Heston Forward.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "zanadu": {}
   },
   "source": [
    "# Heston model: vanilla and forward prices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false,
    "zanadu": {
     "code_type": "",
     "is_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy.integrate import quad\n",
    "from scipy.optimize import bisect\n",
    "import cmath\n",
    "import matplotlib.pyplot as plt\n",
    "from math import isnan\n",
    "from scipy.stats import norm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The Heston model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Recall the following option price formula for a European Call option with maturity $T$ and strike $K$:\n",
    "\\begin{align*}\n",
    "\\mathbb{E}\\left(S_T - \\mathrm{e}^{k}\\right)_+\n",
    " & = S_0\\left(\\frac{1}{2}+\\frac{1}{\\pi}\\int_{0}^{\\infty}\\Re\\left(\\frac{\\Phi_T(\\xi-\\mathrm{i})}{\\mathrm{i}\\xi\\Phi_T(-\\mathrm{i})}\n",
    " \\mathrm{e}^{-\\mathrm{i} k\\xi}\\right)\\mathrm{d} \\xi\\right)\n",
    "-K\\mathrm{e}^{-rT}\\left(\\frac{1}{2}+\\frac{1}{\\pi}\\int_{0}^{\\infty}\\Re\\left(\\frac{\\Phi_T(\\xi)}{\\mathrm{i}\\xi}\\mathrm{e}^{-\\mathrm{i} k\\xi}\\right)\\mathrm{d} \\xi\\right)\\\\\n",
    " & =: S_0 \\Pi_1 - K\\mathrm{e}^{-rT}\\Pi_2,\n",
    "\\end{align*}\n",
    "where $\\Phi_T(\\xi):=\\mathbb{E}\\left(S_T^{\\mathrm{i}\\xi}\\right)$\n",
    "is the characteristic function of the log-stock price at time $T$.\n",
    "\n",
    "\n",
    "We shall use here the slightly-rewritten expression:\n",
    "\\begin{align*}\n",
    "\\mathbb{E}\\left(S_T - K\\right)_+\n",
    " & = S_0\\left(\\frac{1}{2}+\\frac{1}{\\pi}\\int_{0}^{\\infty}\\Re\\left(\\frac{\\Phi_T(\\xi-\\mathrm{i})}{\\mathrm{i}\\xi}\n",
    " \\mathrm{e}^{-\\mathrm{i} k\\xi}\\right)\\mathrm{d} \\xi\\right)\n",
    "-K\\mathrm{e}^{-rT}\\left(\\frac{1}{2}+\\frac{1}{\\pi}\\int_{0}^{\\infty}\\Re\\left(\\frac{\\Phi_T(\\xi)}{\\mathrm{i}\\xi}\\mathrm{e}^{-\\mathrm{i} k\\xi}\\right)\\mathrm{d} \\xi\\right)\\\\\n",
    " & =: S_0 \\Pi_1 - K\\mathrm{e}^{-rT}\\Pi_2,\n",
    "\\end{align*}\n",
    "where $K:=S_0\\mathrm{e}^{k}$.\n",
    "    \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The characteristic function has the following form:\n",
    "$\\Phi_T(\\xi) = \\exp\\Big(C_T(\\xi) + D_T(\\xi)V_0 + \\mathrm{i}\\xi \\log(S_0)\\Big)$, where\n",
    "\\begin{align*}\n",
    "C_T(\\xi) & : = \\mathrm{i}r\\xi T + \\frac{\\kappa\\theta}{\\sigma^2}\\left\\{\\left(\\kappa-\\mathrm{i}\\rho\\sigma\\xi-d_T(\\xi)\\right)T\n",
    "- 2\\log\\left(\\frac{1-\\gamma_T(\\xi)\\mathrm{e}^{-d_T(\\xi)T}}{1-\\gamma_T(\\xi)}\\right)\\right\\},\\\\\n",
    "D_T(\\xi) & : = \\frac{\\kappa - \\mathrm{i}\\rho\\sigma\\xi - d_T(\\xi)}{\\sigma^2}\n",
    "\\left(\\frac{1-\\mathrm{e}^{-d_T(\\xi)T}}{1-\\gamma_T(\\xi)\\mathrm{e}^{-d_T(\\xi)T}}\\right),\\\\\n",
    "\\gamma_T(\\xi) & := \\frac{\\kappa - \\mathrm{i}\\rho\\sigma\\xi - d_T(\\xi)}{\\kappa - \\mathrm{i}\\rho\\sigma\\xi + d_T(\\xi)},\n",
    "\\qquad \n",
    "d_T(\\xi) := \\sqrt{(\\kappa-\\mathrm{i}\\rho\\sigma\\xi)^2 + \\sigma^2(\\mathrm{i}\\xi+\\xi^2)}.\n",
    "\\end{align*}\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Black-Scholes formulae"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def phi(x): ## Gaussian density\n",
    "    return np.exp(-x*x/2.)/np.sqrt(2*np.pi)\n",
    "\n",
    "#### Black Sholes Vega\n",
    "def BlackScholesVegaCore(DF,F,X,T,v):\n",
    "    vsqrt=v*np.sqrt(T)\n",
    "    d1 = (np.log(F/X)+(vsqrt*vsqrt/2.))/vsqrt\n",
    "    return F*phi(d1)*np.sqrt(T)/DF\n",
