u001b[1m expiration \u001b[0m\u001b[1m strike \u001b[0m\u001b[1m call_put \u001b[0m\u001b[1m ask \u001b[0m\u001b[1m \u001b[0m ⋯\n", + "\u001b[1m \u001b[0m│\u001b[90m String3 \u001b[0m\u001b[90m Date \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Date \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m String1 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m \u001b[0m ⋯\n", + "──────┼─────────────────────────────────────────────────────────────────────────\n", + " 1 │ SPX 2017-03-30 2368.06 2017-03-31 1600.0 C 774.0 ⋯\n", + " 2 │ SPX 2017-03-30 2368.06 2017-03-31 1600.0 P 0.05\n", + " 3 │ SPX 2017-03-30 2368.06 2017-03-31 2040.0 P 0.05\n", + " 4 │ SPX 2017-03-30 2368.06 2017-03-31 2050.0 P 0.05\n", + " 5 │ SPX 2017-03-30 2368.06 2017-03-31 2100.0 P 0.05 ⋯\n", + " 6 │ SPX 2017-03-30 2368.06 2017-03-31 2110.0 C 264.5\n", + " 7 │ SPX 2017-03-30 2368.06 2017-03-31 2120.0 P 0.05\n", + " 8 │ SPX 2017-03-30 2368.06 2017-03-31 2150.0 P 0.05\n", + " 9 │ SPX 2017-03-30 2368.06 2017-03-31 2175.0 C 199.5 ⋯\n", + " 10 │ SPX 2017-03-30 2368.06 2017-03-31 2175.0 P 0.05\n", + " 11 │ SPX 2017-03-30 2368.06 2017-03-31 2180.0 C 194.5\n", + " ⋮ │ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱\n", + " 2350 │ SPX 2017-03-30 2368.06 2019-12-20 1900.0 P 111.4\n", + " 2351 │ SPX 2017-03-30 2368.06 2019-12-20 2000.0 P 136.0 ⋯\n", + " 2352 │ SPX 2017-03-30 2368.06 2019-12-20 2200.0 C 345.7\n", + " 2353 │ SPX 2017-03-30 2368.06 2019-12-20 2300.0 C 287.0\n", + " 2354 │ SPX 2017-03-30 2368.06 2019-12-20 2350.0 C 259.7\n", + " 2355 │ SPX 2017-03-30 2368.06 2019-12-20 2375.0 P 264.3 ⋯\n", + " 2356 │ SPX 2017-03-30 2368.06 2019-12-20 2400.0 C 230.7\n", + " 2357 │ SPX 2017-03-30 2368.06 2019-12-20 2400.0 P 275.3\n", + " 2358 │ SPX 2017-03-30 2368.06 2019-12-20 2800.0 C 74.9\n", + " 2359 │ SPX 2017-03-30 2368.06 2019-12-20 3000.0 C 37.5 ⋯\n", + "\u001b[36m 3 columns and 2338 rows omitted\u001b[0m" + ] + } + ], + "source": [ + "vv = (df1.symbol .== \"SPX\") .& (df1.volume .> 0) #rows with SPX, and trade\n", + "\n", + "df2 = df1[vv, :] #create new df, only some rows\n", + "\n", + "show(df2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 2\n", + "\n", + "Create a *group* for each expiration date. These groups can be referred to as `dataG2[key]`.\n", + "\n", + "Hints: `groupby()`" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [], + "source": [ + "dataG2 = groupby(df2,:expiration); #grouped by expiration date" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 3\n", + "\n", + "Print the number of contracts (`nrow`) and the sum of the open interest `:open_interest=>sum` for each of the expiration dates.\n", + "\n", + "Hint: `combine()`" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "

    34 rows × 3 columns

    expirationnrowopen_interest_sum
    DateInt64Int64
    12017-03-311001010183
    22017-04-0394109345
    32017-04-0584102022
    42017-04-07140467105
    52017-04-107351755
    62017-04-126756228
    72017-04-13109303944
    82017-04-177040855
    92017-04-195725690
    102017-04-211661775684
    112017-04-246915085
    122017-04-26566560
    132017-04-28124467161
    142017-05-05129169530
    152017-05-12121225600
    162017-05-19158937452
    172017-05-269512914
    182017-05-31105120944
    192017-06-02340
    202017-06-161261956193
    212017-06-3072136661
    222017-07-213951058
    232017-07-312756511
    242017-08-311415578
    252017-09-1545524556
    262017-09-291745842
    272017-12-1546759858
    282017-12-292235398
    292018-01-191878967
    302018-03-161615051
    ⋮⋮⋮⋮
    " + ], + "text/latex": [ + "\\begin{tabular}{r|ccc}\n", + "\t& expiration & nrow & open\\_interest\\_sum\\\\\n", + "\t\\hline\n", + "\t& Date & Int64 & Int64\\\\\n", + "\t\\hline\n", + "\t1 & 2017-03-31 & 100 & 1010183 \\\\\n", + "\t2 & 2017-04-03 & 94 & 109345 \\\\\n", + "\t3 & 2017-04-05 & 84 & 102022 \\\\\n", + "\t4 & 2017-04-07 & 140 & 467105 \\\\\n", + "\t5 & 2017-04-10 & 73 & 51755 \\\\\n", + "\t6 & 2017-04-12 & 67 & 56228 \\\\\n", + "\t7 & 2017-04-13 & 109 & 303944 \\\\\n", + "\t8 & 2017-04-17 & 70 & 40855 \\\\\n", + "\t9 & 2017-04-19 & 57 & 25690 \\\\\n", + "\t10 & 2017-04-21 & 166 & 1775684 \\\\\n", + "\t11 & 2017-04-24 & 69 & 15085 \\\\\n", + "\t12 & 2017-04-26 & 56 & 6560 \\\\\n", + "\t13 & 2017-04-28 & 124 & 467161 \\\\\n", + "\t14 & 2017-05-05 & 129 & 169530 \\\\\n", + "\t15 & 2017-05-12 & 121 & 225600 \\\\\n", + "\t16 & 2017-05-19 & 158 & 937452 \\\\\n", + "\t17 & 2017-05-26 & 95 & 12914 \\\\\n", + "\t18 & 2017-05-31 & 105 & 120944 \\\\\n", + "\t19 & 2017-06-02 & 34 & 0 \\\\\n", + "\t20 & 2017-06-16 & 126 & 1956193 \\\\\n", + "\t21 & 2017-06-30 & 72 & 136661 \\\\\n", + "\t22 & 2017-07-21 & 39 & 51058 \\\\\n", + "\t23 & 2017-07-31 & 27 & 56511 \\\\\n", + "\t24 & 2017-08-31 & 14 & 15578 \\\\\n", + "\t25 & 2017-09-15 & 45 & 524556 \\\\\n", + "\t26 & 2017-09-29 & 17 & 45842 \\\\\n", + "\t27 & 2017-12-15 & 46 & 759858 \\\\\n", + "\t28 & 2017-12-29 & 22 & 35398 \\\\\n", + "\t29 & 2018-01-19 & 18 & 78967 \\\\\n", + "\t30 & 2018-03-16 & 16 & 15051 \\\\\n", + "\t$\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ \\\\\n", + "\\end{tabular}\n" + ], + "text/plain": [ + "\u001b[1m34×3 DataFrame\u001b[0m\n", + "\u001b[1m Row \u001b[0m│\u001b[1m expiration \u001b[0m\u001b[1m nrow \u001b[0m\u001b[1m open_interest_sum \u001b[0m\n", + "\u001b[1m \u001b[0m│\u001b[90m Date \u001b[0m\u001b[90m Int64 \u001b[0m\u001b[90m Int64 \u001b[0m\n", + "─────┼──────────────────────────────────────\n", + " 1 │ 2017-03-31 100 1010183\n", + " 2 │ 2017-04-03 94 109345\n", + " 3 │ 2017-04-05 84 102022\n", + " 4 │ 2017-04-07 140 467105\n", + " 5 │ 2017-04-10 73 51755\n", + " 6 │ 2017-04-12 67 56228\n", + " 7 │ 2017-04-13 109 303944\n", + " 8 │ 2017-04-17 70 40855\n", + " 9 │ 2017-04-19 57 25690\n", + " 10 │ 2017-04-21 166 1775684\n", + " 11 │ 2017-04-24 69 15085\n", + " ⋮ │ ⋮ ⋮ ⋮\n", + " 25 │ 2017-09-15 45 524556\n", + " 26 │ 2017-09-29 17 45842\n", + " 27 │ 2017-12-15 46 759858\n", + " 28 │ 2017-12-29 22 35398\n", + " 29 │ 2018-01-19 18 78967\n", + " 30 │ 2018-03-16 16 15051\n", + " 31 │ 2018-03-29 9 153\n", + " 32 │ 2018-06-15 17 69132\n", + " 33 │ 2018-12-21 27 179774\n", + " 34 │ 2019-12-20 13 9213\n", + "\u001b[36m 13 rows omitted\u001b[0m" + ] + }, + "execution_count": 6, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "combine(dataG2,nrow,:open_interest=>sum) #same information as above, uncomment to se" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 4 \n", + "Creating two new DataFrames: for expiration date 2017-04-21 and another for 2017-06-16.