Black_Scholes_Merton_Formula_Derived_by_SymPy_Python.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### MIT License (c) 2019 by Andrew Lyasoff"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Jupyter notebook written in Python 3. It illustrates the derivation of the Black-Scholes-Merton formula for the price of an European style call option by using SymPy. In particular, it illustrates the inversion of a function by using 'solve.' The formula is derived with non-zero dividends."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "t=Symbol('t',positive=True)\n",
    "r=Symbol('r',positive=True)\n",
    "S=Symbol('S',positive=True)\n",
    "K=Symbol('K',positive=True)\n",
    "a=Symbol('a',positive=True)\n",
    "x=Symbol('x',real=True)\n",
    "sigma=Symbol('sigma',positive=True)\n",
    "delta=Symbol('delta',positive=True)\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left \\{ x : \\frac{t}{2} \\left(2 \\delta - 2 r + \\sigma^{2}\\right) + \\log{\\left (K \\right )} - \\log{\\left (S \\right )}\\right \\}$$"
      ],
      "text/plain": [
       "⎧     ⎛             2⎞                  ⎫\n",
       "⎪   t⋅⎝2⋅δ - 2⋅r + σ ⎠                  ⎪\n",
       "⎨x: ────────────────── + log(K) - log(S)⎬\n",
       "⎪           2                           ⎪\n",
       "⎩                                       ⎭"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ibound=solve(S*exp(x+(r-delta-Rational(1,2)*sigma**2)*t)-K,x,dict=True)[0]\n",
    "ibound"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{t}{2} \\left(2 \\delta - 2 r + \\sigma^{2}\\right) + \\log{\\left (K \\right )} - \\log{\\left (S \\right )}$$"
      ],
      "text/plain": [
       "  ⎛             2⎞                  \n",
       "t⋅⎝2⋅δ - 2⋅r + σ ⎠                  \n",
       "────────────────── + log(K) - log(S)\n",
       "        2                           "
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ibound[x]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAHlBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACGjDitAAAACXRSTlMAVO8Qq5l2zWYZcMvdAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAHUlEQVQIHWNgAANGZQYGk5DJQDYbqQSr03QPsBkAJYgIYEZbtZEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(exp(-x**2/(2*t*sigma**2))/(sigma*sqrt(2*pi*t)),(x,-oo,+oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$$\\left(- \\frac{K}{2} \\operatorname{erfc}{\\left (\\frac{\\sqrt{2} a}{2 \\sigma \\sqrt{t}} \\right )} - \\frac{\\sqrt{2} S \\sigma \\sqrt{t}}{2 \\pi \\left(- a + \\sigma^{2} t\\right)} \\left(- \\frac{\\sqrt{2} \\pi}{2 \\sigma \\sqrt{t}} \\left(- a + \\sigma^{2} t\\right) e^{\\frac{\\left(- a + \\sigma^{2} t\\right)^{2}}{2 \\sigma^{2} t}} \\operatorname{erf}{\\left (\\frac{\\sqrt{2} \\left(- a + \\sigma^{2} t\\right)}{2 \\sigma \\sqrt{t}} \\right )} - \\frac{\\sqrt{2} \\pi}{2 \\sigma \\sqrt{t}} \\left(- a + \\sigma^{2} t\\right) e^{\\frac{\\left(- a + \\sigma^{2} t\\right)^{2}}{2 \\sigma^{2} t}}\\right) e^{- \\frac{a^{2}}{2 \\sigma^{2} t} + a - \\delta t + r t - 0.5 \\sigma^{2} t}\\right) e^{- r t}$$"
