Spread Options.ipynb

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    "*****\n",
    "# Pricing Spread Options with Python\n",
    "*****\n",
    "<p style=\"text-align: right\"><i>Jesus Perez Colino<br>First version: November 2013<br>Last revision November 2015</i></p>\n",
    "## About this notebook \n",
    "****\n",
    "- Documention prepared by **Jesus Perez Colino**. Version 0.1, Released 25/11/2013, Alpha.\n",
    "\n",
    "\n",
    "- This work is licensed under a [Creative Commons Attribution-ShareAlike 3.0 Unported License](http://creativecommons.org/licenses/by-sa/3.0/deed.en_US). This work is offered for free, with the hope that it will be useful.\n",
    "\n",
    "\n",
    "- **Summary**: This technical note has as main goal to summarize the different methodologies to calculate an spread option, with the implementation in Python. \n",
    "\n",
    "\n",
    "- **Index**:\n",
    "\n",
    "    0. About this notebook\n",
    "    1. Spreads Options: General Framework\n",
    "    2. Spread Options Valuation when $K=0$: Margrabe's Formula\n",
    "    3. Spread Options Valuation when $K \\neq 0$: Kirk's Formula\n",
    "    4. Spread Options Valuation by Monte-Carlo Simulation\n",
    "\n",
    "\n",
    "- **Reference**: [Rene Carmona and Valdo Durrleman \"*Pricing and Hedging Spread Options*\" SIAM Review Vol.45 No.4, pp627-685](http://www.princeton.edu/~rcarmona/download/fe/sirev.pdf)\n",
    "\n",
    "\n",
    "- **Reproducibility conditions**: the content in this technical note can be reproduce in your computer under the following versions of the Python's packages: \n"
   ]
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     "text": [
      "------------- Reproducibility conditions for Pricing Spread Options-------------\n",
      " \n",
      " Python Version: 2.7.10 |Anaconda 2.3.0 (x86_64)| (default, Sep 15 2015, 14:29:08) \n",
      "[GCC 4.2.1 (Apple Inc. build 5577)]\n",
      " Matplotlib Version:1.4.3\n",
      " Numpy Version: 1.9.2\n",
      " \n",
      "--------------------------------------------------------------------------------\n"
     ]
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import sys\n",
    "import numpy as np\n",
    "from matplotlib.pyplot import *\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.stats import norm\n",
    "from abc import ABCMeta, abstractmethod\n",
    "\n",
    "print ' Reproducibility conditions for Pricing Spread Options'.center(80, '-')\n",
    "print ' '\n",
    "print ' Python Version: ' + sys.version\n",
    "print ' Matplotlib Version:' + matplotlib.__version__\n",
    "print ' Numpy Version: ' + np.__version__\n",
    "print ' ' \n",
    "print '-'*80"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "****\n",
    "# Spread Options: General Framework\n",
    "****\n",
    "\n",
    "All over this technical note, we will consider a set of Brownian motions $B_{1}(t),B_{2}(t)$ with correlation $\\rho$ on the probability space $(\\Omega,\\mathcal{F},\\mathbb{P} )$ endowed with the natural (completed and right-continuous) filtration $\\mathbb{F}=(\\mathcal{F_t})_{t \\geq T}$ generated by ${B_{1}(t),B_{2}(t)}$ up to some fixed time horizon $T>0$. \n",
    "\n",
    "We look at the classical setting where besides a riskless bank account with constant interest rate r, our arbitrage-free market model comprises two assets whose prices at time $t$ are denoted by $S_1(t)$ and $S_2(t)$. We assume that their risk-neutral price dynamics are given by the following stochastic differential equations:\n",
    "\n",
    "\n",
    "\\begin{equation}\n",
    "\\begin{array}{lll}\n",
    "\\frac{dS_1(t)}{S_1(t)}& =& \\mu_1 (t,T) dt+\\sigma_{1}(t,T)dB_{1}(t)\\\\\n",
    "\\end{array}\n",
    "\\end{equation}\n",
    "\n",
    "\n",
    "\\begin{equation}\n",
    "\\begin{array}{lll}\n",
    "\\frac{dS_2(t)}{S_2(t)}& =& \\mu_2 (t,T) dt+\\sigma_{2}(t,T)dB_{2}(t)\\\\\n",
    "\\end{array}\n",
    "\\end{equation}\n",
    "\n",
    "\n",
    "\n",
    "where the volatilities $\\sigma_1$ and $\\sigma_2$ are positive constants and $B_1$ and $B_2$ are two Brownian motions with correlation $\\rho$.\n",