\n", + "\n", + "Hint: `dataG2[(expiration = Date(\"2017-04-21\"),)]`" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\n" + ] + } + ], + "source": [ + "#creating two new DataFrames: for two different expiration days\n", + "\n", + "df_20170421 = dataG2[(expiration = Date(\"2017-04-21\"),)]\n", + "df_20170616 = dataG2[(expiration = Date(\"2017-06-16\"),)]\n", + "\n", + "println()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 5\n", + "\n", + "For the expiration date 2017-04-21, calculate the mid price as the average of the `.ask` and `.bid`. \n", + "\n", + "Plot the mid price as a function of the strike price `.strike` for put options. Add a curve another curve for the call options." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [], + "source": [ + "using Plots #this loads the Plots package\n", + "\n", + "#pyplot(size=(600,400)) #choice of plotting backend\n", + "gr(size=(480,320))\n", + "default(fmt = :svg) " + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "vv_P = df_20170421.call_put .== \"P\" #put options\n", + "vv_C = df_20170421.call_put .== \"C\" #call options\n", + "\n", + "midPrice = (df_20170421.ask + df_20170421.bid)/2 #creates a traditional vector\n", + "\n", + "p1 = plot( df_20170421.strike[vv_P],midPrice[vv_P],label=\"puts\",\n", + " linecolor = :red,\n", + " linestyle = :solid,\n", + " linewidth = 1,\n", + " title = \"Option prices $(df_20170421.date[1])\",\n", + " xlabel = \"strike prices\" )\n", + "plot!( df_20170421.strike[vv_C],midPrice[vv_C],label=\"calls\",\n", + " linecolor = :blue,\n", + " linestyle = :dot,\n", + " linewidth = 2 )\n", + "vline!([df_20170421.close[1]],linecolor=:black,line=(:dash,2),label=\"SPX level\")\n", + "display(p1)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "@webio": { + "lastCommId": null, + "lastKernelId": null + }, + "anaconda-cloud": {}, + "kernelspec": { + "display_name": "Julia 1.7.0", + "language": "julia", + "name": "julia-1.7" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.7.0" + }, + "nteract": { + "version": "0.23.1" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Problemsets/Solutions/PS10b_OptionDeltas_Solution.ipynb b/Problemsets/Solutions/PS10b_OptionDeltas_Solution.ipynb new file mode 100644 index 0000000..0c8e134 --- /dev/null +++ b/Problemsets/Solutions/PS10b_OptionDeltas_Solution.ipynb @@ -0,0 +1,645 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Load Packages and Extra Functions" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "printyellow (generic function with 1 method)" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "using Printf, Distributions, OffsetArrays, FiniteDiff\n", + "\n", + "#using FiniteDiff: finite_difference_derivative as derivative #could do this instead\n", + "\n", + "include(\"jlFiles/OptionsCalculations.jl\")\n", + "include(\"jlFiles/printmat.jl\")" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "using Plots\n", + "#pyplot(size=(600,400))\n", + "gr(size=(480,320))\n", + "default(fmt = :svg)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# The Black-Scholes Model\n", + "\n", + "The next cell calculates call `C` and put prices `P` from the Black-Scholes formula.\n", + "\n", + "The key parameters are:\n", + "`(S,K,m,y,σ) = (current undelying price, strike price,time to expiration,interest rate, volatility)`" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "call price at K=42: 2.893 \n", + "\n", + "put price at K=42: 1.856 \n", + "\n" + ] + } + ], + "source": [ + "(S,K,m,y,σ) = (42,42,0.5,0.05,0.2)\n", + "\n", + "C = OptionBlackSPs(S,K,m,y,σ)\n", + "printlnPs(\"call price at K=$K: \",C,\"\\n\")\n", + "\n", + "P = OptionBlackSPs(S,K,m,y,σ,0,true)\n", + "printlnPs(\"put price at K=$K: \",P,\"\\n\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 1\n", + "\n", + "For a range of different prices of the underlying asset `S=30:60`, calculate the call and put prices and plot them (with `S` on the hoizontal axis)." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "S_range = range(30.0,60,step=1) #different spot prices\n", + "C_S_range = OptionBlackSPs.(S_range,K,m,y,σ)\n", + "P_S_range = OptionBlackSPs.(S_range,K,m,y,σ,0,true)\n", + "\n", + "p1 = plot( S_range,[C_S_range P_S_range],\n", + " linecolor = [:red :blue],\n", + " label = [\"call\" \"put\"],\n", + " title = \"Black-Scholes call option price, K = $K\",\n", + " xlabel = \"current asset price\",\n", + " ylabel = \"option price\" )\n", + "display(p1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# The Binomial Option Pricing Model\n", + "\n", + "The next cell contains functions to implement a binomial option pricing model for European style (exercise at expiration only) and American style (exercise any day) options. We use the CRR (Cox-Ross-Rubinstein) parameterisation.\n", + "\n", + "The key parameters are the same as before, but also `n` which is the number of time steps used in the calculations (which defaults to 250)." