1643 lines
343 KiB
Plaintext
1643 lines
343 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {
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"collapsed": true
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},
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"source": [
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"# Some Typical Financial Calculations\n",
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"\n",
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"This notebook (a) estimates CAPM equations and autocorrelations; (b) implements a simple trading strategy; (c) calculates Value at Risk using a simple model for time-varying volatility; (d) calculates the Black-Scholes option price and implied volatility; (e) calculates and draws the mean-variance frontier (w/w.o short selling restrictions)."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Load Packages and Extra Functions\n",
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"\n",
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"The [Roots](https://github.com/JuliaMath/Roots.jl) package solves non-linear equations and the [StatsBase](https://github.com/JuliaStats/StatsBase.jl) package has methods for estimating autocorrelations etc."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"printyellow (generic function with 1 method)"
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]
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},
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"execution_count": 1,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"using Printf, Dates, DelimitedFiles, LinearAlgebra, Roots, Distributions, StatsBase\n",
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"\n",
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"include(\"jlFiles/printmat.jl\")"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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"using Plots\n",
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"\n",
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"#pyplot(size=(600,400)) #use pyplot or gr\n",
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"gr(size=(480,320))\n",
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"default(fmt = :svg)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Load Data"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"1979-01-01 10.190 4.180\n",
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"1979-02-01 -2.820 -3.410\n",
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"1979-03-01 10.900 5.750\n",
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"1979-04-01 2.470 0.050\n",
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"\n"
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]
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}
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],
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"source": [
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"x = readdlm(\"Data/MyData.csv\",',',skipstart=1) #monthly return data\n",
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"ym = round.(Int,x[:,1]) #yearmonth, like 200712\n",
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"Rme = x[:,2] #market excess return\n",
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"Rf = x[:,3] #interest rate\n",
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"R = x[:,4] #return small growth stocks\n",
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"Re = R - Rf #excess returns\n",
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"T = size(Rme,1)\n",
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"\n",
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"dN = Date.(string.(ym),\"yyyymm\") #convert to string and then Julia Date\n",
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"printmat([dN[1:4] Re[1:4] Rme[1:4]])"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# CAPM\n",
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"\n",
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"The CAPM regression is\n",
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"\n",
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"$R_{it}^{e} =\\alpha_{i}+\\beta_{i}R_{mt}^{e}+\\varepsilon_{it}$,\n",
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"\n",
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"where $R_{it}^{e}$ is the excess return of asset $i$ and $R_{mt}^{e}$ is the market excess return. Theory says that $\\alpha=0$, which is easily tested."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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" coeff std t-stat\n",
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"α -0.504 0.304 -1.656\n",
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"β 1.341 0.066 20.427\n",
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"\n",
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" R2: 0.519\n",
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"no. of observations: 388 \n"
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]
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}
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],
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"source": [
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"x = [ones(T) Rme] #regressors\n",
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"y = copy(Re) #to get standard OLS notation\n",
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"b = x\\y #OLS\n",
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"u = y - x*b #residuals\n",
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"covb = inv(x'x)*var(u) #cov(b), see any textbook\n",
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"stdb = sqrt.(diag(covb)) #std(b)\n",
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"R2 = 1 - var(u)/var(y)\n",
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"\n",
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"printmat([b stdb b./stdb],colNames=[\"coeff\",\"std\",\"t-stat\"],rowNames=[\"α\",\"β\"])\n",