      ],
      "text/plain": [
       "⎛                             ⎛                               2               \n",
       "⎜                             ⎜                    ⎛      2  ⎞                \n",
       "⎜                             ⎜                    ⎝-a + σ ⋅t⎠                \n",
       "⎜                             ⎜                    ────────────               \n",
       "⎜                             ⎜                          2         ⎛   ⎛      \n",
       "⎜                             ⎜       ⎛      2  ⎞     2⋅σ ⋅t       ⎜√2⋅⎝-a + σ\n",
       "⎜                             ⎜  √2⋅π⋅⎝-a + σ ⋅t⎠⋅ℯ            ⋅erf⎜──────────\n",
       "⎜        ⎛ √2⋅a ⎞             ⎜                                    ⎝    2⋅σ⋅√t\n",
       "⎜  K⋅erfc⎜──────⎟   √2⋅S⋅σ⋅√t⋅⎜- ─────────────────────────────────────────────\n",
       "⎜        ⎝2⋅σ⋅√t⎠             ⎝                        2⋅σ⋅√t                 \n",
       "⎜- ────────────── - ──────────────────────────────────────────────────────────\n",
       "⎜        2                                                                    \n",
       "⎝                                                                             \n",
       "\n",
       "                                      ⎞                                     ⎞ \n",
       "                                     2⎟                                     ⎟ \n",
       "                          ⎛      2  ⎞ ⎟                                     ⎟ \n",
       "                          ⎝-a + σ ⋅t⎠ ⎟       2                             ⎟ \n",
       "2  ⎞⎞                     ────────────⎟      a                           2  ⎟ \n",
       " ⋅t⎠⎟                           2     ⎟  - ────── + a - δ⋅t + r⋅t - 0.5⋅σ ⋅t⎟ \n",
       "────⎟        ⎛      2  ⎞     2⋅σ ⋅t   ⎟       2                             ⎟ \n",
       "    ⎠   √2⋅π⋅⎝-a + σ ⋅t⎠⋅ℯ            ⎟    2⋅σ ⋅t                           ⎟ \n",
       "───── - ──────────────────────────────⎟⋅ℯ                                   ⎟ \n",
       "                    2⋅σ⋅√t            ⎠                                     ⎟ \n",
       "────────────────────────────────────────────────────────────────────────────⎟⋅\n",
       "     ⎛      2  ⎞                                                            ⎟ \n",
       " 2⋅π⋅⎝-a + σ ⋅t⎠                                                            ⎠ \n",
       "\n",
       "     \n",
       "     \n",
       "     \n",
       "     \n",
       "     \n",
       "     \n",
       "     \n",
       "     \n",
       "     \n",
       " -r⋅t\n",
       "ℯ    \n",
       "     \n",