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "BOPM_American (generic function with 3 methods)" + ] + }, + "execution_count": 5, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "\"\"\"\n", + "CRRparams(σ,m,n,y)\n", + "\n", + " BOPM parameters according to CRR\n", + "\"\"\"\n", + "function CRRparams(σ,m,y,n)\n", + " h = m/n #time step size (in years)\n", + " u = exp(σ*sqrt(h)) #up move\n", + " d = exp(-σ*sqrt(h)) #down move\n", + " p = (exp(y*h) - d)/(u-d) #rn prob of up move\n", + " return h,u,d,p\n", + "end \n", + "\n", + "function BOPM_European(S,K,m,y,σ,isPut=false,n=250)\n", + " (h,u,d,p) = CRRparams(σ,m,y,n)\n", + " STree = BuildSTree(S,n,u,d)\n", + " price = EuOptionPrice(STree,K,y,h,p,isPut)[0][]\n", + " return price\n", + "end\n", + "\n", + "function BOPM_American(S,K,m,y,σ,isPut=false,n=250)\n", + " (h,u,d,p) = CRRparams(σ,m,y,n)\n", + " STree = BuildSTree(S,n,u,d)\n", + " price = AmOptionPrice(STree,K,y,h,p,isPut)[1][0][1]\n", + " return price\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "call (Eu) 2.891\n", + "put (Eu) 1.854\n", + "call (Am) 2.891\n", + "put (Am) 1.954\n", + "\n", + "\u001b[31m\u001b[1mNotice that c_e=c_a but that p_a>=p_e\u001b[22m\u001b[39m\n" + ] + } + ], + "source": [ + "c_e = BOPM_European(S,K,m,y,σ,false) #call price, European style\n", + "p_e = BOPM_European(S,K,m,y,σ,true) #put price, European\n", + "\n", + "c_a = BOPM_American(S,K,m,y,σ,false) #call, American\n", + "p_a = BOPM_American(S,K,m,y,σ,true) #put, American\n", + "\n", + "printmat([c_e,p_e,c_a,p_a],rowNames=[\"call (Eu)\",\"put (Eu)\",\"call (Am)\",\"put (Am)\"])\n", + "\n", + "printred(\"Notice that c_e=c_a but that p_a>=p_e\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Task 2\n", + "\n", + "Redo the calculation of p_e and p_a for `S=30:60` and plot the prices." + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "p_e = BOPM_European.(S_range,K,m,y,σ,true)\n", + "p_a = BOPM_American.(S_range,K,m,y,σ,true)\n", + "\n", + "p1 = plot( S_range,[p_e p_a],\n", + " linecolor = [:red :blue],\n", + " label = [\"European\" \"American\"],\n", + " title = \"BOPM put option prices, K=$K\",\n", + " xlabel = \"current asset price\",\n", + " ylabel = \"option price\" )\n", + "display(p1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Delta\n", + "\n", + "To hedge an option contract (eg. if we are short one option contract), we buy $\\Delta$ units of the underlying asset, where $\\Delta$ is the partial derivative of the option price with respect to the price of the underlying asset.\n", + "\n", + "Calculate $\\Delta$ for both the European and American puts, using the BOPM function. Do the calculation for each value in `S_range` and plot the results.\n", + "\n", + "To calculate the derivatives we use \n", + "`FiniteDiff.finite_difference_derivative(the function,an S value)`" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [], + "source": [ + "L = length(S_range)\n", + "(Δp_e,Δp_a) = [fill(NaN,L) for _=1:2]\n", + "for i = 1:L\n", + " Δp_e[i] = FiniteDiff.finite_difference_derivative( z->BOPM_European(z,K,m,y,σ,true),S_range[i] )\n", + " Δp_a[i] = FiniteDiff.finite_difference_derivative( z->BOPM_American(z,K,m,y,σ,true),S_range[i] )\n", + "end\n", + "\n", + " #this would also work, no need for a loop\n", + " #FiniteDiff.finite_difference_derivative( z->BOPM_European.(z,K,m,y,σ,true),S_range,Val{:central} )" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "p1 = plot( S_range,[Δp_e Δp_a],\n", + " linecolor = [:red :blue],\n", + " label = [\"European\" \"American\"],\n", + " title = \"Δ of BOPM put option prices, K=$K\",\n", + " xlabel = \"current asset price\",\n", + " ylabel = \"Δ\" )\n", + "display(p1)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "@webio": { + "lastCommId": null, + "lastKernelId": null + }, + "anaconda-cloud": {}, + "kernelspec": { + "display_name": "Julia 1.7.0", + "language": "julia", + "name": "julia-1.7" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.7.0" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Problemsets/jlFiles/OlsNW.jl b/Problemsets/jlFiles/OlsNW.jl new file mode 100644 index 0000000..c149561 --- /dev/null +++ b/Problemsets/jlFiles/OlsNW.jl @@ -0,0 +1,112 @@ +#------------------------------------------------------------------------------ +""" + OlsGMFn(Y,X) + +LS of Y on X; for one dependent variable, Gauss-Markov assumptions + +# Usage +(b,u,Yhat,V,R2) = OlsGMFn(Y,X) + +# Input +- `Y::Vector`: T-vector, the dependent variable +- `X::Matrix`: Txk matrix of regressors (including deterministic ones) + +# Output +- `b::Vector`: k-vector, regression coefficients +- `u::Vector`: T-vector, residuals Y - yhat +- `Yhat::Vector`: T-vector, fitted values X*b +- `V::Matrix`: kxk matrix, covariance matrix of b +- `R2::Number`: scalar, R2 value + +""" +function OlsGMFn(Y,X) + + T = size(Y,1) + + b = X\Y + Yhat = X*b + u = Y - Yhat + + σ2 = var(u) + V = inv(X'X)*σ2 + R2 = 1 - σ2/var(Y) + + return b, u, Yhat, V, R2 + +end +#------------------------------------------------------------------------------ + + +#------------------------------------------------------------------------------ +""" + OlsNWFn(Y,X,m=0) + +LS of Y on X; for one dependent variable, using Newey-West covariance matrix + +# Usage +(b,u,Yhat,V,R2) = OlsNWFn(Y,X,m) + +# Input +- `Y::Array`: Tx1, the dependent variable +- `X::Array`: Txk matrix of regressors (including deterministic ones) +- `m::Int`: scalar, bandwidth in Newey-West + +# Output +- `b::Array`: kx1, regression coefficients +- `u::Array`: Tx1, residuals Y - Yhat +- `Yhat::Vector`: Tx1, fitted values X*b +- `V::Array`: kxk matrix, covariance matrix of b +- `R2::Number`: scalar, R2 value + +""" +function OlsNWFn(Y,X,m=0) + + T = size(Y,1) + + b = X\Y + Yhat = X*b + u = Y - Yhat + + S = CovNWFn(X.*u,m) #Newey-West covariance matrix + Sxx = X'X + V = inv(Sxx)'S*inv(Sxx) #covariance matrix of b + R2 = 1 - var(u)/var(Y) + + return b, u, Yhat, V, R2 + +end +#------------------------------------------------------------------------------ + + +#------------------------------------------------------------------------------ + +""" + CovNWFn(g0,m=0) + +Calculates covariance matrix of sample average. + +# Input +- `g0::Matrix`: Txq Matrix of q moment conditions +- `m:int`: scalar, number of lags to use + +# Output +- `S::Matrix`: qxq covariance matrix(average g0) + +""" +function CovNWFn(g0,m=0) + + T = size(g0,1) #g0 is Txq + m = min(m,T-1) #number of lags + + g = g0 .