{
 "metadata": {
  "name": "SABR",
  "zanadu": {
   "authors": [
    "Antoine Jacquier"
   ],
   "category": "6",
   "clearance": "Private",
   "group_name": "Imperial College",
   "md5": "1183684cef13a3fcd64acebbf7f6eeed",
   "notebook_id": "5C5113AE-34DE-4E99-88CD-9599004BF4D2",
   "python_name": null,
   "reviewer_id": null,
   "status": "WIP"
  }
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {
      "zanadu": {}
     },
     "source": "SABR Expansions and arbitrage\n===========\n***"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "import numpy as np\nimport matplotlib.pyplot as plt",
     "language": "python",
     "metadata": {
      "zanadu": {
       "code_type": ""
      }
     },
     "outputs": [],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "zanadu": {}
     },
     "source": "The SABR model is the most popular model on Fixed Income desks. We investigate here some of its pitfalls.\nThe forward process $(F_t)_{t\\geq 0}$ is the unique strong solution to the following stochasic differential equation:\n\\begin{equation*}\n\\begin{array}{rl}\ndF_t & = \\sigma_t F_t^{\\beta}\\left(\\rho dW_t^{1} + \\sqrt{1-\\rho^2}dW_t^{2}\\right),\\\\\nd\\sigma_t & = \\nu \\sigma_t dW_t^{2},\n\\end{array}\n\\end{equation*}\nwith $F_0 = 1$, $\\sigma_0 = \\alpha$, $\\rho \\in [-1,1]$ and $\\beta\\leq 1$.\nWe shall denote by $I_t(x)$, the implied volatility corresponding to European options with maturity $t$ and log strike $x = K/F_0$."
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {
      "zanadu": {}
     },
     "source": "Computing the density"
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "zanadu": {}
     },
     "source": "We compute the density function corresponding to a given smile, at some fixed time horizon $t>0$."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "def density(smile, t, x, eps):\n    ### Computes the density at a point x = log(K/S) given a smile function\n\tw = smile(x)*t\n\twp = smile(x+eps)*smile(x+eps)*t\n\twm = smile(x-eps)*smile(x-eps)*t\n\tw1 = (wp - wm) / (2.*eps)\n\tw2 = (wp - 2.0*w + wm) / (eps*eps)\n\tg = (1. - 0.5*x*w1/w)*(1. - 0.5*x*w1/w) - 0.25*w1*w1*(1./w + 0.25) + 0.5*w2\n\tsqw = np.sqrt(w)\n\treturn g*np.exp(-x-0.5*(-x/sqw - 0.5*sqw)*(-x/sqw - 0.5*sqw)) / np.sqrt(2.*np.pi*w)\n\ndef density2(smile, smileP, smileM, t, x, eps):\n    ### Given three vectors of smile (evaluated at x, e+eps, x-eps), computes the density at x = log(K/S)\n\tw = smile*smile*t\n\twp = smileP*smileP*t\n\twm = smileM*smileM*t\n\tw1 = (wp - wm) / (2.*eps)\n\tw2 = (wp - 2.*w + wm) / (eps*eps)\n\tg = (1. - 0.5*x*w1/w)*(1. - 0.5*x*w1/w) - 0.25*w1*w1*(1./w + 0.25) + 0.5*w2\n\tsqw = np.sqrt(w)\n\treturn g*np.exp(-0.5*(-x/sqw - 0.5*sqw)*(-x/sqw - 0.5*sqw)) / np.sqrt(2.*np.pi*w)\n\ndef plotDensity(smile, smileP, smileM, t, eps, logStrikes, col):\n    ### Plots the density using the function density2 above\n\tmydensity = []\n\tfor n in range(len(logStrikes)):\n\t\tmydensity.append(density2(smile[n], smileP[n], smileM[n], t, logStrikes[n], eps))\n\tplt.plot(logStrikes, mydensity, col, linewidth=2)",
     "language": "python",
     "metadata": {
      "zanadu": {
       "is_hidden": true