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"printlnPs(\"R2: \",R2)\n",
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"printlnPs(\"no. of observations: \",T)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Return Autocorrelation\n",
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"\n",
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"That is, the correlation of $R_{t}^{e}$ and $R_{t-s}^{e}$. \n",
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"\n",
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"It can be shown that the t-stat of an autocorrelation is $\\sqrt{T}$ times the autocorrelation."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Autocorrelations (different lags) of the excess returns in Re\n",
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"lag autocorr t-stat\n",
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"1 0.216 4.253\n",
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"2 0.002 0.046\n",
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"3 -0.018 -0.359\n",
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"4 -0.065 -1.289\n",
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"5 -0.027 -0.536\n",
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"\n"
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]
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}
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],
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"source": [
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"plags = 1:5\n",
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"xCorr = autocor(Re,plags) #using the StatsBase package\n",
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"\n",
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"println(\"Autocorrelations (different lags) of the excess returns in Re\")\n",
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"printmat([xCorr sqrt(T)*xCorr],colNames=[\"autocorr\",\"t-stat\"],rowNames=string.(plags),cell00=\"lag\")"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# A Trading Strategy\n",
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"\n",
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"The next cell implements a very simple momentum trading strategy. \n",
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"\n",
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"1. If $R_{t-1}^{e}\\ge0$, then we hold the market index and shorten the riskfree from $t-1$ to $t$. This means that we will earn $R_{t}^{e}$.\n",
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"\n",
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"2. Instead, if $R_{t-1}^{e}<0$, then we do the opposite. This means that we will earn $-R_{t}^{e}$. \n",
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"\n",
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"This simple strategy could be coded without using a loop, but \"vectorization\" does not speed up much. "
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"The annualized mean excess return of the strategy and a passive portfolio are: 26.733 3.329\n",
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"The annualized Sharpe ratios are: 0.932 0.112\n"
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]
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}
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],
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"source": [
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"(w,Rp) = (fill(NaN,T),fill(NaN,T))\n",
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"for t = 2:T\n",
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" w[t] = (Re[t-1] < 0)*(-1) + (Re[t-1] >= 0)*1 #w is -1 or 1\n",
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" Rp[t] = w[t]*Re[t] \n",
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"end\n",
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"\n",
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"μ = [mean(Rp[2:end]) mean(Re[2:end])]\n",
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"σ = [std(Rp[2:end]) std(Re[2:end])]\n",
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"\n",
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"printlnPs(\"The annualized mean excess return of the strategy and a passive portfolio are: \",μ*12)\n",
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"printlnPs(\"The annualized Sharpe ratios are: \",sqrt(12)*μ./σ)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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||
"# Value at Risk\n",
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"\n",
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"The next cell constructs an simple estimate of $\\sigma_t^2$ as a backward looking moving average (the RiskMetrics approach):\n",
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"\n",
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"$\\sigma_t^2 = \\lambda \\sigma_{t-1}^2 + (1-\\lambda) (R_{t-1} -\\mu)^2$,\n",
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"where $\\mu$ is the average return (for all data).\n",
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"\n",
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"Then, we calculate the 95% VaR by assuming a $N(\\mu,\\sigma_t^2)$ distribution:\n",
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"\n",
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"$\\textrm{VaR}_{t} = - (\\mu-1.64\\sigma_t)$.\n",
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"\n",
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"If the model is correct, then $-R_t > \\text{VaR}_{t}$ should only happen 5% of the times."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [],
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"source": [
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"μ = mean(Rme)\n",
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"\n",
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"λ = 0.95 #weight on old volatility\n",
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"σ² = fill(var(Rme),T) #RiskMetrics approach to estimate variance\n",