    "\n",
    "In any case, the value of an spread option $V$ at time $t$ of the spread option with date of maturity $T$ and strike $K$ is given by the following risk-neutral expectation:\n",
    "\n",
    "\\begin{equation}\n",
    "\\begin{array}{lll}\n",
    "V(t,T)& =& \\mathrm{e}^{-rT} \\mathbb{E_Q}\\{(S_1(T)-S_2(T)-K)^+\\}\\\\\n",
    " & =& \\mathrm{e}^{-rT} \\mathbb{E_Q}\\{(S_1(t)\\mathrm{e}^{(\\mu_1 - \\sigma_1^2/2)T+\\sigma_1 B_1(T) }-S_2(t)\\mathrm{e}^{(\\mu_2 - \\sigma_2^2/2)T+\\sigma_2 B_2(T) }-K)^+\\}\\\\\n",
    " & =& \\mathrm{e}^{-rT} \\int \\int (S_1(t)\\mathrm{e}^{(\\mu_1 - \\sigma_1^2/2)T+\\sigma_1 B_1(T) }-S_2(t)\\mathrm{e}^{(\\mu_2 - \\sigma_2^2/2)T+\\sigma_2 B_2(T) }-K)^+ \\phi_T (B_1,B_2) dB_1 dB_2\\\\\n",
    "\\end{array}\n",
    "\\end{equation}\n",
    "\n",
    "which shows that the value of the spread options $V(t,T)$ is given by the double integral of a function of two variables with respect to a bivariate Gaussian distribution, namely the joint distribution of $B_1(T)$ and $B_2(T)$. \n"
   ]
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   "execution_count": 63,
   "metadata": {
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   "source": [
    "class SpreadOption(object):\n",
    "    \n",
    "    ''' Abstract Base-class for valuation of European Spreads Options (Calls/P.uts).\n",
    "    S1_t : float : initial forward price/index level\n",
    "    S2_t : float : initial forward price/index level\n",
    "    K : float : strike price\n",
    "    T : float : maturity (in year fractions)\n",
    "    r : float : constant risk-free short rate\n",
    "    vol1 : float : volatility factor in diffusion term (std)\n",
    "    vol2 : float : volatility factor in diffusion term (std)\n",
    "    rho : float: \n",
    "    CallPut : integer : 1 for a Call, and -1 for a Put\n",
    "    '''\n",
    "    __metaclass__ = ABCMeta    \n",
    "    def __init__(self, S1_t, S2_t, K, T, r, vol1, vol2, rho, CallPut, model):\n",
    "        try:\n",
    "            \n",
    "            self.S1_t = float(S1_t)\n",
    "            self.S2_t = float(S2_t)\n",
    "            self.K = float(K)\n",
    "            self.T = float(T)\n",
    "            self.r = float(r)\n",
    "            self.vol1 = float(vol1)\n",
    "            self.vol2 = float(vol2)\n",
    "            self.rho = float(rho)\n",
    "            self.CallPut = int(CallPut)\n",
    "            self.model = str(model)\n",
    "            \n",
    "            if T < 0 or r < 0 or S1_t < 0 or S2_t < 0:\n",
    "                raise ValueError('Negative inputs not allowed.')\n",
    "            if vol1 < 0 or vol2 < 0:\n",
    "                raise ValueError('Negative volatilities are not allowed.')\n",
    "            if rho > 1 or rho < -1:\n",
    "                raise ValueError('Correlation out of range')\n",
    "            if CallPut != 1 and CallPut != -1:\n",
    "                raise ValueError('For a Call: CallPut=1, or -1 for a Put')\n",
    "                \n",
    "        except ValueError: \n",
    "            print('Error passing spread option inputs')\n",
    "        \n",
    "    def getmodel(self):\n",
    "        return self.model\n",
    "    \n",
    "    def __str__(self):\n",
    "        return \"This SpreadOption is solved using {0}\".format(self.getmodel())\n",
    "    \n",
    "    @abstractmethod\n",
    "    def price(self):\n",
    "        pass"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "Unfortunately, except in the case of an exchange option (i.e., an option with strike K = 0), the price of the spread option cannot be given by a formula in closed form."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Spread Options Valuation when $K=0$: Margrabe's Formula\n",
    "\n",
    "The most common treatment of spread options is usually restricted to the special case $K = 0$ because\n",
    "when the distributions of the underlying $S_1$ and $S_2$ are log-normal, this is the only case for\n",
    "which one has a solution in a closed form a la `Black-Scholes` for the price of the spread option with\n",