      }
     },
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {
      "zanadu": {}
     },
     "source": "Obloj's Expansion"
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "zanadu": {}
     },
     "source": "Jan Obloj proved the following expansion for the implied volatility in the SABR model for a maturity t (where $\\widetilde{x} = -x = \\log(F_0/K)$):\n$$\nI_t(\\widetilde{x})=I^0(\\widetilde{x})\\left(1+I_H^1(\\widetilde{x})t \\right),\n$$\nwhere\n$$\nI_H^1(\\widetilde{x})=\\frac{(\\beta-1)^2}{24} \\frac{y_0^2}{(fK)^{1-\\beta}}+\\frac{1}{4}\\frac{\\rho \\nu y_0 \\beta}{(f K)^{(1-\\beta)/2}}+\\frac{2-3\\rho^2}{24}\\nu^2.\n$$\nAt the money ($x=0$), the zero-order term reads\n$I^0(0)=y_0K^{\\beta-1}$.\nOtherwise, when $\\widetilde{x}\\neq0$, for $\\nu=0$ the zero-order term is\n$I^0(\\widetilde{x})=\\frac{\\widetilde{x} y_0 (1-\\beta)}{x_0^{1-\\beta}-K^{1-\\beta}}$.\nOtherwise, for $\\nu>0$, when $\\beta < 1$, two zero-order terms exist:\n\n- Hagan et al's:\n$$\nI_H^0(\\widetilde{x}) = \\frac{\\nu \\widetilde{x}\\zeta}{z}\\left(\\log\\left(\\frac{\\sqrt{1-2\\rho \\zeta+\\zeta^2}+\\zeta-\\rho}{1-\\rho}\\right)\\right)^{-1},\n$$\nwith\n$$\nz=\\frac{\\nu}{y_0} \\frac{x_0^{1-\\beta}-K^{1-\\beta}}{1-\\beta}, \\quad \\textrm{and} \\quad \\zeta=\\frac{\\nu}{y_0}\\frac{x_0-K}{(x_0-K)^{\\beta/2}};\n$$\n- Berestycki-Busca-Florent's:\n$$\nI_B^0(\\widetilde{x}) = \\nu \\widetilde{x} / \\log\\left(\\frac{\\sqrt{1-2\\rho z+z^2}+z-\\rho}{1-\\rho}\\right),\n$$\nwith $z$ as above. \n\nHence, for $\\beta < 1$ the two different expansions are\n$I_B(\\widetilde{x})=I_B^0\\left(1+I_H^1(\\widetilde{x})t \\right)$,\nand \n$I_H(\\widetilde{x})=I_H^0\\left(1+I_H^1(\\widetilde{x})t \\right)$.\nWe implemented the BBF one one below.\n\nFor $\\beta=1$ and $\\nu>0$, the zero-order term reads\n$$\nI^0(\\widetilde{x}) = \\nu \\widetilde{x} / \\log\\left(\\frac{\\sqrt{1-2\\rho z+z^2}+z-\\rho}{1-\\rho}\\right),\n$$\nwith $z = \\nu x_0 / y_0$."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "## Note that we normalise the stock price to F_0 = 1\n\ndef sabr_vol_HKLW(y0, nu, beta, rho, x, t):\n    ## returns the implied volatility (square root) using Hagan et al.'s expansion \n    ### and the mapping x -> -x\n    K = np.exp(x)\n    b1 = 1.-beta\n    if abs(x)<1E-7:\n        H0 = y0*pow(K,b1)\n    elif nu<0.00000011:\n        H0 = -x*y0*b1/(1. - pow(K,b1))\n    elif beta<0.999:\n        z = (nu/y0)*(1.0 - pow(K,b1))/b1\n        zeta = (nu/y0)*(1.-K)/pow(K, 0.5*beta)\n        xz = np.log((np.sqrt(1.-2.*rho*z+z*z)+z-rho)/(1.-rho))\n        H0 = -(nu*x*zeta/z)/xz\n    else:\n        z = -nu*x/y0\n        xz = np.log((np.sqrt(1.-2.*rho*z+z*z)+z-rho)/(1.-rho))\n        H0 = -nu*x/xz\n    H1 = (b1*b1/24.) * (y0*y0)/pow(K,b1) + (0.25*rho*nu*beta*y0)/(pow(K,(b1)/2.))+ nu*nu*(2.-3.*rho*rho)/24.\n    return np.sqrt(H0*(1. + H1*t))\n\n\ndef sabr_vol_Obloj(y0, nu, beta, rho, x, t):\n    ## returns the implied volatility (square root of Obloj's formula) using Obloj's expansion \n    ### and the mapping x -> -x\n    K = np.exp(x)\n    b1 = 1.-beta\n    if abs(x)<0.000000001:\n        H0 = y0*pow(K,b1)\n    elif nu<0.00000011:\n        H0 = -x*y0*b1/(1.-pow(K,b1))\n    elif beta<0.999:\n        z = (nu/y0)*(1.-pow(K,b1))/b1\n    else:\n        z = -nu/y0*x \n    xz = np.log((np.sqrt(1.-2.*rho*z+z*z)+z-rho)/(1.-rho))\n    H0 = -(nu*x)/xz\n    H1 = (b1*b1/24.) * (y0*y0)/pow(K,b1) + (0.25*rho*nu*beta*y0)/(pow(K,b1/2.))+ nu*nu*(2.-3.*rho*rho)/24.\n    return np.sqrt(H0*(1. + H1*t))",
