646 lines
182 KiB
Plaintext
646 lines
182 KiB
Plaintext
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{
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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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" 1298.51,627.572 1351.19,581.817 1403.87,535.334 1456.56,488.286 1509.24,440.802 1561.92,392.989 1614.61,344.929 1667.29,296.686 1719.97,248.309 1772.66,199.835 \n",
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" 1825.34,151.29 \n",
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" 244.842,543.758 297.526,591.386 350.209,638.28 402.892,684.113 455.576,728.505 508.259,771.038 560.942,811.292 613.625,848.875 666.309,883.456 718.992,914.794 \n",
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" 771.675,942.749 824.359,967.295 877.042,988.507 929.725,1006.55 982.408,1021.67 1035.09,1034.14 1087.77,1044.28 1140.46,1052.42 1193.14,1058.85 1245.82,1063.88 \n",
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" 1298.51,1067.75 1351.19,1070.71 1403.87,1072.93 1456.56,1074.59 1509.24,1075.82 1561.92,1076.72 1614.61,1077.36 1667.29,1077.83 1719.97,1078.16 1772.66,1078.4 \n",
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" 1825.34,1078.56 \n",
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" \"/>\n",
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" 1556.37,311.756 1816.91,311.756 1816.91,156.236 1556.37,156.236 1556.37,311.756 \n",
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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"
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"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": [
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|
|
"<?xml version=\"1.0\" encoding=\"utf-8\"?>\n",
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"<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"480\" height=\"320\" viewBox=\"0 0 1920 1280\">\n",
|
|||
|
|
"<defs>\n",
|
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|
|
" <clipPath id=\"clip980\">\n",
|
|||
|
|
" <rect x=\"0\" y=\"0\" width=\"1920\" height=\"1280\"/>\n",
|
|||
|
|
" </clipPath>\n",
|
|||
|
|
"</defs>\n",
|
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|
|
"<path clip-path=\"url(#clip980)\" d=\"\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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|||
|
|
"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)"
|
|||
|
|
]
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|||
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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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|||
|
|
"# 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)`"
|
|||
|
|
]
|
|||
|
|
},
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|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 8,
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|||
|
|
"metadata": {},
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|||
|
|
"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} )"
|
|||
|
|
]
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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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]
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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( 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",
|
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|
|
"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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"lastCommId": null,
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"lastKernelId": null
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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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