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"for t = 2:T\n",
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" σ²[t] = λ*σ²[t-1] + (1-λ)*(Rme[t-1]-μ)^2\n",
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"end\n",
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"\n",
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"VaR95 = -(μ .- 1.64*sqrt.(σ²)); #VaR at 95% level"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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" 288.794,508.779 292.841,515.924 296.497,519.941 300.545,514.832 304.462,532.595 308.509,543.255 312.426,550.827 316.474,567.93 320.522,560.458 324.439,575.771 \n",
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" 368.309,451.699 372.226,469.275 376.274,488.285 380.191,442.39 384.239,439.801 388.286,434.81 391.942,454.855 395.99,468.056 399.907,480.797 403.954,499.56 \n",
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" 407.871,510.439 411.919,524.558 415.967,493.919 419.884,457.077 423.931,462.349 427.848,474.837 431.896,478.821 435.943,484.392 439.599,467.128 443.647,480.869 \n",
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" 447.564,494.134 451.612,495.782 455.529,501.087 459.576,507.779 463.624,435.468 467.541,455.253 471.588,387.788 475.505,397.466 479.553,418.444 483.601,432.458 \n",
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" 487.257,450.038 491.304,465.695 495.221,454.99 499.269,474.534 503.186,488.385 507.233,490.159 511.281,507.946 515.198,526.097 519.246,528.589 523.163,543.818 \n",
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" 527.21,555.951 531.258,566.382 535.044,557.841 539.092,574.782 543.009,590.003 547.057,564.68 550.974,580.529 555.021,585.297 559.069,513.592 562.986,529.887 \n",
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" 567.034,545.213 570.951,557.214 574.998,572.985 579.046,540.336 582.702,557.485 586.749,572.632 590.666,586.934 594.714,585.214 598.631,601.159 602.679,615.442 \n",
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" 606.726,628.213 610.643,616.025 614.691,621.198 618.608,603.871 622.655,610.35 626.703,625.928 630.359,603.488 634.407,601.962 638.324,614.147 642.371,613.678 \n",
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" 685.981,488.346 690.028,506.858 693.945,515.782 697.993,523.961 702.041,535.609 705.958,544.316 710.005,164.687 713.922,152.975 717.97,159.193 722.017,178.689 \n",
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" 725.804,195.232 729.852,216.948 733.769,242.519 737.816,266.716 741.733,280.386 745.781,302.118 749.828,314.845 753.745,333.679 757.793,356.086 761.71,372.488 \n",
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" 765.758,393.559 769.805,392.65 773.461,407.712 777.509,427.867 781.426,438.329 785.473,453.214 789.39,470.19 793.438,456.231 797.486,475.134 801.403,492.54 \n",
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" 805.45,496.215 809.367,514.513 813.415,532.216 817.463,492.129 821.118,510.648 825.166,527.595 829.083,530.306 833.131,497.933 837.048,514.059 841.095,527.8 \n",
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" 845.143,454.12 849.06,439.482 853.108,454.356 857.025,450.737 861.072,467.937 865.12,475.448 868.776,460.212 872.823,476.85 876.74,495.306 880.788,506.204 \n",
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" 884.705,499.444 888.753,506.905 892.8,522.869 896.717,536.566 900.765,553.526 904.682,550.229 908.729,485.685 912.777,503.505 916.564,521.64 920.611,529.86 \n",
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" 924.528,547.353 928.576,564.505 932.493,573.66 936.54,581.213 940.588,589.252 944.505,605.258 948.553,620.926 952.47,625.894 956.517,640.274 960.565,654.933 \n",
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" 964.221,669.328 968.268,680.378 972.185,681.313 976.233,689.924 980.15,703.408 984.198,715.65 988.245,717.014 992.162,729.138 996.21,740.519 1000.13,744.161 \n",
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" 1004.17,754.771 1008.22,759.797 1011.88,757.878 1015.93,731.785 1019.84,744.313 1023.89,756.532 1027.81,750.608 1031.85,756.493 1035.9,754.279 1039.82,755.922 \n",
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" 1043.87,767.557 1047.78,750.219 1051.83,762.227 1055.88,772.602 1059.54,772.244 1063.58,780.088 1067.5,788.442 1071.55,792.348 1075.46,797.369 1079.51,795.129 \n",
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" 1083.56,806.032 1087.48,806.724 1091.52,810.225 1095.44,805.27 1099.49,815.675 1103.54,821.318 1107.32,830.961 1111.37,840.971 1115.29,847.179 1119.34,852.303 \n",
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" 1123.25,856.239 1127.3,801.643 1131.35,805.124 1135.26,789.8 1139.31,800.695 1143.23,768.629 1147.28,773.679 1151.32,760.411 1154.98,770.602 1159.03,742.247 \n",
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" 1162.94,741.448 1166.99,708.563 1170.91,707.652 1174.96,670.647 1179,661.076 1182.92,650.471 1186.97,643.912 1190.89,654.201 1194.93,668.171 1198.98,681.94 \n",
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" 1202.64,651.575 1206.69,647.867 1210.6,662.525 1214.65,662.884 1218.57,671.941 1222.61,673.577 1226.66,430.239 1230.58,428.565 1234.63,416.25 1238.54,415.535 \n",
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" 1242.59,414.539 1246.64,428.682 1250.29,431.364 1254.34,445.563 1258.26,453.06 1262.31,465.374 1266.22,471.151 1270.27,476.853 1274.32,492.551 1278.24,502.105 \n",
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" 1282.28,497.391 1286.2,509.57 1290.25,482.362 1294.3,480.784 1298.08,495.826 1302.13,498.883 1306.05,476.356 1310.09,474.846 1314.01,479.68 1318.06,492.119 \n",
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" 1322.11,475.825 1326.02,463.356 1330.07,472.045 1333.99,391.228 1338.04,411.73 1342.08,426.357 1345.74,358.608 1349.79,334.997 1353.7,320.194 1357.75,343.139 \n",
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" 1361.67,360.639 1365.72,377.214 1369.76,365.324 1373.68,317.166 1377.73,337.574 1381.65,325.337 1385.69,347.419 1389.74,365.805 1393.4,381.528 1397.44,392.682 \n",