- mean(g0,dims=1) #normalizing to zero means + + S = g'g #(qxT)*(Txq) + for s = 1:m + Λ_s = g[s+1:T,:]'g[1:T-s,:] #same as Sum[g_t*g_{t-s}',t=s+1,T] + S = S + (1 - s/(m+1))*(Λ_s + Λ_s') + end + + return S + +end +#------------------------------------------------------------------------------ diff --git a/Problemsets/jlFiles/OptionsCalculations.jl b/Problemsets/jlFiles/OptionsCalculations.jl new file mode 100644 index 0000000..709422c --- /dev/null +++ b/Problemsets/jlFiles/OptionsCalculations.jl @@ -0,0 +1,104 @@ +""" + BuildSTree(S,n,u,d) + +Build binomial tree, starting at `S` and having `n` steps with up move `u` and down move `d` + +# Output +- `STree:: Vector or vectors`: each (sub-)vector is for a time step. `STree[0] = [S]` and `STree[n]` is for time period n. + +""" +function BuildSTree(S,n,u,d) + STree = [fill(NaN,i) for i = 1:n+1] #vector of vectors (of different lengths) + STree = OffsetArray(STree,0:n) #convert so the indices are 0:n + STree[0][1] = S #step 0 is in STree[0], element 1 + for i = 1:n #move forward in time + STree[i][1:end-1] = u*STree[i-1] #up move from STree[i-1][1:end] + STree[i][end] = d*STree[i-1][end] #down move from STree[i-1][end] + end + return STree +end + + +""" + EuOptionPrice(STree,K,y,h,p,isPut=false) + +Calculate price of European option from binomial model + +# Output +- `Value:: Vector of vectors`: option values at different nodes, same structure as STree + +""" +function EuOptionPrice(STree,K,y,h,p,isPut=false) #price of European option + Value = similar(STree) #tree for derivative, to fill + n = length(STree) - 1 #number of steps in STree + if isPut + Value[n] = max.(0,K.-STree[n]) #put, at last time node + else + Value[n] = max.(0,STree[n].-K) #call, at last time node + end + for i = n-1:-1:0 #move backward in time + Value[i] = exp(-y*h)*(p*Value[i+1][1:end-1] + (1-p)*Value[i+1][2:end]) + end #p*up + (1-p)*down, discount + return Value +end + +""" + AmOptionPrice(STree,K,y,h,p,isPut=false) + +Calculate price of American option from binomial model + +# Output +- `Value:: Vector of vectors`: option values at different nodes, same structure as STree +- `Exerc::` Vector of vectors`: true if early exercise at the node, same structure as STree + +""" +function AmOptionPrice(STree,K,y,h,p,isPut=false) #price of American option + Value = similar(STree) #tree for derivative, to fill + n = length(STree) - 1 + Exerc = similar(Value,BitArray) #same structure as STree, but BitArrays, empty + if isPut + Value[n] = max.(0,K.-STree[n]) #put, at last time node + else + Value[n] = max.(0,STree[n].-K) #call, at last time node + end + Exerc[n] = Value[n] .> 0 #exercise + for i = n-1:-1:0 #move backward in time + fa = exp(-y*h)*(p*Value[i+1][1:end-1] + (1-p)*Value[i+1][2:end]) + if isPut + Value[i] = max.(K.-STree[i],fa) #put + else + Value[i] = max.(STree[i].-K,fa) #call + end + Exerc[i] = Value[i] .> fa #early exercise + end + return Value, Exerc +end + + +""" + Φ(x) + +Calculate Pr(z<=x) for N(0,1) variable z +""" +function Φ(x) + Pr = cdf(Normal(0,1),x) + return Pr +end + + +""" + OptionBlackSPs(S,K,m,y,σ,δ=0,PutIt=false) + +Calculate Black-Scholes European call or put option price, continuous dividends of δ +""" +function OptionBlackSPs(S,K,m,y,σ,δ=0,PutIt=false) + d1 = ( log(S/K) + (y-δ+0.5*σ^2)*m ) / (σ*sqrt(m)) + d2 = d1 - 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znV}Wz?RP-9CNwPE!q5q;dF=mFp~X8Do=7;4HiZ7-7lhV?9!E!EV<4nr4PDxTe{EeGIKEEzKlbX<["1";"4"],:width=>10,:prec=>3,:NoPrinting=>false,:StringFmt=>"") +printmat(x;colNames=["a","b"],opt...) #notice ; and ... +``` +(not all keywords are needed) + +# Requires +- fmtNumPs + +# Notice +- The prefixN and suffixN could potentially be made function inputs. This would allow +a fairly flexible way to format tables. + + +Paul.Soderlind@unisg.ch + +""" +function printmat(fh::IO,x...;colNames=[],rowNames=[], + width=10,prec=3,NoPrinting=false,StringFmt="",cell00="") + + isempty(x) && return nothing #do nothing is isempty(x) + + typeTestQ = any(!=(eltype(x[1])),[eltype(z) for z in x]) #test if eltype(x[i]) differs + if typeTestQ #create matrix from tuple created by x... + x = hcat(Matrix{Any}(hcat(x[1])),x[2:end]...) #preserving types of x[i] + else + x = hcat(x...) + end + + (m,n) = (size(x,1),size(x,2)) + + (length(rowNames) == 1 < m) && (rowNames = [string(rowNames[1],i) for i = 1:m]) #"ri" + (length(colNames) == 1 < n) && (colNames = [string(colNames[1],i) for i = 1:n]) #"ci" + + if StringFmt == "csv" + (prefixN,suffixN) = (fill("",n),vcat(fill(",",n-1),"")) #prefix and suffix for column 1:n + (prefixC0,suffixC0) = ("",",") #prefix and suffix for column 0 + else + (prefixN,suffixN) = (fill("",n),fill("",n)) + (prefixC0,suffixC0) = ("","") + end + + if length(rowNames) == 0 #width of column 0 (cell00 and rowNames) + col0Width = 0 + else + col0Width = maximum(length,vcat(cell00,rowNames)) + length(prefixC0) + length(suffixC0) + end + + colWidth = [width + length(prefixN[j]) + length(suffixN[j]) for j=1:n] #widths of column 1:n + + iob = IOBuffer() + + if !isempty(colNames) #print (cell00,colNames), if any + !isempty(cell00) ? txt0 = string(prefixC0,cell00,suffixC0) : txt0 = "" + print(iob,rpad(txt0,col0Width)) + for j = 1:n #loop over columns + print(iob,lpad(string(prefixN[j],colNames[j],suffixN[j]),colWidth[j])) + end + print(iob,"\n") + end + #print rowNames and x + (i0,j0) = (1 - first(axes(x,1)),1 - first(axes(x,2))) #i+i0,j+j0 give traditional indices + for i in axes(x,1) #loop over rows + !isempty(rowNames) && print(iob,rpad(string(prefixC0,rowNames[i+i0],suffixC0),col0Width)) + for j in axes(x,2) #loop over columns + print(iob,fmtNumPs(x[i,j],width,prec,"right",prefix=prefixN[j+j0],suffix=suffixN[j+j0])) + end + print(iob,"\n") + end + str = String(take!(iob)) + + if NoPrinting #no printing, just return str + return str + else #print, return nothing + print(fh,str,"\n") + return nothing + end + +end + #when fh is not supplied: printing to screen +printmat(x...;colNames=[],rowNames=[],width=10,prec=3,NoPrinting=false,StringFmt="",cell00="") = + printmat(stdout::IO,x...;colNames,rowNames,width,prec,NoPrinting,StringFmt,cell00) +#------------------------------------------------------------------------------ + + +#------------------------------------------------------------------------------ +""" + printlnPs([fh::IO],z...;width=10,prec=3) + +Subsitute for println, with predefined formatting. + + +# Input +- `fh::IO`: (optional) file handle. If not supplied, prints to screen +- `z::String`: string, numbers and arrays to print + +Paul.Soderlind@unisg.ch + +""" +function printlnPs(fh::IO,z...;width=10,prec=3) + + for x in z #loop over inputs in z... + if isa(x,AbstractArray) + iob = IOBuffer() + for i = 1:length(x) + print(iob,fmtNumPs(x[i],width,prec,"right")) + end + print(fh,String(take!