    "\n",
    "#### Black Sholes Function\n",
    "def BlackScholesCore(CallPutFlag,DF,F,X,T,v):\n",
    "    ## DF: discount factor\n",
    "    ## F: Forward\n",
    "    ## X: strike\n",
    "    vsqrt=v*np.sqrt(T)\n",
    "    d1 = (np.log(F/X)+(vsqrt*vsqrt/2.))/vsqrt\n",
    "    d2 = d1-vsqrt\n",
    "    if CallPutFlag:\n",
    "        return DF*(F*norm.cdf(d1)-X*norm.cdf(d2))\n",
    "    else:\n",
    "        return DF*(X*norm.cdf(-d2)-F*norm.cdf(-d1))\n",
    "    \n",
    "##  Black-Scholes Pricing Function\n",
    "def BlackScholes(CallPutFlag,S,X,T,r,d,v):\n",
    "    ## r, d: continuous interest rate and dividend\n",
    "    return BlackScholesCore(CallPutFlag,np.exp(-r*T),np.exp((r-d)*T)*S,X,T,v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Pricing Vanilla options"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def Heston_P_Value(hestonParams,r,T,s0,K,typ):\n",
    "    kappa, theta, sigma, rho, v0 = hestonParams\n",
    "    return 0.5+(1./np.pi)*quad(lambda xi: Int_Function_1(xi,kappa,theta, sigma,rho,v0,r,T,s0,K,typ),0.,500.)[0]\n",
    "\n",
    "def Int_Function_1(xi,kappa,theta,sigma,rho,v0,r,T,s0,K,typ):\n",
    "    return (cmath.e**(-1j*xi*np.log(K))*Int_Function_2(xi,kappa,theta,sigma,rho,v0,r,T,s0,typ)/(1j*xi)).real\n",
    "\n",
    "def Int_Function_2(xi,kappa,theta,sigma,rho,v0,r,T,s0,typ):\n",
    "    if typ == 1:\n",
    "        w = 1.\n",
    "        b = kappa - rho*sigma\n",
    "    else:\n",
    "        w = -1.\n",
    "        b = kappa\n",
    "    ixi = 1j*xi\n",
    "    d = cmath.sqrt((rho*sigma*ixi-b)*(rho*sigma*ixi-b) - sigma*sigma*(w*ixi-xi*xi))\n",
    "    g = (b-rho*sigma*ixi-d) / (b-rho*sigma*ixi+d)\n",
    "    ee = cmath.e**(-d*T)\n",
    "    C = r*ixi*T + kappa*theta/(sigma*sigma)*((b-rho*sigma*ixi-d)*T - 2.*cmath.log((1.0-g*ee)/(1.-g)))\n",
    "    D = ((b-rho*sigma*ixi-d)/(sigma*sigma))*(1.-ee)/(1.-g*ee)\n",
    "    return cmath.e**(C + D*v0 + ixi*np.log(s0))\n",
    "\n",
    "def heston_EuropeanCall(hestonParams,r,T,s0,K):\n",
    "    a = s0*Heston_P_Value(hestonParams,r,T,s0,K,1)\n",
    "    b = K*np.exp(-r*T)*Heston_P_Value(hestonParams,r,T,s0,K,2)\n",
    "    return a-b\n",
    "\n",
    "def heston_Impliedvol(hestonParams,r,T,s0,K):\n",
    "    myPrice = heston_EuropeanCall(hestonParams,r,T,s0,K)\n",
    "    ## Bisection algorithm when the Lee-Li algorithm breaks down\n",
    "    def smileMin(vol, *args):\n",
    "        K, s0, T, r, price = args\n",
    "        return price - BlackScholes(True, s0, K, T, r, 0., vol)\n",
    "    vMin = 0.000001\n",
    "    vMax = 10.\n",
    "    return bisect(smileMin, vMin, vMax, args=(K, s0, T, r, myPrice), xtol=1e-20, rtol=1e-15, full_output=False, disp=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "v0,kappa,theta,sigma,rho = 0.1, 1., 0.25, 0.3, -0.2\n",
    "hestonParams = kappa, theta, sigma, rho, v0 \n",
    "r, s0, K, T = 0., 1., 1., 3."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Call option price:  0.29762404877689647\n",
      "Implied volatility:  0.4412285751258531\n"
     ]
    }
   ],
   "source": [
    "print(\"Call option price: \", heston_EuropeanCall(hestonParams,r,T,s0,K))\n",
    "print(\"Implied volatility: \", heston_Impliedvol(hestonParams,r,T,s0,K))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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UkQpvGtDCzJqZWQ1gEDAx8wR3b+buTd29KZGsnOruD7n758ACM2uVnHoQ8A6A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\niYyZmZmZmRXO/wEXf7Z2y0xZbAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 1008x576 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xx = np.linspace(-1.5, 1.5, 20)\n",
