Num_Program_US_Call_Julia.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<style>.container { width:100% !important; }</style>"
      ],
      "text/plain": [
       "HTML{String}(\"<style>.container { width:100% !important; }</style>\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(HTML(\"<style>.container { width:100% !important; }</style>\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### MIT license (c) 2019 by Andrew Lyasoff"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Jupyter notebook written in Julia 1.1.0. It implements on a CPUs the numerical program for computing the exercise boundary and the price of an American-style call option described in Sec. 18.4 of \"Stochastic Methods in Asset Pricing\" (pp. 514-516). The same program is implemented in Python in Appendix B.3 of SMAP."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note the formula $\\int_a^bf(t)dt={1\\over2}(b-a)\\int_{-1}^{+1}f\\bigl({1\\over2}(b-a)s+{1\\over2}(a+b)\\bigr)d s$ to use with FastGaussQuadratures."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5000"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "## the number of grid points over the time domain is set below\n",
    "## 5000 is excessive and is meant to test Julia's capabilities\n",
    "total_grid_no=5000 #this number must be divisible by pnmb"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Introduce the parameters in the model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "KK=40.0; # strike\n",
    "sigma=0.3; # volatility\n",
    "delta=0.07; # dividend\n",
    "rr=0.02; # interest\n",
    "TT=0.5; # time to maturity\n",
    "DLT=TT/(total_grid_no) # distance between two grid points in the time domain\n",
    "ABSC=range(0.0,length=total_grid_no+1,stop=TT) # the entire grid on the time domain\n",
    "VAL=[max(KK,KK*(rr/delta)) for x in ABSC]; # first guess for the exercise boundary"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "using SpecialFunctions\n",
    "using Interpolations\n",
    "using Roots\n",
    "using FastGaussQuadrature\n",
    "nodes,weights=gausslegendre( 200 );\n",
    "\n",
    "function EUcall(S::Float64,t::Float64,K::Float64,σ::Float64,r::Float64,δ::Float64)\n",
    "        local v1,v2\n",
    "        v1=(δ-r+σ^2/2.0)*t+log(K/S)\n",
    "        v2=(-δ + r + σ^2/2.0)*t - log(K/S)\n",
    "        v3=σ*sqrt(2.0*t)\n",
    "        return -K*(exp(-t*r)/2)+K*(exp(-t*r)/2)*erf(v1/v3)+S*(exp(-t*δ)/2)*erf(v2/v3)+S*(exp(-t*δ)/2)\n",
    "    end;\n",
    "\n",
    "function F(ϵ::Int64,t::Float64,u::Float64,v::Float64,r::Float64,δ::Float64,σ::Float64)\n",
    "        v1=(r-δ+ϵ*σ^2/2)*(u-t)-log(v)\n",
    "        v2=σ*sqrt(2*(u-t))\n",
    "        return 1.0+erf(v1/v2)\n",
    "    end;\n",
    "\n",
    "function ah(t::Float64,z::Float64,r::Float64,δ::Float64,σ::Float64,f)\n",
    "        return (exp(-δ*(z-t))*(δ/2)*F(1,t,z,f(z)/f(t),r,δ,σ))\n",
    "    end;\n",
    "\n",
    "function bh(t::Float64,z::Float64,r::Float64,δ::Float64,σ::Float64,K::Float64,f)\n",
    "        return (exp(-r*(z-t))*(r*K/2)*F(-1,t,z,f(z)/f(t),r,δ,σ))\n",
    "    end;\n",
    "\n",
    "function make_grid0(step::Float64,size::Int64)\n",
    "        return 0.0:step:(step+(size-1)*step)\n",
    "    end;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "mainF (generic function with 1 method)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function mainF(start_iter::Int64,nmb_of_iter::Int64,conv_tol::Float64,K::Float64,σ::Float64,δ::Float64,r::Float64,T::Float64,Δ::Float64,nmb_grd_pnts::Int64,vls::Array{Float64,1},nds::Array{Float64,1},wghts::Array{Float64,1})\n",
    "    local no_iter,conv_check,absc,absc0,valPrev,val,loc,f\n",
    "    absc=range(0.0,length=nmb_grd_pnts+1,stop=T)\n",
    "    absc0=range(0.0,length=nmb_grd_pnts,stop=(T-Δ));\n",
    "    val=vls;\n",
    "    f=CubicSplineInterpolation(absc,val,extrapolation_bc = Interpolations.Line())\n",
    "    no_iter=start_iter;\n",
    "    conv_check=100.0\n",
    "    while no_iter<nmb_of_iter&&conv_check>conv_tol\n",
    "        no_iter+=1\n",
    "        loc=[max(K,K*(r/δ))]\n",
    "        for ttt=Iterators.reverse(absc0)\n",