(iob))) + else + print(fh,fmtNumPs(x,width,prec,"right")) + end + end + + print(fh,"\n") + +end + #when fh is not supplied: printing to screen +printlnPs(z...;width=10,prec=3) = printlnPs(stdout::IO,z...;width,prec) +#------------------------------------------------------------------------------ + + +#------------------------------------------------------------------------------ +""" + fmtNumPs(z,width=10,prec=2,justify="right";prefix="",suffix="") + +Create a formatted string of a float (eg, "%10.4f"), nothing (""), +while other values are passed through. Strings are right (or left) justified +and can optionally be given prefix and suffix (eg, ",") + +# Notice +- With prec > 0 and isa(z,Integer), then the string is padded with 1+prec spaces +to align with the printing of floats with the same prec. + +# Requires +- Printf (for 1.6-), fmtNumPsC (for < 1.6) + +""" +function fmtNumPs(z,width=10,prec=2,justify="right";prefix="",suffix="") + + isa(z,Bool) && (z = convert(Int,z)) #Bool -> Int + + if isa(z,AbstractFloat) #example: 101.0234, prec=3 + if VERSION < v"1.6-" + fmt = "%$(width).$(prec)f" + zRound = round(z,digits=prec) + strLR = fmtNumPsC(fmt,zRound) #C fallback solution + else + fmt = Printf.Format("%$(width).$(prec)f") + strLR = Printf.format(fmt,z) + end + elseif isa(z,Nothing) + strLR = "" + elseif isa(z,Integer) && prec > 0 #integer followed by (1+prec spaces) + strLR = string(z," "^(1+prec)) + else #Int, String, Date, Missing, etc + strLR = string(z) + end + + strLR = string(prefix,strLR,suffix) + + if justify == "left" #justification + strLR = rpad(strLR,width+length(prefix)+length(suffix)) + else + strLR = lpad(strLR,width+length(prefix)+length(suffix)) + end + + return strLR + +end +#------------------------------------------------------------------------------ + + +#------------------------------------------------------------------------------ +""" + fmtNumPsC(fmt,z) + +c fallback solution for formatting of floating point number. Used if VERSION < v"1.6-" +""" +function fmtNumPsC(fmt,z) #c fallback solution + if ismissing(z) || isnan(z) || isinf(z) #asprintf does not work for these cases + str = string(z) + else + strp = Ref{Ptr{Cchar}}(0) + len = ccall(:asprintf,Cint,(Ptr{Ptr{Cchar}},Cstring,Cdouble...),strp,fmt,z) + str = unsafe_string(strp[],len) + Libc.free(strp[]) + end + return str +end +#------------------------------------------------------------------------------ + + +#------------------------------------------------------------------------------ +function printblue(x...) + foreach(z->printstyled(z,color=:blue,bold=true),x) + print("\n") +end +function printred(x...) + foreach(z->printstyled(z,color=:red,bold=true),x) + print("\n") +end +function printmagenta(x...) + foreach(z->printstyled(z,color=:magenta,bold=true),x) + print("\n") +end +function printyellow(x...) + foreach(z->printstyled(z,color=:yellow,bold=true),x) + print("\n") +end 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170407P00011500,04/07/17,11.5,P,A,0.05,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170407C00012000,04/07/17,12,C,A,0.73,0.67,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170407P00012000,04/07/17,12,P,A,0.05,0,0,32,12.69 +XEG,TSX,03/30/17,12.69,XEG 170407C00012500,04/07/17,12.5,C,A,0.28,0.25,0,32,12.69 +XEG,TSX,03/30/17,12.69,XEG 170407P00012500,04/07/17,12.5,P,A,0.09,0.05,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170407C00013000,04/07/17,13,C,A,0.07,0.01,0,10,12.69 +XEG,TSX,03/30/17,12.69,XEG 170407P00013000,04/07/17,13,P,A,0.38,0.32,0,30,12.69 +XEG,TSX,03/30/17,12.69,XEG 170407C00013500,04/07/17,13.5,C,A,0.05,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170407P00013500,04/07/17,13.5,P,A,0.84,0.79,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170407C00014000,04/07/17,14,C,A,0.05,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170407P00014000,04/07/17,14,P,A,1.34,1.28,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170407C00014500,04/07/17,14.5,C,A,0.05,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170407P00014500,04/07/17,14.5,P,A,1.84,1.78,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170407C00015000,04/07/17,15,C,A,0.05,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170407P00015000,04/07/17,15,P,A,2.34,2.28,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170413C00010500,04/13/17,10.5,C,A,2.22,2.17,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170413P00010500,04/13/17,10.5,P,A,0.05,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170413C00011000,04/13/17,11,C,A,1.72,1.67,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170413P00011000,04/13/17,11,P,A,0.05,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170413C00011500,04/13/17,11.5,C,A,1.22,1.17,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170413P00011500,04/13/17,11.5,P,A,0.05,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170413C00012000,04/13/17,12,C,A,0.74,0.69,0,10,12.69 +XEG,TSX,03/30/17,12.69,XEG 170413P00012000,04/13/17,12,P,A,0.07,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170413C00012500,04/13/17,12.5,C,A,0.32,0.27,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170413P00012500,04/13/17,12.5,P,A,0.13,0.09,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170413C00013000,04/13/17,13,C,A,0.08,0.04,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170413P00013000,04/13/17,13,P,A,0.4,0.36,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170413C00013500,04/13/17,13.5,C,A,0.06,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170413P00013500,04/13/17,13.5,P,A,0.85,0.79,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170413C00014000,04/13/17,14,C,A,0.05,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170413P00014000,04/13/17,14,P,A,1.34,1.29,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170413C00014500,04/13/17,14.5,C,A,0.05,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170413P00014500,04/13/17,14.5,P,A,1.84,1.78,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170413C00015000,04/13/17,15,C,A,0.05,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170413P00015000,04/13/17,15,P,A,2.34,2.28,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