    "zero strike. This formula was first derived by Margrabe as early as in 1978, and it bears his\n",
    "name:\n",
    "\n",
    "\n",
    "\\begin{equation}\n",
    "\\begin{array}{lll}\n",
    "V(t,T)_{Margrabe}& =& S_2(t)\\phi(d_1) - S_1(t)\\phi(d_0)\\\\\n",
    "\\end{array}\n",
    "\\end{equation}\n",
    "where \n",
    "\\begin{equation}\n",
    "\\begin{array}{lll}\n",
    "d_0 & =& \\frac{\\ln(S_2(t)/S_1(t))}{\\sigma \\sqrt{T}}-\\frac{1}{2} \\sigma \\sqrt{T}\\\\\n",
    "\\end{array}\n",
    "\\end{equation}\n",
    "\n",
    "\n",
    "\\begin{equation}\n",
    "\\begin{array}{lll}\n",
    "d_1 & =& \\frac{\\ln(S_2(t)/S_1(t))}{\\sigma \\sqrt{T}}+\\frac{1}{2} \\sigma \\sqrt{T}\\\\\n",
    "\\end{array}\n",
    "\\end{equation}\n",
    "\n",
    "\n",
    "and \n",
    "\n",
    "\n",
    "\\begin{equation}\n",
    "\\begin{array}{lll}\n",
    "\\sigma & =& \\sqrt{\\sigma_1^2 + \\sigma_2^2 - 2 \\rho \\sigma_1 \\sigma_2}\n",
    "\\end{array}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "class margrabe(SpreadOption):\n",
    "    def __init__ (self, S1_t, S2_t, K, T, r, vol1, vol2, rho, CallPut):\n",
    "        SpreadOption.__init__(self, S1_t, S2_t, K, T, r, vol1, vol2, rho, CallPut, \"Margrabe\")\n",
    "        if K != 0:\n",
    "            raise ValueError('Strike should be null to use Margrabe')\n",
    "            \n",
    "    @property\n",
    "    def price(self):\n",
    "        vol = np.sqrt(self.vol1 ** 2 + self.vol2 ** 2 - 2 * self.rho * self.vol1 * self.vol2)\n",
    "        d1 = ( np.log(self.S2_t / self.S1_t) / (vol * np.sqrt(self.T)) + 0.5 * vol * np.sqrt(self.T))\n",
    "        d2 = d1 - vol * np.sqrt(self.T)\n",
    "        price = (self.CallPut * (self.S2_t  * norm.cdf(self.CallPut * d1, 0, 1) \n",
    "                                  - self.S1_t * norm.cdf(self.CallPut * d2, 0, 1)))\n",
    "        return price  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "This SpreadOption is solved using Margrabe\n",
      "3.77584707495\n"
     ]
    }
   ],
   "source": [
    "SpreadCALL = margrabe(40, 40, 0., .75, 0., .5, .5, .85, 1)\n",
    "print SpreadCALL\n",
    "print SpreadCALL.price"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def margrabe_experiment():\n",
    "    for S1_t in (36., 40., 44.):  # initial stock price values\n",
    "        for S2_t in (44., 40., 36.):\n",
    "            print '-'*90\n",
    "            for rho in (.01, .5, .99):\n",
    "                SpreadCALL = margrabe(S1_t, S2_t, 0, 1, .0, 0.5, 0.5, rho, 1)\n",
    "                print (\"Initial prices: {0}, Sigmas: {1}, Correlation: {2} --> Option Value: {3:.2f}\"\n",
    "                       .format((S1_t, S2_t), (0.5, 0.5), rho, SpreadCALL.price))\n",
    "            "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": false,
    "scrolled": false
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     "name": "stdout",
     "output_type": "stream",
     "text": [
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (36.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 15.42\n",
      "Initial prices: (36.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 12.51\n",
      "Initial prices: (36.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 8.00\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (36.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 12.56\n",
      "Initial prices: (36.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 9.66\n",
      "Initial prices: (36.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 4.08\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (36.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 9.90\n",
      "Initial prices: (36.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 7.11\n",
      "Initial prices: (36.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 1.02\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (40.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 13.65\n",
      "Initial prices: (40.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 10.44\n",
