646 lines
182 KiB
Plaintext
646 lines
182 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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"source": [
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"## Load Packages and Extra Functions"
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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, Distributions, OffsetArrays, FiniteDiff\n",
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"\n",
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"#using FiniteDiff: finite_difference_derivative as derivative #could do this instead\n",
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"\n",
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"include(\"jlFiles/OptionsCalculations.jl\")\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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"#pyplot(size=(600,400))\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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"# The Black-Scholes Model\n",
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"\n",
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"The next cell calculates call `C` and put prices `P` from the Black-Scholes formula.\n",
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"\n",
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"The key parameters are:\n",
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"`(S,K,m,y,σ) = (current undelying price, strike price,time to expiration,interest rate, volatility)`"
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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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"call price at K=42: 2.893 \n",
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"\n",
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"put price at K=42: 1.856 \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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"(S,K,m,y,σ) = (42,42,0.5,0.05,0.2)\n",
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"\n",
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"C = OptionBlackSPs(S,K,m,y,σ)\n",
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"printlnPs(\"call price at K=$K: \",C,\"\\n\")\n",
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"\n",
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"P = OptionBlackSPs(S,K,m,y,σ,0,true)\n",
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"printlnPs(\"put price at K=$K: \",P,\"\\n\")"
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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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"# Task 1\n",
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"\n",
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"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)."
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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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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
|
||
],
|
||
"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"
|
||
]
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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": [
|
||
{
|
||
"name": "stdout",
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||
"output_type": "stream",
|
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"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\")"
|
||
]
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||
},
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{
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"cell_type": "markdown",
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"metadata": {},
|
||
"source": [
|
||
"# Task 2\n",
|
||
"\n",
|
||
"Redo the calculation of p_e and p_a for `S=30:60` and plot the prices."
|
||
]
|
||
},
|
||
{
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"cell_type": "code",
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"execution_count": 7,
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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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"p_e = BOPM_European.(S_range,K,m,y,σ,true)\n",
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"p_a = BOPM_American.(S_range,K,m,y,σ,true)\n",
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"\n",
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"p1 = plot( S_range,[p_e p_a],\n",
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" linecolor = [:red :blue],\n",
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||
" label = [\"European\" \"American\"],\n",
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" title = \"BOPM put option prices, K=$K\",\n",
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||
" xlabel = \"current asset price\",\n",
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||
" ylabel = \"option price\" )\n",
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"display(p1)"
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]
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},
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{
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"metadata": {},
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"source": [
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"# Delta\n",
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"\n",
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"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",
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||
"\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",
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||
"\n",
|
||
"To calculate the derivatives we use \n",
|
||
"`FiniteDiff.finite_difference_derivative(the function,an S value)`"
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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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"metadata": {},
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"outputs": [],
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"source": [
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"L = length(S_range)\n",
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||
"(Δp_e,Δp_a) = [fill(NaN,L) for _=1:2]\n",
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||
"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",
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||
"end\n",
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||
"\n",
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||
" #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} )"
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]
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},
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{
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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( 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)"
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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": null,
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"@webio": {
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},
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"anaconda-cloud": {},
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"kernelspec": {
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"display_name": "Julia 1.7.0",
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"language": "julia",
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"name": "julia-1.7"
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},
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"language_info": {
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"file_extension": ".jl",
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"mimetype": "application/julia",
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"name": "julia",
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"version": "1.7.0"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 4
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}
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