421C00010000,04/21/17,10,C,A,2.72,2.67,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421P00010000,04/21/17,10,P,A,0.03,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421C00011000,04/21/17,11,C,A,1.72,1.67,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421P00011000,04/21/17,11,P,A,0.04,0,0,77,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421C00011250,04/21/17,11.25,C,A,1.47,1.42,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421P00011250,04/21/17,11.25,P,A,0.04,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421C00011500,04/21/17,11.5,C,A,1.23,1.17,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421P00011500,04/21/17,11.5,P,A,0.05,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421C00011750,04/21/17,11.75,C,A,0.99,0.94,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421P00011750,04/21/17,11.75,P,A,0.06,0.01,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421C00012000,04/21/17,12,C,A,0.76,0.72,0,20,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421P00012000,04/21/17,12,P,A,0.07,0.03,0,100369,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421C00012250,04/21/17,12.25,C,A,0.55,0.5,0,30,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421P00012250,04/21/17,12.25,P,A,0.11,0.07,0,201,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421C00012500,04/21/17,12.5,C,A,0.36,0.32,120,156,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421P00012500,04/21/17,12.5,P,A,0.17,0.14,0,144,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421C00012750,04/21/17,12.75,C,A,0.21,0.18,0,84,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421P00012750,04/21/17,12.75,P,A,0.28,0.24,0,10,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421C00013000,04/21/17,13,C,A,0.12,0.08,0,756,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421P00013000,04/21/17,13,P,A,0.43,0.39,0,57,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421C00013250,04/21/17,13.25,C,A,0.07,0.03,0,324,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421P00013250,04/21/17,13.25,P,A,0.64,0.59,0,20,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421C00013500,04/21/17,13.5,C,A,0.06,0.01,0,11000,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421P00013500,04/21/17,13.5,P,A,0.85,0.8,0,10,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421C00013750,04/21/17,13.75,C,A,0.04,0,0,20,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421P00013750,04/21/17,13.75,P,A,1.09,1.05,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421C00014000,04/21/17,14,C,A,0.04,0,0,152,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421P00014000,04/21/17,14,P,A,1.35,1.29,0,242,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421C00014250,04/21/17,14.25,C,A,0.03,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421P00014250,04/21/17,14.25,P,A,1.6,1.54,0,100,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421C00014500,04/21/17,14.5,C,A,0.03,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421P00014500,04/21/17,14.5,P,A,1.84,1.79,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421C00014750,04/21/17,14.75,C,A,0.03,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421P00014750,04/21/17,14.75,P,A,2.09,2.04,0,50,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421C00015000,04/21/17,15,C,A,0.03,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421P00015000,04/21/17,15,P,A,2.34,2.29,0,50,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421C00015250,04/21/17,15.25,C,A,0.03,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421P00015250,04/21/17,15.25,P,A,2.59,2.54,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421C00015500,04/21/17,15.5,C,A,0.03,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421P00015500,04/21/17,15.5,P,A,2.84,2.79,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421C00015750,04/21/17,15.75,C,A,0.03,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421P00015750,04/21/17,15.75,P,A,3.09,3.04,0,50,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421C00016000,04/21/17,16,C,A,0.03,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170421P00016000,04/21/17,16,P,A,3.34,3.29,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170428C00011000,04/28/17,11,C,A,1.73,1.68,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170428P00011000,04/28/17,11,P,A,0.05,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170428C00011500,04/28/17,11.5,C,A,1.24,1.19,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170428P00011500,04/28/17,11.5,P,A,0.06,0.01,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170428C00012000,04/28/17,12,C,A,0.78,0.74,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170428P00012000,04/28/17,12,P,A,0.1,0.05,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170428C00012500,04/28/17,12.5,C,A,0.4,0.36,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170428P00012500,04/28/17,12.5,P,A,0.21,0.17,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170428C00013000,04/28/17,13,C,A,0.15,0.11,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170428P00013000,04/28/17,13,P,A,0.48,0.42,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170428C00013500,04/28/17,13.5,C,A,0.07,0.02,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170428P00013500,04/28/17,13.5,P,A,0.87,0.83,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170428C00014000,04/28/17,14,C,A,0.06,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170428P00014000,04/28/17,14,P,A,1.35,1.3,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170428C00014500,04/28/17,14.5,C,A,0.05,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170428P00014500,04/28/17,14.5,P,A,1.86,1.79,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170428C00015000,04/28/17,15,C,A,0.05,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170428P00015000,04/28/17,15,P,A,2.35,2.29,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519C00010000,05/19/17,10,C,A,2.73,2.67,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519P00010000,05/19/17,10,P,A,0.05,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519C00011000,05/19/17,11,C,A,1.74,1.68,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