      "Initial prices: (40.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 4.12\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (40.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 11.00\n",
      "Initial prices: (40.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 7.90\n",
      "Initial prices: (40.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 1.13\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (40.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 8.56\n",
      "Initial prices: (40.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 5.66\n",
      "Initial prices: (40.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 0.08\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (44.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 12.10\n",
      "Initial prices: (44.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 8.69\n",
      "Initial prices: (44.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 1.24\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (44.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 9.65\n",
      "Initial prices: (44.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 6.44\n",
      "Initial prices: (44.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 0.12\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (44.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 7.42\n",
      "Initial prices: (44.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 4.51\n",
      "Initial prices: (44.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 0.00\n",
      "------------------------------------------------------------------------------------------\n",
      "This SpreadOption is solved using Margrabe\n",
      "Duration in Seconds 0.00490212440491\n"
     ]
    }
   ],
   "source": [
    "from time import time\n",
    "t0 = time()\n",
    "margrabe_experiment()\n",
    "t1 = time(); d1 = t1 - t0\n",
    "print '-'*90\n",
    "print SpreadCALL\n",
    "print \"Duration in Seconds {0}\".format(d1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Spread Options Valuation when $K\\neq0$: Kirk's Formula\n",
    "\n",
    "The **Margrabe**(1978) approach above works nicely because we are considering the difference between two lognormal quantities, namely $S_1(t) − S_2(t)$. The approximation of **Kirk** (1995) is that when $K << S_2(t)$ we can regard $S_2(t) + K$ as being approximately lognormal. \n",
    "\n",
    "The ratio of two lognormal processes $X_t$ and $Y_t$ is itself lognormal; suppose $\\ln X_t ∼ N (\\mu_X , \\sigma_X^2 )$ and $\\ln Yt ∼ N(\\mu_Y,\\sigma_Y^2)$ and $Cov(\\ln Xt,\\ln Yt) = \\rho \\sigma_X \\sigma_Y$. Note that $\\rho$ is the terminal correlation between $\\ln X_t$ and $ln Y_t$, not the correlation between $S_1(t)$ and $S_2(t)$. \n",
    "\n",
    "Therefore, we can construct the ratio $\\ln(X_t/Y_t) = \\ln X_t − \\ln Y_t$ which gives $\\ln(X_t/Y_t)∼N(\\mu_X −\\mu_Y, \\sigma^2_X +\\sigma^2_Y − 2 \\rho \\sigma_x \\sigma_Y)$. That is the basis of **Kirk**(1995) and **Venkatramanam and Alexander**(2011), from where we take the following pricing formula:\n",
    "\n",
    "\\begin{equation}\n",
    "\\begin{array}{lll}\n",
    "V(t,T)_{Kirk}& =& S_2(t)\\phi(d_1) - (S_1(t)+K\\mathrm{e}^{-rT})\\phi(d_0)\\\\\n",
    "\\end{array}\n",
    "\\end{equation}\n",
    "where \n",
    "\\begin{equation}\n",
    "\\begin{array}{lll}\n",
    "d_0 & =& \\frac{\\ln(S_2(t)/(S_1(t)+K\\mathrm{e}^{-rT}))}{\\sigma \\sqrt{T}}-\\frac{1}{2} \\sigma \\sqrt{T}\\\\\n",
    "\\end{array}\n",
    "\\end{equation}\n",
    "\n",
    "\n",
    "\\begin{equation}\n",
    "\\begin{array}{lll}\n",
    "d_1 & =& \\frac{\\ln(S_2(t)/(S_1(t)+K\\mathrm{e}^{-rT}))}{\\sigma \\sqrt{T}}+\\frac{1}{2} \\sigma \\sqrt{T}\\\\\n",
    "\\end{array}\n",
    "\\end{equation}\n",
    "\n",
    "and \n",
    "\n",
    "\n",
    "\\begin{equation}\n",
    "\\begin{array}{lll}\n",
    "\\sigma & =& \\sqrt{\\sigma_1^2 \\frac{S_1}{S_1(t)+K\\mathrm{e}^{-rT}} + \\sigma_2^2 - 2 \\rho \\sigma_1 \\sigma_2 \\frac{S_1}{S_1(t)+K\\mathrm{e}^{-rT}}}\n",
    "\\end{array}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "class kirk(SpreadOption):\n",