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" 1401.36,389.756 1405.41,408.563 1409.33,384.595 1413.37,349.669 1417.42,371.878 1421.34,313.187 1425.39,305.749 1429.3,309.473 1433.35,308.209 1437.4,325.207 \n",
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" 1441.05,344.598 1445.1,366.862 1449.02,347.957 1453.07,347.777 1456.98,369.426 1461.03,389.187 1465.08,407.912 1469,426.721 1473.04,424.905 1476.96,444.453 \n",
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" 1481.01,452.424 1485.06,469.901 1488.84,488.418 1492.89,504.499 1496.81,514.527 1500.85,532.074 1504.77,547.708 1508.82,546.781 1512.87,563.822 1516.78,578.929 \n",
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" 1520.83,594.238 1524.75,594.144 1528.8,602.708 1532.84,606.917 1536.5,620.331 1540.55,629.284 1544.46,632.937 1548.51,638.855 1552.43,653.637 1556.48,655.061 \n",
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" 1560.52,667.117 1564.44,681.259 1568.49,685.201 1572.41,687.763 1576.45,701.036 1580.5,703.441 1584.16,715.25 1588.2,727.133 1592.12,739.644 1596.17,730.599 \n",
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" 1600.09,741.802 1604.13,752.261 1608.18,761.377 1612.1,772.01 1616.15,773.652 1620.06,782.633 1624.11,793.873 1628.16,803.677 1631.81,806.111 1635.86,816.659 \n",
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" 1639.78,814.037 1643.83,812.159 1647.74,813.569 1651.79,798.564 1655.84,809.377 1659.76,805.215 1663.8,811.843 1667.72,773.468 1671.77,782.623 1675.82,729.07 \n",
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" 1679.6,731.018 1683.65,739.46 1687.57,728.291 1691.61,737.69 1695.53,662.989 1699.58,672.537 1703.63,686.404 1707.54,583.873 1711.59,310.937 1715.51,278.666 \n",
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" 1719.56,301.278 1723.6,278.855 1727.26,230.061 1731.31,214.593 1735.22,175.205 1739.27,179.789 1743.19,205.839 1747.24,196.745 1751.28,218.811 1755.2,234.879 \n",
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" 1759.25,252.574 1763.17,260.484 1767.21,281.52 1771.26,293.266 1774.92,311.193 1778.96,311.447 1782.88,333.257 1786.93,305.665 1790.85,306.395 1794.89,301.52 \n",
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" 1798.94,308.235 1802.86,282.17 1806.91,299.02 1810.82,322.513 1814.87,319.541 1818.92,341.153 1822.57,356.179 1826.62,378.152 \n",
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" \"/>\n",
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
|
||
"xTicksLoc = [Date(1980);Date(1990);Date(2000);Date(2010)]\n",
|
||
"xTicksLab = Dates.format.(xTicksLoc,\"Y\")\n",
|
||
"\n",
|
||
"p1 = plot( dN,VaR95,\n",
|
||
" color = :blue,\n",
|
||
" legend = false,\n",
|
||
" xticks = (xTicksLoc,xTicksLab),\n",
|
||
" ylim = (0,11),\n",
|
||
" title = \"1-month Value at Risk (95%)\",\n",
|
||
" ylabel = \"%\",\n",
|
||
" annotation = (Date(1982),1,text(\"(for US equity market)\",8,:left)) )\n",
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"display(p1)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
|
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"# Options"
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]
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||
},
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{
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"cell_type": "markdown",
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"metadata": {},
|
||
"source": [
|
||
"## Black-Scholes Option Price\n",
|
||
"\n",
|
||
"Let $S$ be the the current spot price of an asset and $y$ be the interest rate.\n",
|
||
"\n",
|
||
"The Black-Scholes formula for a European call option with strike price $K$ and time to expiration $m$ is\n",
|
||
"\n",
|
||
"$C =S\\Phi(d_{1}) -e^{-ym}K\\Phi(d_{2})$, where\n",
|
||
"\n",
|
||
"$d_{1} =\\frac{\\ln(S/K)+(y+\\sigma^{2}/2)m}{\\sigma\\sqrt{m}} \\ \\text{ and } \\ d_{2}=d_{1}-\\sigma\\sqrt{m}$ \n",
|
||
"\n",
|
||
"and where $\\Phi(d)$ denotes the probability of $x\\leq d$ when $x$ has an $N(0,1)$ distribution. All variables except the volatility ($\\sigma$) are directly observable. "
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||
]
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},
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{
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"cell_type": "code",
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||
"execution_count": 9,
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||
"metadata": {},
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||
"outputs": [
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||
{
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"data": {
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"text/plain": [
|
||
"OptionBlackSPs"
|
||
]
|
||
},
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"execution_count": 9,
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||
"metadata": {},
|
||
"output_type": "execute_result"
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||
}
|
||
],
|
||
"source": [
|
||
"\"\"\"\n",
|
||
"Pr(z<=x) for N(0,1)\n",
|
||
"\"\"\"\n",
|
||
"Φ(x) = cdf(Normal(0,1),x) #a one-line function\n",
|
||
"\n",
|
||
"\"\"\"\n",
|
||
"Calculate Black-Scholes european call option price\n",
|
||
"\"\"\"\n",
|
||
"function OptionBlackSPs(S,K,m,y,σ)\n",
|
||
"\n",
|
||
" d1 = ( log(S/K) + (y+1/2*σ^2)*m ) / (σ*sqrt(m))\n",
|
||
" d2 = d1 - σ*sqrt(m)\n",
|
||
" c = S*Φ(d1) - exp(-y*m)*K*Φ(d2)\n",
|
||
" return c\n",
|
||
"end"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 10,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
" \n",
|
||
"call price according to Black-Scholes: 1.358\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"σ = 0.4\n",
|
||
"c1 = OptionBlackSPs(10,10,0.5,0.1,σ)\n",
|
||
"printlnPs(\"\\n\",\"call price according to Black-Scholes: \",c1)\n",
|
||
"\n",
|
||
"K = range(7,stop=13,length=51)\n",
|
||
"c = OptionBlackSPs.(10,K,0.5,0.1,σ);"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 11,
|
||
"metadata": {},
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"outputs": [
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" 1825.82,1078.56 \n",