    "calls = [heston_EuropeanCall(hestonParams,r,T,s0,s0*np.exp(x)) for x in xx]\n",
    "ivs = [heston_Impliedvol(hestonParams,r,T,s0,s0*np.exp(x)) for x in xx]\n",
    "\n",
    "fig = plt.figure(figsize=(14,8))\n",
    "plt.subplot(2, 2, 1)\n",
    "plt.plot(xx, calls, 'b-', label=\"Params 1\", linewidth=2)\n",
    "plt.title(\"Heston Call prices\", fontsize=12, fontweight='bold')\n",
    "plt.xlabel(u'log-moneyness', fontsize=12)\n",
    "plt.legend(loc=1)\n",
    "plt.subplot(2, 2, 2)\n",
    "plt.plot(xx, ivs, 'b-', label=\"Params 1\", linewidth=2)\n",
    "plt.title(\"Heston implied volatility\", fontsize=12, fontweight='bold')\n",
    "plt.xlabel(u'log-moneyness', fontsize=12)\n",
    "plt.legend(loc=1)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "zanadu": {}
   },
   "source": [
    "# Forward-start pricing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We are now interested in forward-start Call otions, the value of which is given, at inception, by\n",
    "$$\n",
    "\\mathbb{E}\\left[\\left(\\frac{S_{t+\\tau}}{S_t} - K\\right)_+\\right],\n",
    "$$\n",
    "for some forward-starting date $t\\geq 0$ and some remaining maturity $\\tau\\geq 0$.\n",
    "Obviously this coincides with the vanilla option whenever $t=0$. \n",
    "The functions below are normalised with $S_0=1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "zanadu": {}
   },
   "source": [
    "## Forward characteristic function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false,
    "zanadu": {
     "is_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "def LittleB(t, hestonParams):\n",
    "    return ((xi*xi)/(4.0*kappa))*(1. - np.exp(-kappa*t))\n",
    "    \n",
    "def d(u, hestonParams):\n",
    "    kappa, theta, xi, rho, v0 = hestonParams\n",
    "    return cmath.sqrt(((kappa - rho*xi*u)*(kappa - rho*xi*u)) + u*(1. - u)*xi*xi)\n",
    "\n",
    "def LittleGamma(u, hestonParams):\n",
    "    kappa, theta, xi, rho, v0 = hestonParams\n",
    "    dd = d(u, hestonParams)\n",
    "    return (kappa - rho*xi*u - dd)/(kappa-rho*xi*u + dd)\n",
    "\n",
    "def BigA(u, tau, hestonParams):\n",
    "    kappa, theta, xi, rho, v0 = hestonParams\n",
    "    ddd = d(u,hestonParams)\n",
    "    g = LittleGamma(u,hestonParams)\n",
    "    LL = cmath.log((1.0-g*cmath.e**(-ddd*tau))/(1.-g))\n",
    "    return ((kappa*theta)/(xi*xi))*((kappa-rho*xi*u-ddd)*tau-2.*LL)\n",
    "\n",
    "def BigB(u, tau, hestonParams):\n",
    "    kappa, theta, xi, rho, v0 = hestonParams\n",
    "    ddd = d(u,hestonParams)\n",
    "    ee = cmath.e**(-ddd*tau)\n",
    "    return (1./(xi*xi))*(kappa-rho*xi*u-ddd)*((1.-ee)/(1.-LittleGamma(u, hestonParams)*ee))\n",
    "\n",
    "def cfHestonFwd(u, t, tau, hestonParams):\n",
    "    kappa, theta, xi, rho, v0 = hestonParams\n",
    "    BBB = BigB(u, tau, hestonParams)\n",
    "    BBb = LittleB(t, hestonParams)\n",
    "    return cmath.e**(BigA(u, tau, hestonParams)+(BBB/(1.-2.*BBb*BBB))*(v0*(np.exp(-kappa*t)))-((2.*kappa*theta)/(xi*xi))*cmath.log(1.-2.*BBb*BBB))\n",
    "\n",
    "def intPhi(w, alpha, k, t, tau, hestonParams):\n",
    "    return (cmath.e**(-k*(alpha+1j*w))*cfHestonFwd(alpha+1j*w,t,tau,hestonParams)/((alpha+1j*w)*(1.0-alpha-1j*w))).real"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false,
    "zanadu": {
     "code_type": "importable"
    }
   },
   "outputs": [],
   "source": [
    "def hestonForwardCallPrice(K, t, tau, hestonParams, precision=20.):\n",