     "language": "python",
     "metadata": {
      "zanadu": {
       "is_hidden": true
      }
     },
     "outputs": [],
     "prompt_number": 16
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {
      "zanadu": {}
     },
     "source": "Numerics"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "logStrikeMin, logStrikeMax, nbStrikes  = -10., 10., 200\nvMin, vMax = 1E-5, 10.\nnu, beta, y0, rho, t = 0.1, 0.4, 0.25, -0.33, 15.1\n\nlogStrikes = np.linspace(logStrikeMin, logStrikeMax, num=nbStrikes)\noblojformula, hklwformula = [], []\n\nfor k in logStrikes: \n    expansion = sabr_vol_Obloj(y0,nu,beta,rho,k,t)\n    expansion2 = sabr_vol_HKLW(y0,nu,beta,rho,k,t)\n    oblojformula.append(expansion)\n    hklwformula.append(expansion2)\n    \nplt.figure(figsize=(8,4))\nplt.plot(logStrikes, oblojformula, 'g', linewidth=2, label=\"Obloj\")\nplt.plot(logStrikes, hklwformula, 'b', linewidth=2, label=\"Hagan\")\nplt.title('Implied volatilities')\nplt.legend(loc=1)\nplt.show()\n\neps = 1E-4\noblojSmile, oblojSmileP, oblojSmileM, haganSmile, haganSmileP, haganSmileM = [], [], [], [], [], []\n\nfor logStrike in logStrikes:\n    oblojSmile.append(sabr_vol_Obloj(y0,nu,beta,rho,logStrike,t))\n    oblojSmileP.append(sabr_vol_Obloj(y0,nu,beta,rho,(logStrike+eps),t))\n    oblojSmileM.append(sabr_vol_Obloj(y0,nu,beta,rho,(logStrike-eps),t))\n    haganSmile.append(sabr_vol_HKLW(y0,nu,beta,rho,logStrike,t))\n    haganSmileP.append(sabr_vol_HKLW(y0,nu,beta,rho,(logStrike+eps),t))\n    haganSmileM.append(sabr_vol_HKLW(y0,nu,beta,rho,(logStrike-eps),t))\n\nplt.figure(figsize=(8,4))\nplotDensity(oblojSmile, oblojSmileP, oblojSmileM, t, eps, logStrikes, 'b')\nplotDensity(haganSmile, haganSmileP, haganSmileM, t, eps, logStrikes, 'k')\nplt.plot(logStrikes, [0. for k in logStrikes], 'k')\nplt.title('Density')\nplt.show()\n",
     "language": "python",
     "metadata": {
      "zanadu": {}
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
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eey3h4eH069ePMWPGMHTo0ApdZGjboXww6QOcTE4s+nYRL+94udIB21PDhvDp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\ngrOTM9nZ8P33sG2bUevduRO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       "text": "<matplotlib.figure.Figure at 0x7fd59050cbd0>"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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ikJUry/Pcc7qWJgQgLXMhclRiInz4IdSurQV5yZKwYAHs3Zu1II+Li2PUqFEA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\nQoj8Ljg4mODgYON2orKpZs2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       "text": "<matplotlib.figure.Figure at 0x7fd5904d90d0>"
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "",
     "language": "python",
     "metadata": {
      "zanadu": {}
     },
     "outputs": [],
     "prompt_number": 36
    }
   ],
   "metadata": {}
  }
 ]
}