3.3 Pricing with the NIG Process.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Normal Inverse Gaussian Process\n",
    "\n",
    "\n",
    "## Contents\n",
    "   - [Numerical methods for option pricing](#sec1)\n",
    "   - [The NIG PIDE method](#sec2)\n",
    "     - [The Lévy measure](#sec2.1)\n",
    "     - [The PIDE](#sec2.2) \n",
    "\n",
    "The methods used in this notebook are basically the same presented in the notebook **3.2**.     \n",
    "As you probably have noticed, reading the notebook **5.1**, the NIG process is very similar to the VG process.      They are both obtained by Brownian subordination. They both depend on three variables. Their distribution is quite similar.   \n",
    "\n",
    "Of course, the numerical methods for option pricing must be identical as well."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy as scp\n",
    "import scipy.stats as ss\n",
    "import scipy.special as scps\n",
    "import matplotlib.pyplot as plt\n",
    "from functools import partial\n",
    "from scipy.integrate import quad\n",
    "\n",
    "from functions.probabilities import Q1, Q2\n",
    "from functions.CF import cf_NIG"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "T = 1                    # terminal time \n",
    "N= 10000000              # number of generated random variables\n",
    "S0 = 100.0               # initial price\n",
    "K = 100.0                # strike \n",
    "k = np.log(K/S0)         # log moneyness\n",
    "\n",
    "r = 0.1\n",
    "theta = -0.11          # drift of the Brownian motion \n",
    "sigma = 0.2            # volatility of the Brownian motion\n",
    "kappa = 0.3            # variance of the IG process\n",
    "lam = T**2 / kappa     # scale\n",
    "mu_s = T / lam         # scaled mu\n",
    "\n",
    "w = ( 1 - np.sqrt( 1 - 2*theta*kappa -kappa*sigma**2) )/kappa  # Martingale correction\n",
    "dev_X = np.sqrt(sigma**2 + theta**2 * kappa)                   # std dev NIG process"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let me recall the derivation of some formulas:\n",
    "\n",
    "##### Mean and Variance: \n",
    "Using the Brownian subordination expression for the NIG process (where $T_t \\sim IG(t,\\kappa t)$), we get\n",
    "\n",
    "$$ \\mathbb{E}[X_t] = \\mathbb{E}[\\theta T_t + \\sigma W_{T_t}] = \\theta \\mathbb{E}[T_t] + 0 = \\theta t.$$\n",
    "\n",
    "where I used the [Tower property](https://en.wikipedia.org/wiki/Law_of_total_expectation), and\n",
    "\n",
    "$$ \\mbox{Var}[X_t] = \\mbox{Var}[\\theta T_t + \\sigma W_{T_t}] = \\theta^2 \\mbox{Var}[T_t] + \n",
    "\\sigma^2 \\mbox{Var}[W_{T_t}] = (\\theta^2 \\kappa + \\sigma^2) t. $$\n",
    "\n",
    "where for the term $\\mbox{Var}[W_{T_t}]$ I used the [conditional variance](https://en.wikipedia.org/wiki/Law_of_total_variance) formula.\n",
    "\n",
    "##### Martingale correction:\n",
    "\n",
    "The idea is to find the correction term $w$ such that the process $e^{-rt}S_t$ is a martingale. The stock process has dynamics $S_t = S_0 e^{rt - wt + X_t}$.     \n",
    "We have that\n",
    "\n",
    "$$ \\mathbb{E}[ e^{-rt} S_0 e^{rt - wt + X_t} ] = S_0$$\n",
    "\n",
    "$$ \\mathbb{E}[ e^{-wt + X_t}] = 1$$\n",
    "\n",
    "$$ e^{wt} = \\mathbb{E}[ e^{X_t}] $$\n",
    "\n",
    "$$ w = \\log( \\phi(-i) ) \\quad \\quad \\mbox{for} \\quad t=1  $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The martingale correction w=-0.088817 can be computed from the CF as: \n",
      "(-0.088817+0j)\n"
     ]
    }
   ],
   "source": [
    "print(\"The martingale correction w={0:.6f} can be computed from the CF as: \".format(w))\n",
    "print( np.log(cf_NIG( -1j, t=1, mu=0, theta=theta, sigma=sigma, kappa=kappa )).round(6) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec1'></a>\n",
    "\n",
    "# Numerical methods\n",
    "\n",
    "##### Monte Carlo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Monte Carlo, call: 13.429721691441594, std err: 0.0049326098983063895\n",
      "Monte Carlo, put: 3.915092184682041, std_err: 0.0025466749195472587\n"
     ]
    }
   ],