    "    alpha = 0.5\n",
    "    pp = cfHestonFwd(1., t, tau, hestonParams).real\n",
    "    return (pp-K/(2.*np.pi)*quad(lambda w: intPhi(w, alpha,np.log(K),t,tau,hestonParams),-precision,precision)[0]).real\n",
    "\n",
    "def heston_Impliedvol(hestonParams,r,T,s0,K):\n",
    "    myPrice = heston_EuropeanCall(hestonParams,r,T,s0,K)\n",
    "    ## Bisection algorithm when the Lee-Li algorithm breaks down\n",
    "    def smileMin(vol, *args):\n",
    "        K, s0, T, r, price = args\n",
    "        return price - BlackScholes(True, s0, K, T, r, 0., vol)\n",
    "    vMin = 0.000001\n",
    "    vMax = 10.\n",
    "    return bisect(smileMin, vMin, vMax, args=(K, s0, T, r, myPrice), xtol=1e-20, rtol=1e-15, full_output=False, disp=True)\n",
    "\n",
    "def hestonImpliedVol_Fwd(K, t, tau,hestonParams, precision=200.):\n",
    "    myPrice = hestonForwardCallPrice(K, t, tau, hestonParams, precision)\n",
    "    def smileMin(vol, *args):\n",
    "        K, s0, tau, r, price = args\n",
    "        return price - BlackScholes(True, 1., K, tau, r, 0., vol)\n",
    "    vMin = 0.000001\n",
    "    vMax = 10.\n",
    "    return bisect(smileMin, vMin, vMax, args=(K, 1., tau, r, myPrice), xtol=1e-20, rtol=1e-15, full_output=False, disp=True)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "zanadu": {}
   },
   "source": [
    "# Comparison"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false,
    "zanadu": {
     "code_type": ""
    }
   },
   "outputs": [],
   "source": [
    "K, v0, kappa, theta, xi, rho = 0.7, 0.05, 2.1, 0.03, 0.1, -0.2\n",
    "t, tau = 1., 1.\n",
    "hestonParams = kappa, theta, xi, rho, v0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Forward Call price:  0.30151947721202876\n",
      "Forward implied volatility:  0.18371308966181205\n"
     ]
    }
   ],
   "source": [
    "print(\"Forward Call price: \", hestonForwardCallPrice(K,t,tau,hestonParams, 100.))\n",
    "print(\"Forward implied volatility: \", hestonImpliedVol_Fwd(K,t,tau,hestonParams))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": false,
    "zanadu": {
     "code_type": "",
     "is_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
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Eer2ewYMHM3jw4Ca3276ITpkyhdraWjIzM8nKyiIrK4uLFy+Sn59v7QwHmqjc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\nExQtABqNRjNB+f9je+4GkUjcBgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "KK = np.linspace(0.7, 1.3, 10)\n",
    "tt = [0.05, 0.3, 0.5, 1.]\n",
    "ivs = [heston_Impliedvol(hestonParams,r,tau,1.,K) for K in KK]\n",
    "plt.figure(figsize=(6,4))\n",
    "plt.plot(KK, ivs, 'k--', label=\"Smile\", linewidth=2)\n",
    "for t in tt:\n",
    "    ivfwds = [hestonImpliedVol_Fwd(K,t,tau,hestonParams) for K in KK]\n",
    "    plt.plot(KK, ivfwds, label=\"Forward smile, t=%.1f\" %t)\n",
    "plt.legend(loc=\"best\")\n",
    "plt.title(\"Vanilla and forward smiles for $\\tau=$ %.1f\" %tau)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false,
    "zanadu": {
     "code_type": ""
    }
   },
   "outputs": [],
   "source": [
    "    "
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.0"
  },
  "zanadu": {
   "authors": [
    "Antoine Jacquier"
   ],
   "bibliography_data": {},
   "category": "3",
   "clearance": "Internal",
   "group_name": "Imperial College",
   "md5": "1c40e1601e25226bb9768c41295a3bcf",
   "notebook_id": "5422DBB7-BC65-4AC5-97CC-14D428E112E8",
   "python_name": "hestonForward",
   "reviewer_id": null,
   "status": "WIP"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}