    "    def __init__ (self, S1_t, S2_t, K, T, r, vol1, vol2, rho, CallPut):\n",
    "        SpreadOption.__init__(self, S1_t, S2_t, K, T, r, vol1, vol2, rho, CallPut, \"Kirk\")\n",
    "            \n",
    "    @property\n",
    "    def price(self):\n",
    "        z = self.S1_t / (self.S1_t + self.K * np.exp(-1. * self.r * self.T))\n",
    "        vol = np.sqrt( self.vol1 ** 2 * z ** 2 + self.vol2 ** 2 - 2 * self.rho* self.vol1 * self.vol2 * z )\n",
    "        d1 = (np.log(self.S2_t / (self.S1_t + self.K * np.exp(-self.r * self.T)))\n",
    "              / (vol * np.sqrt(self.T)) + 0.5 * vol * np.sqrt(self.T))\n",
    "        d2 = d1 - vol * np.sqrt(self.T)\n",
    "        price = (self.CallPut * (self.S2_t  * norm.cdf(self.CallPut * d1, 0, 1) \n",
    "                                 - (self.S1_t + self.K * np.exp(-self.r * self.T)) \n",
    "                                 * norm.cdf(self.CallPut * d2, 0, 1)))\n",
    "        return price"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "This SpreadOption is solved using Kirk\n",
      "3.77584707495\n"
     ]
    }
   ],
   "source": [
    "SpreadCALL = kirk(40, 40, 0., .75, 0., .5, .5, .85, 1)\n",
    "print SpreadCALL\n",
    "print SpreadCALL.price"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def kirk_experiment():\n",
    "    for S1_t in (36., 40., 44.):  # initial stock price values\n",
    "        for S2_t in (44., 40., 36.):\n",
    "            print '-'*90\n",
    "            for rho in (.01, .5, .99):\n",
    "                SpreadCALL = kirk(S1_t, S2_t, 0., 1, .0, 0.5, 0.5, rho, 1)\n",
    "                print (\"Initial prices: {0}, Sigmas: {1}, Correlation: {2} --> Option Value: {3:.2f}\"\n",
    "                       .format((S1_t, S2_t), (0.5, 0.5), rho, SpreadCALL.price))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (36.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 15.42\n",
      "Initial prices: (36.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 12.51\n",
      "Initial prices: (36.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 8.00\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (36.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 12.56\n",
      "Initial prices: (36.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 9.66\n",
      "Initial prices: (36.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 4.08\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (36.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 9.90\n",
      "Initial prices: (36.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 7.11\n",
      "Initial prices: (36.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 1.02\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (40.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 13.65\n",
      "Initial prices: (40.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 10.44\n",
      "Initial prices: (40.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 4.12\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (40.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 11.00\n",
      "Initial prices: (40.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 7.90\n",
      "Initial prices: (40.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 1.13\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (40.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 8.56\n",
      "Initial prices: (40.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 5.66\n",
      "Initial prices: (40.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 0.08\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (44.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 12.10\n",
      "Initial prices: (44.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 8.69\n",
      "Initial prices: (44.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 1.24\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (44.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 9.65\n",
      "Initial prices: (44.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 6.44\n",