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" \"/>\n",
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"</svg>\n"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
|
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"p1 = plot( K,c,\n",
|
||
" color = :red,\n",
|
||
" legend = false,\n",
|
||
" title = \"Black-Scholes call option price\",\n",
|
||
" xlabel = \"strike price\",\n",
|
||
" ylabel = \"option price\" )\n",
|
||
"display(p1)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
|
||
"# Implied Volatility\n",
|
||
"\n",
|
||
"is the $\\sigma$ value that makes the Black-Scholes equation give the same option price as observed on the market. It is often interpreted as the \"market uncertainty.\"\n",
|
||
"\n",
|
||
"The next cell uses the call option price calculated above as the market price. The implied volatility should then equal the volatility used above (this is a way to check your coding).\n",
|
||
"\n",
|
||
"The next few cells instead use some data on options on German government bonds. "
|
||
]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
|
||
"Implied volatility: 0.400, compare with: 0.4\n"
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]
|
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}
|
||
],
|
||
"source": [
|
||
" #solve for implied vol\n",
|
||
"iv = find_zero(σ->OptionBlackSPs(10,10,0.5,0.1,σ)-c1,(0.00001,5))\n",
|
||
"\n",
|
||
"printlnPs(\"Implied volatility: \",iv,\", compare with: $σ\")"
|
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]
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},
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{
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"cell_type": "code",
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"execution_count": 13,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"21"
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]
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},
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"execution_count": 13,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
|
||
"# LIFFE Bunds option data, trade date April 6, 1994\n",
|
||
"K = [ #strike prices; Mx1 vector\n",
|
||
" 92.00; 94.00; 94.50; 95.00; 95.50; 96.00; 96.50; 97.00;\n",
|
||
" 97.50; 98.00; 98.50; 99.00; 99.50; 100.0; 100.5; 101.0;\n",
|
||
" 101.5; 102.0; 102.5; 103.0; 103.5 ];\n",
|
||
"C = [ #call prices; Mx1 vector\n",
|
||
" 5.13; 3.25; 2.83; 2.40; 2.00; 1.64; 1.31; 1.02;\n",
|
||
" 0.770; 0.570; 0.400; 0.280; 0.190; 0.130; 0.0800; 0.0500;\n",
|
||
" 0.0400; 0.0300; 0.0200; 0.0100; 0.0100 ];\n",
|
||
"S = 97.05 #spot price\n",
|
||
"m = 48/365 #time to expiry in years\n",
|
||
"y = 0.0 #Interest rate: LIFFE=>no discounting\n",
|
||
"N = length(K)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 14,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"Strike and iv for data: \n",
|
||
" 92.000 0.094\n",
|
||
" 94.000 0.081\n",
|
||
" 94.500 0.081\n",
|
||
" 95.000 0.078\n",
|
||
" 95.500 0.075\n",
|
||
" 96.000 0.074\n",
|
||
" 96.500 0.072\n",
|
||
" 97.000 0.071\n",
|
||
" 97.500 0.070\n",
|
||
" 98.000 0.069\n",
|
||
" 98.500 0.068\n",
|
||
" 99.000 0.067\n",
|
||
" 99.500 0.067\n",
|
||
" 100.000 0.068\n",
|
||
" 100.500 0.067\n",
|
||
" 101.000 0.067\n",
|
||
" 101.500 0.070\n",
|
||
" 102.000 0.073\n",
|
||
" 102.500 0.074\n",
|
||
" 103.000 0.072\n",
|
||
" 103.500 0.077\n",
|
||
"\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"iv = fill(NaN,N) #looping over strikes\n",
|
||
"for i = 1:N\n",
|
||
" iv[i] = find_zero(sigma->OptionBlackSPs(S,K[i],m,y,sigma)-C[i],(0.00001,5))\n",
|
||
"end\n",
|
||
"\n",
|
||
"println(\"Strike and iv for data: \")\n",
|
||
"printmat([K iv])"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
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"execution_count": 15,
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
|
||
"p1 = plot( K,iv,\n",
|
||
" color = :red,\n",
|
||
" legend =false,\n",
|
||
" title = \"Implied volatility\",\n",
|
||
" xlabel = \"strike price\",\n",
|
||
" annotation = (98,0.09,text(\"Bunds options April 6, 1994\",8,:left)) )\n",
|
||
"display(p1)"
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||
]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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||
"source": [
|
||
"# Mean-Variance Frontier\n",
|
||
"\n",
|
||
"Given a vector of average returns ($\\mu$) and a variance-covariance matrix ($\\Sigma$), the mean-variance frontier shows the lowest possible portfolio uncertainty for a given expected portfolio return (denoted $\\mu\\text{star}$ below).\n",
|
||
"\n",
|
||
"It is thus the solution to a quadratic minimization problem. The cells below will use the explicit (matrix) formulas for this solution, but we often have to resort to numerical methods when there are portfolio restrictions.\n",
|
||
"\n",
|
||
"It is typically plotted with the portfolio standard deviation on the horizontal axis and the portfolio expected return on the vertical axis.\n",
|
||
"\n",
|
||
"We calculate and plot two different mean-variance frontiers: (1) when we only consider risky assets; (2) when we also consider a risk-free asset."
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||
]
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||
},
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{
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"cell_type": "code",
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||
"execution_count": 16,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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||
"output_type": "stream",
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"text": [
|
||
"μ: \n",
|
||
" 0.115\n",
|
||
" 0.095\n",
|
||
" 0.060\n",
|
||
"\n",
|
||
"Σ: \n",
|
||
" 0.017 0.003 0.006\n",
|
||
" 0.003 0.006 0.000\n",
|
||
" 0.006 0.000 0.010\n",
|
||
"\n",
|
||
"Rf: \n",
|
||
" 0.030\n",
|
||
"\n"
|
||
]
|
||
}
|
||
],
|
||
"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",
|
||
"Rf = 0.03 #riskfree return (an interest rate)\n",
|