       "     "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A=exp(-r*t)*integrate(exp(-x**2/(2*t*sigma**2))*(S*exp(x+(r-delta-(1/2)*sigma**2)*t)-K)/(sigma*sqrt(2*pi*t)),(x,a,+oo))\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{1}{2} \\left(- K e^{\\delta t} \\operatorname{erfc}{\\left (\\frac{\\sqrt{2}}{4 \\sigma \\sqrt{t}} \\left(2 \\delta t - 2 r t + \\sigma^{2} t + \\log{\\left (\\frac{K^{2}}{S^{2}} \\right )}\\right) \\right )} + S e^{r t} \\operatorname{erf}{\\left (\\frac{\\sqrt{2}}{4 \\sigma \\sqrt{t}} \\left(- 2 \\delta t + 2 r t + \\sigma^{2} t + \\log{\\left (\\frac{S^{2}}{K^{2}} \\right )}\\right) \\right )} + S e^{r t}\\right) e^{- t \\left(\\delta + r\\right)}$$"
      ],
      "text/plain": [
       "⎛             ⎛   ⎛                          ⎛ 2⎞⎞⎞             ⎛   ⎛         \n",
       "⎜             ⎜   ⎜                 2        ⎜K ⎟⎟⎟             ⎜   ⎜         \n",
       "⎜             ⎜√2⋅⎜2⋅δ⋅t - 2⋅r⋅t + σ ⋅t + log⎜──⎟⎟⎟             ⎜√2⋅⎜-2⋅δ⋅t + \n",
       "⎜             ⎜   ⎜                          ⎜ 2⎟⎟⎟             ⎜   ⎜         \n",
       "⎜     δ⋅t     ⎜   ⎝                          ⎝S ⎠⎠⎟      r⋅t    ⎜   ⎝         \n",
       "⎜- K⋅ℯ   ⋅erfc⎜───────────────────────────────────⎟ + S⋅ℯ   ⋅erf⎜─────────────\n",
       "⎝             ⎝               4⋅σ⋅√t              ⎠             ⎝             \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                             2                \n",
       "\n",
       "                  ⎛ 2⎞⎞⎞         ⎞            \n",
       "         2        ⎜S ⎟⎟⎟         ⎟            \n",
       "2⋅r⋅t + σ ⋅t + log⎜──⎟⎟⎟         ⎟            \n",
       "                  ⎜ 2⎟⎟⎟         ⎟            \n",
       "                  ⎝K ⎠⎠⎟      r⋅t⎟  -t⋅(δ + r)\n",
       "───────────────────────⎟ + S⋅ℯ   ⎟⋅ℯ          \n",
       "  4⋅σ⋅√t               ⎠         ⎠            \n",
       "──────────────────────────────────────────────\n",
       "                                              "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.subs(a,ibound[x]).simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$$\\frac{1}{2} \\left(- K e^{\\delta t} \\operatorname{erfc}{\\left (\\frac{\\sqrt{2}}{4 \\sigma \\sqrt{t}} \\left(2 \\delta t - 2 r t + \\sigma^{2} t + \\log{\\left (\\frac{K^{2}}{S^{2}} \\right )}\\right) \\right )} + S e^{r t} \\operatorname{erf}{\\left (\\frac{\\sqrt{2}}{4 \\sigma \\sqrt{t}} \\left(- 2 \\delta t + 2 r t + \\sigma^{2} t + \\log{\\left (\\frac{S^{2}}{K^{2}} \\right )}\\right) \\right )} + S e^{r t}\\right) e^{- t \\left(\\delta + r\\right)}$$"
      ],
      "text/plain": [
       "⎛             ⎛   ⎛                          ⎛ 2⎞⎞⎞             ⎛   ⎛         \n",
       "⎜             ⎜   ⎜                 2        ⎜K ⎟⎟⎟             ⎜   ⎜         \n",
       "⎜             ⎜√2⋅⎜2⋅δ⋅t - 2⋅r⋅t + σ ⋅t + log⎜──⎟⎟⎟             ⎜√2⋅⎜-2⋅δ⋅t + \n",
       "⎜             ⎜   ⎜                          ⎜ 2⎟⎟⎟             ⎜   ⎜         \n",
       "⎜     δ⋅t     ⎜   ⎝                          ⎝S ⎠⎠⎟      r⋅t    ⎜   ⎝         \n",
       "⎜- K⋅ℯ   ⋅erfc⎜───────────────────────────────────⎟ + S⋅ℯ   ⋅erf⎜─────────────\n",
       "⎝             ⎝               4⋅σ⋅√t              ⎠             ⎝             \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                             2                \n",