      "Initial prices: (44.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 0.12\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (44.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 7.42\n",
      "Initial prices: (44.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 4.51\n",
      "Initial prices: (44.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 0.00\n",
      "------------------------------------------------------------------------------------------\n",
      "This SpreadOption is solved using Kirk\n",
      "Duration in Seconds  0.005\n"
     ]
    }
   ],
   "source": [
    "from time import time\n",
    "t0 = time()\n",
    "kirk_experiment()\n",
    "t1 = time(); d1 = t1 - t0\n",
    "print '-'*90\n",
    "print SpreadCALL\n",
    "print \"Duration in Seconds %6.3f\" % d1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Spread Options Valuation by Monte-Carlo Simulation\n",
    "\n",
    "Usually, the most common way to compute an expectation is to use traditional Monte-Carlo methods. The idea is to simulate randomly values of the variables $S_1(t),S_2(t)$ for every $t \\in [0,T]$. That will generate multiple trayectories or sample paths, and for each of these sample paths, we compute the value of the function of the path whose expectation we evaluate, and then to average these values over the sample paths. The principle of the method is simple, the computation of the sample average is usually quite straightforward, the only difficulty is to quantify and control the error. \n",
    "\n",
    "If we assume that the coefficients of the SDE that charectarize the dynamics of the underlyings are deterministic, then the situation become much simpler.  Indeed, in such a case the joint distribution of the underlying indexes at the terminal time can be computed explicitly, and there is no need to simulate entire sample paths: one can simulate samples from the terminal distribution directly. This distribution is well-known. As explained earlier, it is a joint log-normal distribution. The simulation of samples from the underlying index values at maturity is very easy. Indeed, if the coefficients are constant, the pair $(S_1(t), S_2(t))$ of indexes at maturity can be written in the form:\n",
    "\n",
    "\\begin{equation}\n",
    "\\begin{array}{lll}\n",
    "S_1(T)& =& S_1(t) \\exp \\left[ \\left( \\mu_1 - \\frac{\\sigma^2_1}{2} \\right) T \n",
    "+ \\sigma_1 dB_{1}(t) \\right]\\\\\n",
    "S_2(T)& =& S_2(t)\\exp \\left[ \\left( \\mu_2 - \\frac{\\sigma^2_2}{2} \\right) T \n",
    "+ \\sigma_2 \\left( \\rho dB_{1}(t)+  \\sqrt{1-\\rho^2} dB_{2}(t)\\right)  \\right]\\\\\n",
    "\\end{array}\n",
    "\\end{equation}\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "class montecarlo(SpreadOption):\n",
    "    \n",
    "    def __init__ (self, S1_t, S2_t, K, T, r, vol1, vol2, rho, CallPut, simulations):\n",
    "        SpreadOption.__init__(self, S1_t, S2_t, K, T, r, vol1, vol2, rho, CallPut, \"Monte Carlo\")\n",
    "        self.simulations = int(simulations)\n",
    "        try: \n",
    "            if self.simulations > 0 :\n",
    "                assert isinstance(self.simulations, int)\n",
    "        except:\n",
    "            raise ValueError(\"Simulation's number has to be positive integer\")\n",
    "        \n",
    "    def generate_spreads(self, seed = 12345678 ):    \n",
    "        try:\n",
    "            if seed is not None:\n",
    "                assert isinstance(seed, int)\n",
    "        except:\n",
    "            print 'Error passing seed'\n",
    "        np.random.seed(seed)\n",
    "        B1 = np.sqrt(self.T) * np.random.randn(self.simulations, 1)\n",
    "        B2 = np.sqrt(self.T) * np.random.randn(self.simulations, 1) \n",
    "        S1_T = self.S1_t * np.exp ((self.r - 0.5 * self.vol1 ** 2) * self.T + self.vol1 *  B1)\n",
    "        S2_T = self.S2_t * np.exp ((self.r - 0.5 * self.vol2 ** 2) * self.T + \n",
    "                                    self.vol2 * ( self.rho * B1 + np.sqrt(1 - self.rho ** 2) * B2)) \n",
    "        if self.CallPut == 1 :\n",