||
"\n",
|
||
"println(\"μ: \")\n",
|
||
"printmat(μ)\n",
|
||
"println(\"Σ: \")\n",
|
||
"printmat(Σ)\n",
|
||
"println(\"Rf: \")\n",
|
||
"printmat(Rf)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 17,
|
||
"metadata": {},
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"outputs": [
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||
{
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"data": {
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"text/plain": [
|
||
"MVCalcRf"
|
||
]
|
||
},
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||
"execution_count": 17,
|
||
"metadata": {},
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||
"output_type": "execute_result"
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||
}
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||
],
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||
"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",
|
||
"# Input\n",
|
||
"- `μstar::Vector`: K vector, mean returns to calculate results for\n",
|
||
"- `μ::Vector`: n vector, mean returns\n",
|
||
"- `Σ::Matrix`: nxn, covariance matrix of returns, can contain riskfree assets\n",
|
||
"\n",
|
||
"# Output\n",
|
||
"- `StdRp::Vector`: K vector, standard deviation of mean-variance portfolio (risky only) with mean μstar\n",
|
||
"- `w_p::Matrix`: Kxn, portfolio weights of \"\"\n",
|
||
"\n",
|
||
"\"\"\"\n",
|
||
"function MVCalc(μstar,μ,Σ)\n",
|
||
"\n",
|
||
" (K,n) = (length(μstar),length(μ))\n",
|
||
"\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",
|
||
"\n",
|
||
" (w_p,StdRp) = (fill(NaN,K,n),fill(NaN,K))\n",
|
||
" for i = 1:K\n",
|
||
" λ = (c*μstar[i] - b)/(a*c-b^2)\n",
|
||
" δ = (a-b*μstar[i])/(a*c-b^2)\n",
|
||
" w = Σ_1 *(μ*λ.+δ)\n",
|
||
" StdRp[i] = sqrt(w'Σ*w)\n",
|
||
" w_p[i,:] = w\n",
|
||
" end\n",
|
||
"\n",
|
||
" return StdRp,w_p\n",
|
||
"\n",
|
||
"end\n",
|
||
"\n",
|
||
"\n",
|
||
"\"\"\"\n",
|
||
"Calculate the std of a portfolio (with mean μstar) on MVF of (Risky,Riskfree)\n",
|
||
"\"\"\"\n",
|
||
"function MVCalcRf(μstar,μ,Σ,Rf)\n",
|
||
"\n",
|
||
" (K,n) = (length(μstar),length(μ))\n",
|
||
"\n",
|
||
" μᵉ = μ .- Rf #expected excess returns\n",
|
||
" Σ_1 = inv(Σ)\n",
|
||
"\n",
|
||
" (w_p,StdRp) = (fill(NaN,K,n),fill(NaN,K))\n",
|
||
" for i = 1:K\n",
|
||
" w = (μstar[i]-Rf)/(μᵉ'Σ_1*μᵉ) * Σ_1*μᵉ\n",
|
||
" StdRp[i] = sqrt(w'Σ*w)\n",
|
||
" w_p[i,:] = w\n",
|
||
" end\n",
|
||
"\n",
|
||
" return StdRp,w_p\n",
|
||
"\n",
|
||
"end"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 18,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"μstar = range(Rf,stop=0.15,length=201)\n",
|
||
"L = length(μstar)\n",
|
||
"\n",
|
||
"StdRp = MVCalc(μstar,μ,Σ)[1] #risky assets only, [1] to get the first output\n",
|
||
"println()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 19,
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" 1631.87,909.798 1619.14,905.867 1606.44,901.935 1593.79,898.003 1581.18,894.072 1568.6,890.14 1556.07,886.209 1543.59,882.277 1531.15,878.345 1518.75,874.414 \n",
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" 1506.41,870.482 1494.11,866.55 1481.87,862.619 1469.67,858.687 1457.53,854.755 1445.44,850.824 1433.42,846.892 1421.44,842.961 1409.53,839.029 1397.69,835.097 \n",
|
||
" 1385.9,831.166 1374.18,827.234 1362.53,823.302 1350.95,819.371 1339.44,815.439 1328,811.508 1316.64,807.576 1305.36,803.644 1294.16,799.713 1283.04,795.781 \n",
|
||
" 1272.01,791.849 1261.07,787.918 1250.22,783.986 1239.47,780.054 1228.81,776.123 1218.25,772.191 1207.8,768.26 1197.45,764.328 1187.21,760.396 1177.09,756.465 \n",
|
||
" 1167.08,752.533 1157.2,748.601 1147.44,744.67 1137.81,740.738 1128.31,736.807 1118.94,732.875 1109.72,728.943 1100.64,725.012 1091.72,721.08 1082.94,717.148 \n",
|
||
" 1074.33,713.217 1065.88,709.285 1057.6,705.353 1049.49,701.422 1041.56,697.49 1033.81,693.559 1026.25,689.627 1018.88,685.695 1011.72,681.764 1004.75,677.832 \n",
|
||
" 998.001,673.9 991.462,669.969 985.144,666.037 979.051,662.106 973.189,658.174 967.563,654.242 962.179,650.311 957.04,646.379 952.154,642.447 947.523,638.516 \n",
|
||
" 943.154,634.584 939.05,630.653 935.217,626.721 931.658,622.789 928.378,618.858 925.38,614.926 922.667,610.994 920.244,607.063 918.112,603.131 916.275,599.199 \n",
|
||
" 914.734,595.268 913.492,591.336 912.55,587.405 911.91,583.473 911.572,579.541 911.536,575.61 911.803,571.678 912.372,567.746 913.243,563.815 914.414,559.883 \n",
|
||
" 915.885,555.952 917.652,552.02 919.715,548.088 922.07,544.157 924.714,540.225 927.645,536.293 930.859,532.362 934.353,528.43 938.122,524.498 942.162,520.567 \n",
|
||
" 946.469,516.635 951.038,512.704 955.865,508.772 960.944,504.84 966.271,500.909 971.841,496.977 977.648,493.045 983.687,489.114 989.952,485.182 996.439,481.251 \n",
|
||
" 1003.14,477.319 1010.06,473.387 1017.18,469.456 1024.5,465.524 1032.01,461.592 1039.71,457.661 1047.6,453.729 1055.67,449.797 1063.91,445.866 1072.32,441.934 \n",
|
||
" 1080.9,438.003 1089.63,434.071 1098.53,430.139 1107.57,426.208 1116.76,422.276 1126.09,418.344 1135.55,414.413 1145.15,410.481 1154.88,406.55 1164.74,402.618 \n",
|
||
" 1174.72,398.686 1184.81,394.755 1195.02,390.823 1205.35,386.891 1215.77,382.96 1226.31,379.028 1236.94,375.097 1247.68,371.165 1258.5,367.233 1269.42,363.302 \n",
|
||
" 1280.43,359.37 1291.53,355.438 1302.71,351.507 1313.97,347.575 1325.32,343.643 1336.73,339.712 1348.23,335.78 1359.79,331.849 1371.43,327.917 1383.13,323.985 \n",
|
||
" 1394.9,320.054 1406.73,316.122 1418.63,312.19 1430.59,308.259 1442.6,304.327 1454.67,300.396 1466.8,296.464 1478.98,292.532 1491.22,288.601 1503.5,284.669 \n",
|
||
" 1515.84,280.737 1528.22,276.806 1540.65,272.874 1553.13,268.942 1565.65,265.011 1578.21,261.079 1590.81,257.148 1603.46,253.216 1616.14,249.284 1628.86,245.353 \n",
|
||
" 1641.63,241.421 1654.42,237.489 1667.26,233.558 1680.12,229.626 1693.02,225.695 1705.96,221.763 1718.93,217.831 1731.92,213.9 1744.95,209.968 1758.01,206.036 \n",
|
||
" 1771.1,202.105 1784.21,198.173 1797.36,194.241 1810.53,190.31 1823.72,186.378 1836.94,182.447 1850.19,178.515 1863.46,174.583 1876.76,170.652 1890.08,166.72 \n",