519P00011000,05/19/17,11,P,A,0.07,0.02,0,29,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519C00011250,05/19/17,11.25,C,A,1.5,1.46,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519P00011250,05/19/17,11.25,P,A,0.08,0.03,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519C00011500,05/19/17,11.5,C,A,1.27,1.22,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519P00011500,05/19/17,11.5,P,A,0.1,0.05,0,20,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519C00011750,05/19/17,11.75,C,A,1.05,1.01,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519P00011750,05/19/17,11.75,P,A,0.12,0.08,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519C00012000,05/19/17,12,C,A,0.85,0.8,0,53,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519P00012000,05/19/17,12,P,A,0.16,0.12,0,535,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519C00012250,05/19/17,12.25,C,A,0.66,0.61,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519P00012250,05/19/17,12.25,P,A,0.22,0.19,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519C00012500,05/19/17,12.5,C,A,0.49,0.45,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519P00012500,05/19/17,12.5,P,A,0.3,0.27,0,31,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519C00012750,05/19/17,12.75,C,A,0.34,0.3,51,51,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519P00012750,05/19/17,12.75,P,A,0.41,0.37,0,31,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519C00013000,05/19/17,13,C,A,0.22,0.18,10,481,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519P00013000,05/19/17,13,P,A,0.54,0.51,20,81,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519C00013250,05/19/17,13.25,C,A,0.15,0.11,0,31,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519P00013250,05/19/17,13.25,P,A,0.72,0.68,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519C00013500,05/19/17,13.5,C,A,0.1,0.07,0,81,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519P00013500,05/19/17,13.5,P,A,0.92,0.87,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519C00013750,05/19/17,13.75,C,A,0.07,0.04,0,31,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519P00013750,05/19/17,13.75,P,A,1.14,1.09,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519C00014000,05/19/17,14,C,A,0.07,0.01,0,28,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519P00014000,05/19/17,14,P,A,1.37,1.32,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519C00014250,05/19/17,14.25,C,A,0.06,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519P00014250,05/19/17,14.25,P,A,1.61,1.56,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519C00014500,05/19/17,14.5,C,A,0.05,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519P00014500,05/19/17,14.5,P,A,1.85,1.8,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519C00014750,05/19/17,14.75,C,A,0.05,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519P00014750,05/19/17,14.75,P,A,2.1,2.04,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519C00015000,05/19/17,15,C,A,0.04,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519P00015000,05/19/17,15,P,A,2.35,2.3,0,3,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519C00015250,05/19/17,15.25,C,A,0.05,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519P00015250,05/19/17,15.25,P,A,2.6,2.55,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519C00015500,05/19/17,15.5,C,A,0.05,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519P00015500,05/19/17,15.5,P,A,2.85,2.79,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519C00015750,05/19/17,15.75,C,A,0.05,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170519P00015750,05/19/17,15.75,P,A,3.1,3.05,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170616C00010000,06/17/17,10,C,A,2.74,2.68,0,49,12.69 +XEG,TSX,03/30/17,12.69,XEG 170616P00010000,06/17/17,10,P,A,0.06,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170616C00011000,06/17/17,11,C,A,1.79,1.72,0,7,12.69 +XEG,TSX,03/30/17,12.69,XEG 170616P00011000,06/17/17,11,P,A,0.1,0.05,0,30,12.69 +XEG,TSX,03/30/17,12.69,XEG 170616C00012000,06/17/17,12,C,A,0.93,0.87,10,193,12.69 +XEG,TSX,03/30/17,12.69,XEG 170616P00012000,06/17/17,12,P,A,0.24,0.19,0,3102,12.69 +XEG,TSX,03/30/17,12.69,XEG 170616C00013000,06/17/17,13,C,A,0.33,0.29,0,394,12.69 +XEG,TSX,03/30/17,12.69,XEG 170616P00013000,06/17/17,13,P,A,0.66,0.6,135,1682,12.69 +XEG,TSX,03/30/17,12.69,XEG 170616C00014000,06/17/17,14,C,A,0.1,0.05,0,81,12.69 +XEG,TSX,03/30/17,12.69,XEG 170616P00014000,06/17/17,14,P,A,1.43,1.36,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170616C00015000,06/17/17,15,C,A,0.04,0,0,5,12.69 +XEG,TSX,03/30/17,12.69,XEG 170616P00015000,06/17/17,15,P,A,2.38,2.3,0,20,12.69 +XEG,TSX,03/30/17,12.69,XEG 170616C00016000,06/17/17,16,C,A,0.04,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170616P00016000,06/17/17,16,P,A,3.35,3.29,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170616C00017000,06/17/17,17,C,A,0.04,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170616P00017000,06/17/17,17,P,A,4.36,4.29,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170915C00010000,09/16/17,10,C,A,2.79,2.7,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170915P00010000,09/16/17,10,P,A,0.13,0.06,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170915C00011000,09/16/17,11,C,A,1.91,1.8,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170915P00011000,09/16/17,11,P,A,0.25,0.16,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170915C00012000,09/16/17,12,C,A,1.16,1.06,0,50,12.69 +XEG,TSX,03/30/17,12.69,XEG 170915P00012000,09/16/17,12,P,A,0.5,0.43,0,230,12.69 +XEG,TSX,03/30/17,12.69,XEG 170915C00013000,09/16/17,13,C,A,0.62,0.52,0,281,12.69 +XEG,TSX,03/30/17,12.69,XEG 170915P00013000,09/16/17,13,P,A,0.97,0.89,0,364,12.69 +XEG,TSX,03/30/17,12.69,XEG 170915C00014000,09/16/17,14,C,A,0.3,0.21,0,194,12.69 +XEG,TSX,03/30/17,12.69,XEG 170915P00014000,09/16/17,14,P,A,1.69,1.56,0,320,12.69 +XEG,TSX,03/30/17,12.69,XEG 170915C00015000,09/16/17,15,C,A,0.16,0.07,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170915P00015000,09/16/17,15,P,A,2.53,2.4,0,50,12.69 +XEG,TSX,03/30/17,12.69,XEG 