       "\n",
       "                  ⎛ 2⎞⎞⎞         ⎞            \n",
       "         2        ⎜S ⎟⎟⎟         ⎟            \n",
       "2⋅r⋅t + σ ⋅t + log⎜──⎟⎟⎟         ⎟            \n",
       "                  ⎜ 2⎟⎟⎟         ⎟            \n",
       "                  ⎝K ⎠⎠⎟      r⋅t⎟  -t⋅(δ + r)\n",
       "───────────────────────⎟ + S⋅ℯ   ⎟⋅ℯ          \n",
       "  4⋅σ⋅√t               ⎠         ⎠            \n",
       "──────────────────────────────────────────────\n",
       "                                              "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(-K*exp(delta*t)*erfc(sqrt(2)*(2*delta*t - 2*r*t + sigma**2*t + log(K**2/S**2))/(4*sigma*sqrt(t))) + S*exp(r*t)*erf(sqrt(2)*(-2*delta*t + 2*r*t + sigma**2*t + log(S**2/K**2))/(4*sigma*sqrt(t))) + S*exp(r*t))*exp(-t*(delta + r))/2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Note: ${\\rm erfc}(x)=1-{\\rm erf}(x)=1+{\\rm erf}(-x)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$$\\frac{1}{2} \\left(K \\operatorname{erfc}{\\left (\\frac{\\sqrt{2}}{4 \\sigma \\sqrt{t}} \\left(2 r t - \\sigma^{2} t + \\log{\\left (\\frac{S^{2}}{K^{2}} \\right )}\\right) \\right )} - 2 K + S e^{r t} \\operatorname{erf}{\\left (\\frac{\\sqrt{2}}{4 \\sigma \\sqrt{t}} \\left(2 r t + \\sigma^{2} t + \\log{\\left (\\frac{S^{2}}{K^{2}} \\right )}\\right) \\right )} + S e^{r t}\\right) e^{- r t}$$"
      ],
      "text/plain": [
       "⎛      ⎛   ⎛                  ⎛ 2⎞⎞⎞                   ⎛   ⎛                  \n",
       "⎜      ⎜   ⎜         2        ⎜S ⎟⎟⎟                   ⎜   ⎜         2        \n",
       "⎜      ⎜√2⋅⎜2⋅r⋅t - σ ⋅t + log⎜──⎟⎟⎟                   ⎜√2⋅⎜2⋅r⋅t + σ ⋅t + log\n",
       "⎜      ⎜   ⎜                  ⎜ 2⎟⎟⎟                   ⎜   ⎜                  \n",
       "⎜      ⎜   ⎝                  ⎝K ⎠⎠⎟            r⋅t    ⎜   ⎝                  \n",
       "⎜K⋅erfc⎜───────────────────────────⎟ - 2⋅K + S⋅ℯ   ⋅erf⎜──────────────────────\n",
       "⎝      ⎝           4⋅σ⋅√t          ⎠                   ⎝           4⋅σ⋅√t     \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                 2                            \n",
       "\n",
       "⎛ 2⎞⎞⎞         ⎞      \n",
       "⎜S ⎟⎟⎟         ⎟      \n",
       "⎜──⎟⎟⎟         ⎟      \n",
       "⎜ 2⎟⎟⎟         ⎟      \n",
       "⎝K ⎠⎠⎟      r⋅t⎟  -r⋅t\n",
       "─────⎟ + S⋅ℯ   ⎟⋅ℯ    \n",
       "     ⎠         ⎠      \n",
       "──────────────────────\n",
       "                      "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.subs(a,ibound[x]).subs(delta,0).simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$$\\frac{1}{2} \\left(K \\operatorname{erfc}{\\left (\\frac{\\sqrt{2}}{4 \\sigma \\sqrt{t}} \\left(2 r t - \\sigma^{2} t + \\log{\\left (\\frac{S^{2}}{K^{2}} \\right )}\\right) \\right )} - 2 K + S e^{r t} \\operatorname{erf}{\\left (\\frac{\\sqrt{2}}{4 \\sigma \\sqrt{t}} \\left(2 r t + \\sigma^{2} t + \\log{\\left (\\frac{S^{2}}{K^{2}} \\right )}\\right) \\right )} + S e^{r t}\\right) e^{- r t}$$"