|
||
" 1903.42,162.788 1916.78,158.857 1930.16,154.925 1943.57,150.994 1957,147.062 1970.44,143.13 1983.91,139.199 1997.4,135.267 2010.9,131.335 2024.43,127.404 \n",
|
||
" 2037.97,123.472 \n",
|
||
" \"/>\n",
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"source": [
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"p1 = plot( StdRp*100,μstar*100,\n",
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" linecolor = :red,\n",
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" ylim = (0,15),\n",
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" label = \"MVF\",\n",
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" legend = :topleft,\n",
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" title = \"MVF, only risky assets\",\n",
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" xlabel = \"Std(Rp), %\",\n",
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" ylabel = \"ERp, %\" )\n",
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"scatter!(sqrt.(diag(Σ))*100,μ*100,color=:red,label=\"assets\")\n",
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"outputs": [
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{
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"text": [
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"\n"
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}
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"source": [
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"StdRpRf = MVCalcRf(μstar,μ,Σ,Rf)[1] #with riskfree too\n",
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"println()"
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},
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" 1631.87,909.798 1619.14,905.867 1606.44,901.935 1593.79,898.003 1581.18,894.072 1568.6,890.14 1556.07,886.209 1543.59,882.277 1531.15,878.345 1518.75,874.414 \n",
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" 1506.41,870.482 1494.11,866.55 1481.87,862.619 1469.67,858.687 1457.53,854.755 1445.44,850.824 1433.42,846.892 1421.44,842.961 1409.53,839.029 1397.69,835.097 \n",
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" 1385.9,831.166 1374.18,827.234 1362.53,823.302 1350.95,819.371 1339.44,815.439 1328,811.508 1316.64,807.576 1305.36,803.644 1294.16,799.713 1283.04,795.781 \n",
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||
" 1272.01,791.849 1261.07,787.918 1250.22,783.986 1239.47,780.054 1228.81,776.123 1218.25,772.191 1207.8,768.26 1197.45,764.328 1187.21,760.396 1177.09,756.465 \n",
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" 1167.08,752.533 1157.2,748.601 1147.44,744.67 1137.81,740.738 1128.31,736.807 1118.94,732.875 1109.72,728.943 1100.64,725.012 1091.72,721.08 1082.94,717.148 \n",
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" 1074.33,713.217 1065.88,709.285 1057.6,705.353 1049.49,701.422 1041.56,697.49 1033.81,693.559 1026.25,689.627 1018.88,685.695 1011.72,681.764 1004.75,677.832 \n",
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||
" 998.001,673.9 991.462,669.969 985.144,666.037 979.051,662.106 973.189,658.174 967.563,654.242 962.179,650.311 957.04,646.379 952.154,642.447 947.523,638.516 \n",
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||
" 943.154,634.584 939.05,630.653 935.217,626.721 931.658,622.789 928.378,618.858 925.38,614.926 922.667,610.994 920.244,607.063 918.112,603.131 916.275,599.199 \n",
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||
" 914.734,595.268 913.492,591.336 912.55,587.405 911.91,583.473 911.572,579.541 911.536,575.61 911.803,571.678 912.372,567.746 913.243,563.815 914.414,559.883 \n",
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||
" 915.885,555.952 917.652,552.02 919.715,548.088 922.07,544.157 924.714,540.225 927.645,536.293 930.859,532.362 934.353,528.43 938.122,524.498 942.162,520.567 \n",
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||
" 946.469,516.635 951.038,512.704 955.865,508.772 960.944,504.84 966.271,500.909 971.841,496.977 977.648,493.045 983.687,489.114 989.952,485.182 996.439,481.251 \n",
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||
" 1003.14,477.319 1010.06,473.387 1017.18,469.456 1024.5,465.524 1032.01,461.592 1039.71,457.661 1047.6,453.729 1055.67,449.797 1063.91,445.866 1072.32,441.934 \n",
|
||
" 1080.9,438.003 1089.63,434.071 1098.53,430.139 1107.57,426.208 1116.76,422.276 1126.09,418.344 1135.55,414.413 1145.15,410.481 1154.88,406.55 1164.74,402.618 \n",
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" 1174.72,398.686 1184.81,394.755 1195.02,390.823 1205.35,386.891 1215.77,382.96 1226.31,379.028 1236.94,375.097 1247.68,371.165 1258.5,367.233 1269.42,363.302 \n",
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}
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],
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"source": [
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"p1 = plot( [StdRp StdRpRf]*100,[μstar μstar]*100,\n",
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" linestyle = [:solid :dash],\n",
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},
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{
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"metadata": {},
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"source": [
|
||
"# Mean-Variance Frontier without Short Selling (extra)\n",
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||
"\n",
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||
"The code below solves (numerically) the following minimization problem \n",
|
||
"\n",
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"$\\min \\text{Var}(R_p) \\: \\text{ s.t. } \\: \\text{E}R_p = \\mu^*$,\n",
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" \n",
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"and where we also require $w_i\\ge 0$ and $\\sum_{i=1}^{n}w_{i}=1$.\n",
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"\n",
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||