170915C00016000,09/16/17,16,C,A,0.09,0.02,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170915P00016000,09/16/17,16,P,A,3.45,3.35,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170915C00017000,09/16/17,17,C,A,0.07,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 170915P00017000,09/16/17,17,P,A,4.42,4.33,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 171215C00010000,12/16/17,10,C,A,2.86,2.74,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 171215P00010000,12/16/17,10,P,A,0.2,0.1,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 171215C00011000,12/16/17,11,C,A,2.04,1.9,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 171215P00011000,12/16/17,11,P,A,0.38,0.26,0,40,12.69 +XEG,TSX,03/30/17,12.69,XEG 171215C00012000,12/16/17,12,C,A,1.34,1.21,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 171215P00012000,12/16/17,12,P,A,0.69,0.57,0,90,12.69 +XEG,TSX,03/30/17,12.69,XEG 171215C00013000,12/16/17,13,C,A,0.81,0.69,20,8,12.69 +XEG,TSX,03/30/17,12.69,XEG 171215P00013000,12/16/17,13,P,A,1.17,1.05,90,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 171215C00014000,12/16/17,14,C,A,0.48,0.35,0,5,12.69 +XEG,TSX,03/30/17,12.69,XEG 171215P00014000,12/16/17,14,P,A,1.86,1.68,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 171215C00015000,12/16/17,15,C,A,0.26,0.17,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 171215P00015000,12/16/17,15,P,A,2.65,2.49,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 180316C00005000,03/16/18,5,C,A,7.74,7.65,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 180316P00005000,03/16/18,5,P,A,0.09,0,0,25,12.69 +XEG,TSX,03/30/17,12.69,XEG 180316C00006000,03/16/18,6,C,A,6.74,6.64,0,1,12.69 +XEG,TSX,03/30/17,12.69,XEG 180316P00006000,03/16/18,6,P,A,0.09,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 180316C00007000,03/16/18,7,C,A,5.74,5.64,0,40,12.69 +XEG,TSX,03/30/17,12.69,XEG 180316P00007000,03/16/18,7,P,A,0.09,0,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 180316C00008000,03/16/18,8,C,A,4.74,4.66,0,48,12.69 +XEG,TSX,03/30/17,12.69,XEG 180316P00008000,03/16/18,8,P,A,0.12,0.03,0,66,12.69 +XEG,TSX,03/30/17,12.69,XEG 180316C00009000,03/16/18,9,C,A,3.81,3.68,0,221,12.69 +XEG,TSX,03/30/17,12.69,XEG 180316P00009000,03/16/18,9,P,A,0.17,0.07,0,65,12.69 +XEG,TSX,03/30/17,12.69,XEG 180316C00010000,03/16/18,10,C,A,2.92,2.77,0,162,12.69 +XEG,TSX,03/30/17,12.69,XEG 180316P00010000,03/16/18,10,P,A,0.28,0.18,0,4,12.69 +XEG,TSX,03/30/17,12.69,XEG 180316C00011000,03/16/18,11,C,A,2.15,2,0,196,12.69 +XEG,TSX,03/30/17,12.69,XEG 180316P00011000,03/16/18,11,P,A,0.49,0.39,0,1205,12.69 +XEG,TSX,03/30/17,12.69,XEG 180316C00012000,03/16/18,12,C,A,1.44,1.33,0,12704,12.69 +XEG,TSX,03/30/17,12.69,XEG 180316P00012000,03/16/18,12,P,A,0.84,0.75,25,627,12.69 +XEG,TSX,03/30/17,12.69,XEG 180316C00013000,03/16/18,13,C,A,0.94,0.82,0,1095,12.69 +XEG,TSX,03/30/17,12.69,XEG 180316P00013000,03/16/18,13,P,A,1.34,1.24,0,225,12.69 +XEG,TSX,03/30/17,12.69,XEG 180316C00014000,03/16/18,14,C,A,0.59,0.48,0,251,12.69 +XEG,TSX,03/30/17,12.69,XEG 180316P00014000,03/16/18,14,P,A,2,1.88,0,320,12.69 +XEG,TSX,03/30/17,12.69,XEG 180316C00015000,03/16/18,15,C,A,0.34,0.26,0,526,12.69 +XEG,TSX,03/30/17,12.69,XEG 180316P00015000,03/16/18,15,P,A,2.78,2.62,0,240,12.69 +XEG,TSX,03/30/17,12.69,XEG 190315C00009000,03/15/19,9,C,A,4.05,3.78,0,35,12.69 +XEG,TSX,03/30/17,12.69,XEG 190315P00009000,03/15/19,9,P,A,0.4,0.26,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 190315C00010000,03/15/19,10,C,A,3.27,3.02,0,0,12.69 +XEG,TSX,03/30/17,12.69,XEG 190315P00010000,03/15/19,10,P,A,0.64,0.43,0,30,12.69 +XEG,TSX,03/30/17,12.69,XEG 190315C00011000,03/15/19,11,C,A,2.61,2.4,0,7,12.69 +XEG,TSX,03/30/17,12.69,XEG 190315P00011000,03/15/19,11,P,A,0.92,0.74,0,3274,12.69 +XEG,TSX,03/30/17,12.69,XEG 190315C00012000,03/15/19,12,C,A,1.88,1.79,0,3068,12.69 +XEG,TSX,03/30/17,12.69,XEG 190315P00012000,03/15/19,12,P,A,1.29,1.07,0,139,12.69 +XEG,TSX,03/30/17,12.69,XEG 190315C00013000,03/15/19,13,C,A,1.37,1.22,0,99,12.69 +XEG,TSX,03/30/17,12.69,XEG 190315P00013000,03/15/19,13,P,A,1.84,1.61,20,20,12.69 +XEG,TSX,03/30/17,12.69,XEG 190315C00014000,03/15/19,14,C,A,1.1,0.89,0,24,12.69 +XEG,TSX,03/30/17,12.69,XEG 190315P00014000,03/15/19,14,P,A,2.46,2.19,0,2,12.69 +XEG,TSX,03/30/17,12.69,XEG 190315C00015000,03/15/19,15,C,A,0.81,0.57,0,55,12.69 +XEG,TSX,03/30/17,12.69,XEG 190315P00015000,03/15/19,15,P,A,3.18,2.9,0,14,12.69 +XIU,TSX,03/30/17,23.06,XIU 170331C00020500,03/31/17,20.5,C,A,2.58,2.54,0,0,23.06 +XIU,TSX,03/30/17,23.06,XIU 170331P00020500,03/31/17,20.5,P,A,0.05,0,0,0,23.06 +XIU,TSX,03/30/17,23.06,XIU 170331C00021000,03/31/17,21,C,A,2.08,2.04,0,0,23.06 +XIU,TSX,03/30/17,23.06,XIU 170331P00021000,03/31/17,21,P,A,0.05,0,0,0,23.06 +XIU,TSX,03/30/17,23.06,XIU 170331C00021500,03/31/17,21.5,C,A,1.58,1.54,0,0,23.06 +XIU,TSX,03/30/17,23.06,XIU 170331P00021500,03/31/17,21.5,P,A,0.05,0,0,0,23.06 +XIU,TSX,03/30/17,23.06,XIU 170331C00022000,03/31/17,22,C,A,1.08,1.04,0,0,23.06 +XIU,TSX,03/30/17,23.06,XIU 170331P00022000,03/31/17,22,P,A,0.05,0,0,50,23.06 +XIU,TSX,03/30/17,23.06,XIU 170331C00022500,03/31/17,22.5,C,A,0.59,0.55,0,0,23.06 +XIU,TSX,03/30/17,23.06,XIU 170331P00022500,03/31/17,22.5,P,A,0.05,0,0,0,23.06 +XIU,TSX,03/30/17,23.06,XIU 170331C00023000,03/31/17,23,C,A,0.11,0.06,6,122,23.06 +XIU,TSX,03/30/17,23.06,XIU 170331P00023000,03/31/17,23,P,A,0.06,0.01,0,99,23.06 +XIU,TSX,03/30/17,23.06,XIU 170331C00023500,03/31/17,23.5,C,A,0.05,0,0,0,23.06 +XIU,TSX,03/30/17,23.06,XIU 170331P00023500,03/31/17,23.5,P,A,0.46,0.43,0,10,23.06 +XIU,TSX,03/30/17,23.06,XIU 170331C00024000,03/31/17,24,C,A,0.09,0,0,0,23.06 +XIU,TSX,03/30/17,23.06,XIU 170331P00024000,03/31/17,24,P,A,0.96,0.93,0,10,23.06 +XIU,TSX,03/30/17,23.06,XIU 170331C00024500,03/31/17,24.5,C,A,0.09,0,0,0,23.06 +XIU,TSX,03/30/17,23.06,XIU 170331P00024500,03/31/17,24.5,P,A,1.46,1.43,0,0,23.06 +XIU,TSX,03/30/17,23.06,XIU 170331C00025000,03/31/17,25,C,A,0.09,0,0,0,23.06 +XIU,TSX,03/30/17,23.06,XIU 170331P00025000,03/31/17,25,P,A,1.96,1.93,0,0,23.06 +XIU,TSX,03/30/17,23.06,XIU 170407C00020500,04/07/17,20.5,C,A,2.59,2.56,0,0,23.06 +XIU,TSX,03/30/17,23.06,XIU 170407P00020500,04/07/17,20.5,P,A,0.05,0,0,0,23.06 +XIU,TSX,03/30/17,23.06,XIU 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