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    "# Black-Scholes option pricing theory\n",
    "\n",
    "## Hedging and the partial differential equation\n",
    "\n",
    "The original derivation of the Black-Scholes partial differential equation was via stochastic calculus, Ito's lemma and a simple hedging argument (Black & Scholes, 1973).\n",
    "\n",
    "Assume that the underlying follows a lognormal random walk\n",
    "$$\n",
    "\tdS = \\mu S dt + \\sigma S dX.\n",
    "$$\n",
    "Use $\\Pi$ to denote the value of a portfolio of one long option position and a short position in some quantity $\\Delta$ of the underlying:\n",
    "$$ \n",
    "\t\\Pi = V(S,t) - \\Delta S.\n",
    "$$\n",
    "\n",
    "The first term on the right is the option and the second term is the short asset position.\n",
    "\n",
    "Ask .... $t$ to $t+dt$.\n",
    "$$\n",
    "\td\\Pi = dV -\n",
    "$$\n",
    "\n",
    "From Ito's lemma we have\n",
    "$$\n",
    "\td\\Pi = \\frac{\\partial V}{\\partial t}dt + \\frac{\\partial V}{\\partial S}dS +\n",
    "\t\\frac{1}{2} \\sigma^2 S^2 \\frac{\\partial^2 V}{\\partial S^2}dt -\n",
    "$$\n",
    "\n",
    ".....\n",
    "\n",
    "$$\n",
    "\td\\Pi = \\Big( \\frac{\\partial V}{\\partial t} + ...\\Big)dt.\n",
    "$$\n",
    "\n",
    "$$\n",
    "\tV(S,T)=\\max(S-K,0)\n",
    "$$"
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      "8.916035060662303\n"
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    "from math import *\n",
    "# Cumulative normal distribution\n",
    "\n",
    "def CND(X):\n",
    "\n",
    "    (a1,a2,a3,a4,a5) = (0.31938153, -0.356563782, 1.781477937, \n",
    "\n",
    "     -1.821255978, 1.330274429)\n",
    "    L = abs(X)\n",
    "\n",
    "    K = 1.0 / (1.0 + 0.2316419 * L)\n",
    "\n",
    "    w = 1.0 - 1.0 / sqrt(2*pi)*exp(-L*L/2.) * (a1*K + a2*K*K + a3*pow(K,3) +\n",
    "\n",
    "    a4*pow(K,4) + a5*pow(K,5))\n",
    "    if X<0:\n",
    "\n",
    "        w = 1.0-w\n",
    "\n",
    "    return w\n",
    "# Black Sholes Function\n",
    "\n",
    "def BlackSholes(CallPutFlag,S,X,T,r,v):\n",
    "\n",
    "    d1 = (log(S/X)+(r+v*v/2.)*T)/(v*sqrt(T))\n",
    "\n",
    "    d2 = d1-v*sqrt(T)\n",
    "    if CallPutFlag=='c':\n",
    "\n",
    "        return S*CND(d1)-X*exp(-r*T)*CND(d2)\n",
    "\n",
    "    else:\n",
    "\n",
    "        return X*exp(-r*T)*CND(-d2)-S*CND(-d1)\n",
    "    \n",
    "call = BlackSholes('c',100,100,1,0.02,0.2)\n",
    "\n",
    "print (call)"
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