    "            payoff = np.maximum((S2_T - S1_T - self.K), 0)\n",
    "        else:\n",
    "            payoff = np.maximum((self.K - S2_T - S1_T), 0)\n",
    "        return np.exp(-1. * self.r * self.T) * payoff\n",
    "    \n",
    "    @property\n",
    "    def price(self):\n",
    "        price = np.sum(self.generate_spreads()) / float(self.simulations)\n",
    "        return price"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "--------------------------------------------------\n",
      "This SpreadOption is solved using Kirk\n",
      "3.05801412278\n",
      "--------------------------------------------------\n",
      "This SpreadOption is solved using Monte Carlo\n",
      "3.06645630782\n",
      "--------------------------------------------------\n"
     ]
    }
   ],
   "source": [
    "print '-'*50\n",
    "SpreadCALL1 = kirk(40, 40, 3., 1., 0., .5, .5, .85, 1)\n",
    "print SpreadCALL1\n",
    "print SpreadCALL1.price\n",
    "print '-'*50\n",
    "SpreadCALL2 = montecarlo(40, 40, 3., 1., 0., .5, .5, .85, 1, 1000000)\n",
    "print SpreadCALL2\n",
    "print SpreadCALL2.price\n",
    "print '-'*50"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def montecarlo_experiment():\n",
    "    for S1_t in (36., 40., 44.):  # initial stock price values\n",
    "        for S2_t in (44., 40., 36.):\n",
    "            print '-'*90\n",
    "            for rho in (.01, .5, .99):\n",
    "                SpreadCALL = montecarlo(S1_t, S2_t, 0., 1, .0, 0.5, 0.5, rho, 1, 1000000)\n",
    "                print (\"Initial prices: {0}, Sigmas: {1}, Correlation: {2} --> Option Value: {3:.2f}\"\n",
    "                       .format((S1_t, S2_t), (0.5, 0.5), rho, SpreadCALL.price))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (36.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 15.42\n",
      "Initial prices: (36.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 12.52\n",
      "Initial prices: (36.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 8.00\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (36.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 12.56\n",
      "Initial prices: (36.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 9.67\n",
      "Initial prices: (36.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 4.08\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (36.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 9.89\n",
      "Initial prices: (36.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 7.11\n",
      "Initial prices: (36.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 1.02\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (40.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 13.65\n",
      "Initial prices: (40.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 10.44\n",
      "Initial prices: (40.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 4.12\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (40.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 10.99\n",
      "Initial prices: (40.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 7.90\n",
      "Initial prices: (40.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 1.13\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (40.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 8.55\n",
      "Initial prices: (40.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 5.67\n",
      "Initial prices: (40.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 0.08\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (44.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 12.09\n",
      "Initial prices: (44.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 8.69\n",
      "Initial prices: (44.0, 44.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 1.24\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (44.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 9.64\n",
      "Initial prices: (44.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 6.44\n",
      "Initial prices: (44.0, 40.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 0.12\n",