      ],
      "text/plain": [
       "⎛      ⎛   ⎛                  ⎛ 2⎞⎞⎞                   ⎛   ⎛                  \n",
       "⎜      ⎜   ⎜         2        ⎜S ⎟⎟⎟                   ⎜   ⎜         2        \n",
       "⎜      ⎜√2⋅⎜2⋅r⋅t - σ ⋅t + log⎜──⎟⎟⎟                   ⎜√2⋅⎜2⋅r⋅t + σ ⋅t + log\n",
       "⎜      ⎜   ⎜                  ⎜ 2⎟⎟⎟                   ⎜   ⎜                  \n",
       "⎜      ⎜   ⎝                  ⎝K ⎠⎠⎟            r⋅t    ⎜   ⎝                  \n",
       "⎜K⋅erfc⎜───────────────────────────⎟ - 2⋅K + S⋅ℯ   ⋅erf⎜──────────────────────\n",
       "⎝      ⎝           4⋅σ⋅√t          ⎠                   ⎝           4⋅σ⋅√t     \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                 2                            \n",
       "\n",
       "⎛ 2⎞⎞⎞         ⎞      \n",
       "⎜S ⎟⎟⎟         ⎟      \n",
       "⎜──⎟⎟⎟         ⎟      \n",
       "⎜ 2⎟⎟⎟         ⎟      \n",
       "⎝K ⎠⎠⎟      r⋅t⎟  -r⋅t\n",
       "─────⎟ + S⋅ℯ   ⎟⋅ℯ    \n",
       "     ⎠         ⎠      \n",
       "──────────────────────\n",
       "                      "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(K*erfc(sqrt(2)*(2*r*t - sigma**2*t + log(S**2/K**2))/(4*sigma*sqrt(t))) - 2*K + S*exp(r*t)*erf(sqrt(2)*(2*r*t + sigma**2*t + log(S**2/K**2))/(4*sigma*sqrt(t))) + S*exp(r*t))*exp(-r*t)/2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$$\\frac{K}{2} e^{- r t} \\operatorname{erfc}{\\left (\\frac{\\sqrt{2} r \\sqrt{t}}{2 \\sigma} - \\frac{\\sqrt{2} \\sigma}{4} \\sqrt{t} - \\frac{\\sqrt{2} \\log{\\left (K \\right )}}{2 \\sigma \\sqrt{t}} + \\frac{\\sqrt{2} \\log{\\left (S \\right )}}{2 \\sigma \\sqrt{t}} \\right )} - K e^{- r t} + \\frac{S}{2} \\operatorname{erf}{\\left (\\frac{\\sqrt{2} r \\sqrt{t}}{2 \\sigma} + \\frac{\\sqrt{2} \\sigma}{4} \\sqrt{t} - \\frac{\\sqrt{2} \\log{\\left (K \\right )}}{2 \\sigma \\sqrt{t}} + \\frac{\\sqrt{2} \\log{\\left (S \\right )}}{2 \\sigma \\sqrt{t}} \\right )} + \\frac{S}{2}$$"
      ],
      "text/plain": [
       "   -r⋅t     ⎛√2⋅r⋅√t   √2⋅σ⋅√t   √2⋅log(K)   √2⋅log(S)⎞                  ⎛√2⋅r\n",
       "K⋅ℯ    ⋅erfc⎜─────── - ─────── - ───────── + ─────────⎟             S⋅erf⎜────\n",
       "            ⎝  2⋅σ        4        2⋅σ⋅√t      2⋅σ⋅√t ⎠      -r⋅t        ⎝  2⋅\n",
       "─────────────────────────────────────────────────────── - K⋅ℯ     + ──────────\n",
       "                           2                                                  \n",
       "\n",
       "⋅√t   √2⋅σ⋅√t   √2⋅log(K)   √2⋅log(S)⎞    \n",
       "─── + ─────── - ───────── + ─────────⎟    \n",
       "σ        4        2⋅σ⋅√t      2⋅σ⋅√t ⎠   S\n",
       "────────────────────────────────────── + ─\n",
       "             2                           2"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.subs(a,ibound[x]).subs(delta,0).simplify().expand()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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