    "            an=[(1/2)*(T-ttt)*ah(ttt,(1/2)*(T-ttt)*s+(1/2)*(ttt+T),r,δ,σ,f) for s in nodes]\n",
    "            bn=[(1/2)*(T-ttt)*bh(ttt,(1/2)*(T-ttt)*s+(1/2)*(ttt+T),r,δ,σ,K,f) for s in nodes]\n",
    "            aaa=weights'*an\n",
    "            bbb=weights'*bn;\n",
    "            LRT=find_zero(x->x-K-EUcall(x,T-ttt,K,σ,r,δ)-aaa*x+bbb,(K-10,K+20));\n",
    "            pushfirst!(loc,LRT)\n",
    "        end\n",
    "        valPrev=val\n",
    "        val=loc\n",
    "        f=CubicSplineInterpolation(absc,val,extrapolation_bc = Interpolations.Line())\n",
    "        conv_check=maximum(abs.(valPrev-val))\n",
    "    end\n",
    "    return absc,val,conv_check,no_iter\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Run the main routine. The third argument is an upper bound on the total iterations to run. \n",
    "The last two arguments control FastGaussQuadrature.\n",
    "\n",
    "masterF(prnmb,start_iter,nmb_of_iter,conv_tol,K,σ,δ,r,T,Δ,no_grd_pnts,vls,nds,wghts)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "mainF(start_iter, nmb_of_iter, conv_tol, $K$, $\\sigma$, $\\delta$, $r$, $T$, $\\Delta$, no_grd_pnts, vls, nds, wghts)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " 13.992447 seconds (406.23 M allocations: 6.942 GiB, 3.71% gc time)\n"
     ]
    }
   ],
   "source": [
    "@time ABSC,VAL,conv,iterations=mainF(0,100,1.0e-5,KK,sigma,delta,rr,TT,DLT,total_grid_no,VAL,nodes,weights);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The call to 'masterF' can be repeated with the most recent VAL and the second argument set to the number of already performed iterations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(9.63110706209136e-6, 23)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "conv,iterations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "f=CubicSplineInterpolation(ABSC,VAL,extrapolation_bc = Interpolations.Line());"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Plots.PyPlotBackend()"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "pyplot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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"
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plotgrid=ABSC[1]:.001:ABSC[end];\n",
    "pval=[f(x) for x in plotgrid];\n",
    "plot(plotgrid,pval,label=\"exercise boundary\")\n",
    "xlabel!(\"real time\")\n",
    "ylabel!(\"underlying spot price\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "53.58270125587188"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The price of the EU call option at the money ($S_0=K$) with $T={1\\over2}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.8378265020118754"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "EUcall(KK,TT,KK,sigma,rr,delta)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The early exercise premium for at the money option at $t=0$ is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.10081943488743587"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "an=[(0.5*(TT-0.0))*exp(-delta*((1/2)*(TT-0.0)*s+(1/2)*(0.0+TT)-0.0))*(KK*delta/2)*F(1,0.0,(1/2)*(TT-0.0)*s+(1/2)*(0.0+TT),f((1/2)*(TT-0.0)*s+(1/2)*(0.0+TT))/KK,rr,delta,sigma) for s in nodes]\n",
    "bn=[(0.5*(TT-0.0))*exp(-rr*((1/2)*(TT-0.0)*s+(1/2)*(0.0+TT)-0.0))*(rr*KK/2)*F(-1,0.0,(1/2)*(TT-0.0)*s+(1/2)*(0.0+TT),f((1/2)*(TT-0.0)*s+(1/2)*(0.0+TT))/KK,rr,delta,sigma)  for s in nodes]\n",
    "aaa=weights'*an\n",
    "bbb=weights'*bn;\n",
    "EEP=aaa-bbb\n",
    "EEP"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The price of an American at the money call with 6 months to expiry is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.938645936899311"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "EUcall(KK,TT,KK,sigma,rr,delta)+EEP"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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   "display_name": "Julia 1.1.0",
   "language": "julia",
   "name": "julia-1.1"
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  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
   "version": "1.1.0"
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