"To solve this, we use the packages [Convex.jl](https://github.com/jump-dev/Convex.jl) (for the interface) and [SCS.jl](https://github.com/jump-dev/SCS.jl) (for the optimization algorithm). To check for convergence, we also need a function from the [MathOptInterface.jl](https://github.com/jump-dev/MathOptInterface.jl) package."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 22,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
|
||
"MathOptInterface"
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||
]
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||
},
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"execution_count": 22,
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"metadata": {},
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"output_type": "execute_result"
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||
}
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],
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"source": [
|
||
"using Convex, SCS\n",
|
||
"import MathOptInterface\n",
|
||
"const MOI = MathOptInterface"
|
||
]
|
||
},
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{
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"cell_type": "code",
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"execution_count": 23,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
|
||
"MeanVarNoSSPs"
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]
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},
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"execution_count": 23,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"\"\"\"\n",
|
||
" MeanVarNoSSPs(μ,Σ,μstar)\n",
|
||
"\n",
|
||
"Calculate mean variance frontier when short sales are not allowed, using Convex and SCS\n",
|
||
"\n",
|
||
"\n",
|
||
"# Input\n",
|
||
"- `μ::Vector`: n vector, mean returns\n",
|
||
"- `Σ::Matrix`: nxn, covariance matrix of returns, can contain riskfree assets\n",
|
||
"- `μstar::Vector`: K vector, mean returns to calculate results for\n",
|
||
"\n",
|
||
"# Output\n",
|
||
"- `StdRp::Vector`: K vector, standard deviation of mean-variance portfolio (risky only) with mean μstar\n",
|
||
"- `w_p::Matrix`: Kxn, portfolio weights of \"\"\n",
|
||
"\n",
|
||
"# Requires\n",
|
||
"- Convex and SCS\n",
|
||
"\n",
|
||
"\"\"\"\n",
|
||
"function MeanVarNoSSPs(μ,Σ,μstar) #MV with no short-sales, numerical minimization\n",
|
||
"\n",
|
||
" (K,n) = (length(μstar),length(μ))\n",
|
||
"\n",
|
||
" n = length(μ)\n",
|
||
" vv = findall( minimum(μ) .<= μstar .<= maximum(μ) ) #solve only if feasible\n",
|
||
"\n",
|
||
" w = Variable(n)\n",
|
||
" Varp = quadform(w,Σ)\n",
|
||
" c1 = sum(w) == 1\n",
|
||
" c3 = 0 <= w #replace 0 and 1 to get other restrictions\n",
|
||
" c4 = w <= 1\n",
|
||
"\n",
|
||
" (w_p,StdRp) = (fill(NaN,K,n),fill(NaN,K))\n",
|
||
" for i in vv #loop over (feasible) μstar elements\n",
|
||
" c2 = dot(w,μ) == μstar[i]\n",
|
||
" problem = minimize(Varp,c1,c2,c3,c4)\n",
|
||
" Convex.solve!(problem,()->SCS.Optimizer(verbose=false))\n",
|
||
" if problem.status == MOI.OPTIMAL #check if solution has been found\n",
|
||
" w_p[i,:] = evaluate(w)\n",
|
||
" StdRp[i] = sqrt(evaluate(Varp))\n",
|
||
" end\n",
|
||
" end\n",
|
||
"\n",
|
||
" return StdRp, w_p\n",
|
||
"\n",
|
||
"end"
|
||
]
|
||
},
|
||
{
|
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"cell_type": "code",
|
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"execution_count": 24,
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"metadata": {},
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"outputs": [
|
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{
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||
"name": "stdout",
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||
"output_type": "stream",
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"text": [
|
||
"\n"
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]
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}
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],
|
||
"source": [
|
||
"Std_no_ss = MeanVarNoSSPs(μ,Σ,μstar)[1]\n",
|
||
"println()"
|
||
]
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||
},
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||
{
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"cell_type": "code",
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" 1631.87,909.798 1619.14,905.867 1606.44,901.935 1593.79,898.003 1581.18,894.072 1568.6,890.14 1556.07,886.209 1543.59,882.277 1531.15,878.345 1518.75,874.414 \n",
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" 1506.41,870.482 1494.11,866.55 1481.87,862.619 1469.67,858.687 1457.53,854.755 1445.44,850.824 1433.42,846.892 1421.44,842.961 1409.53,839.029 1397.69,835.097 \n",
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" 1385.9,831.166 1374.18,827.234 1362.53,823.302 1350.95,819.371 1339.44,815.439 1328,811.508 1316.64,807.576 1305.36,803.644 1294.16,799.713 1283.04,795.781 \n",
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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||
"p1 = plot( [StdRp Std_no_ss]*100,[μstar μstar]*100,\n",
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||
" linecolor = [:red :green],\n",
|
||
" linestyle = [:solid :dash],\n",
|
||
" linewidth = 2, \n",
|
||
" label = [\"no constraints\" \"no short sales\"],\n",
|
||
" xlim = (0,15),\n",
|
||
" ylim = (0,15),\n",
|
||
" legend = :topleft,\n",
|
||
" title = \"MVF (with/without constraints)\",\n",
|
||
" xlabel = \"Std(Rp), %\",\n",
|
||
" ylabel = \"ERp, %\" )\n",
|
||
"scatter!(sqrt.(diag(Σ))*100,μ*100,color=:red,label=\"assets\")\n",
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"display(p1)"
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]
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},
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{
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"source": []
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}
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],
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"language": "julia",
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"name": "julia-1.6"
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