      "------------------------------------------------------------------------------------------\n",
      "Initial prices: (44.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.01 --> Option Value: 7.41\n",
      "Initial prices: (44.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.5 --> Option Value: 4.51\n",
      "Initial prices: (44.0, 36.0), Sigmas: (0.5, 0.5), Correlation: 0.99 --> Option Value: 0.00\n",
      "------------------------------------------------------------------------------------------\n",
      "Duration in Seconds  3.133\n"
     ]
    }
   ],
   "source": [
    "from time import time\n",
    "t0 = time()\n",
    "montecarlo_experiment()\n",
    "t1 = time(); d1 = t1 - t0\n",
    "print '-'*90\n",
    "print \"Duration in Seconds %6.3f\" % d1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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S1AGGJUmSJKmTpHVrST/9Lun+u6hc+m7iiOPLLknbwbAkSZIkdYI07RGq3/0q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\nI0v99+3J6dwAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1068b5ed0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "set_corr = [i/100. for i in range(99) ]\n",
    "result_kirk = []\n",
    "result_montecarlo = []\n",
    "for rho in set_corr:\n",
    "    SpreadCALL1 = kirk(40, 40, 3., 1., 0., .5, .5, rho, 1)\n",
    "    result_kirk.append(SpreadCALL1.price)\n",
    "    SpreadCALL2 = montecarlo(40, 40, 3., 1., 0., .5, .5, rho, 1, 100000)\n",
    "    result_montecarlo.append(SpreadCALL2.price)\n",
    "    \n",
    "plt.figure(num=None, figsize=(14, 6))\n",
    "plt.style.use('ggplot')\n",
    "plt.plot(set_corr, 100*(np.array(result_montecarlo) - np.array(result_kirk))/np.array(result_kirk))\n",
    "plt.title('Relative Difference Spreads prices MonteCarlo - Kirk')\n",
    "plt.xlabel('Correlation')\n",
    "plt.ylabel('% error');\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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d9KPskjlSyatkGreSuf9xmdAwr6MhH6EcAQAAwBP2xDHZpfNlF82UDh+UubGl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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10b5c5e10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "set_strikes = range(-25, 25)\n",
    "result_kirk = []\n",
    "result_montecarlo = []\n",
    "for strike in set_strikes:\n",
    "    SpreadCALL1 = kirk(50, 50, strike, 1., 0., .5, .5, 0.5, 1)\n",
    "    result_kirk.append(SpreadCALL1.price)\n",
    "    SpreadCALL2 = montecarlo(50, 50, strike, 1., 0., .5, .5, 0.5, 1, 100000)\n",
    "    result_montecarlo.append(SpreadCALL2.price)\n",
    "    \n",
    "plt.figure(num=None, figsize=(14, 6))\n",
    "plt.style.use('ggplot')\n",
    "plt.plot(set_strikes, 100*(np.array(result_montecarlo) - np.array(result_kirk))/np.array(result_kirk))\n",
    "plt.title('Relative Difference Spreads prices MonteCarlo - Kirk')\n",
    "plt.xlabel('Strikes')\n",
    "plt.ylabel('% error');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Z1r/MHjsqbVovu3aF7KIfZd9/UzJGptzFOpxYTbZ4Wd/wwGw8KyMAAMgc/DUB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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10b191290>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "vol1 = [i/100. for i in range(1, 50)]\n",
    "vol2 = [i/100. for i in range(50, 1,-1)]\n",
    "result_kirk = []\n",
    "result_montecarlo = []\n",
    "for i in xrange(len(vol1)):\n",
    "    SpreadCALL1 = kirk(50, 50, 0, 1., 0., vol1[i], vol2[i], 0.5, 1)\n",
    "    result_kirk.append(SpreadCALL1.price)\n",
    "    SpreadCALL2 = montecarlo(50, 50, 0, 1., 0., vol1[i], vol2[i], 0.5, 1, 100000)\n",
    "    result_montecarlo.append(SpreadCALL2.price)\n",
    "    \n",
    "plt.figure(num=None, figsize=(14, 6))\n",
    "plt.style.use('ggplot')\n",
    "dif_vols = [vol1[i]-vol2[i] for i in range(len(vol1))]\n",
    "plt.plot(dif_vols, 100*(np.array(result_montecarlo) - np.array(result_kirk))/np.array(result_kirk))\n",
    "plt.title('Relative Difference Spreads prices MonteCarlo - Kirk')\n",
    "plt.xlabel('vol1-vol2')\n",
    "plt.ylabel('% error');"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}