   "source": [
    "np.random.seed(seed=42) \n",
    "IG = ss.invgauss.rvs(mu=mu_s, scale=lam, size=N)    # The IG RV\n",
    "Norm = ss.norm.rvs(0,1,N)                           # The normal RV  \n",
    "X = theta * IG + sigma * np.sqrt(IG) * Norm         # NIG random vector\n",
    "S_T = S0 * np.exp( (r-w)*T + X )                    # exponential dynamics\n",
    "\n",
    "call_payoff = np.maximum(S_T-K,0)\n",
    "put_payoff = np.maximum(K-S_T,0)\n",
    "call = np.exp(-r*T) * scp.mean( call_payoff )\n",
    "put = np.exp(-r*T) * scp.mean( put_payoff )\n",
    "call_err = np.exp(-r*T) * ss.sem( call_payoff )\n",
    "put_err = np.exp(-r*T) * ss.sem( put_payoff )\n",
    "print(\"Monte Carlo, call: {}, std err: {}\".format(call,call_err))\n",
    "print(\"Monte Carlo, put: {}, std_err: {}\".format(put,put_err))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Fourier inversion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Fourier inversion call:  13.427965\n",
      "Fourier inversion put:  3.911707\n"
     ]
    }
   ],
   "source": [
    "# function binding\n",
    "cf_NIG_b = partial(cf_NIG, t=T, mu=r-w, theta=theta, sigma=sigma, kappa=kappa ) \n",
    "\n",
    "# price\n",
    "call_F = S0 * Q1(k, cf_NIG_b, np.inf) - K * np.exp(-r*T) * Q2(k, cf_NIG_b, np.inf)\n",
    "put_F = -S0 * (1-Q1(k, cf_NIG_b, np.inf)) + K * np.exp(-r*T) * (1-Q2(k, cf_NIG_b, np.inf))\n",
    "\n",
    "print(\"Fourier inversion call: \", call_F.round(6))\n",
    "print(\"Fourier inversion put: \", put_F.round(6))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec2'></a>\n",
    "\n",
    "# The NIG PIDE method \n",
    "\n",
    "<a id='sec2.1'></a>\n",
    "## Curiosities about the NIG Lévy measure\n",
    "\n",
    "The NIG process is a pure jump process with triplet $(b,0,\\nu)$. (For more information on the Lévy triplet see Appendix **A.3**)\n",
    "\n",
    "The expression of the drift $b$ in [2] and [3] is given for the parameters:\n",
    "\n",
    "$$ \\beta = \\frac{\\theta}{\\sigma^2}, \\quad \\quad \\alpha = \\sqrt{ \\beta^2 + \\frac{1}{\\kappa \\sigma^2} } \\quad \\quad\n",
    " \\delta = \\frac{T \\sigma}{\\sqrt{\\kappa}} $$\n",
    "\n",
    "and is:\n",
    "\n",
    "$$ b = \\frac{2 \\delta \\alpha}{\\pi} \\int_0^1 \\sinh(\\beta x) K_1(\\alpha x) dx $$\n",
    "\n",
    "The Lévy measure is:\n",
    "\n",
    "$$ \\nu(x) = \\frac{C}{|x|} e^{A x} K_1(B|x|) $$\n",
    "\n",
    "with\n",
    "\n",
    "$$ A = \\frac{\\theta}{\\sigma^2} \\quad B = \\frac{\\sqrt{\\theta^2 + \\sigma^2/\\kappa}}{\\sigma^2} \n",
    "\\quad C = \\frac{\\sqrt{\\theta^2 + \\sigma^2/\\kappa}}{\\sigma \\pi \\sqrt{\\kappa}} $$\n",
    "\n",
    "The function $K_1$ is a Bessel function of second kind with 1 degree of freedom.\n",
    "\n",
    "Let us recall from the Levy processes theory that:\n",
    "\n",
    "$$ \\mathbb{E}[X_t] = t \\biggl( b +  \\int_{|x|>1} x \\nu(dx) \\biggr)$$\n",
    "\n",
    "$$ \\text{Var}[X_t] = t \\int_{\\mathbb{R}} x^2 \\nu(dx) $$\n",
    "\n",
    "Let's verify it is correct:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "def NIG_measure(x):\n",
    "    A = theta/(sigma**2)\n",
    "    B = np.sqrt( theta**2 + sigma**2/kappa ) / sigma**2\n",
    "    C = np.sqrt(theta**2 + sigma**2/kappa) /(np.pi*sigma * np.sqrt(kappa))\n",
    "    return C/np.abs(x) * np.exp(A*(x)) * scps.kv(1, B*np.abs(x) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.concatenate( (np.linspace(-0.1,-0.001,200), np.linspace(0.001, 0.1, 200) ))\n",
    "plt.plot(x, NIG_measure(x)); plt.title(\"NIG Lévy measure\"); plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### mean:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The theoretical mean is theta:  -0.11\n",
      "The drift component is:  -0.10992734491736127\n",
      "The mean is the sum of the drift and the integration over |x>1| i.e. :  -0.11000000000015916\n"
     ]
    }
   ],
   "source": [
    "beta = theta/(sigma**2)\n",
    "alpha = np.sqrt( beta**2 + 1/(kappa*sigma**2) )\n",
    "delta = sigma/np.sqrt(kappa)\n",
    "int_b = lambda y: 2/np.pi * delta * alpha * np.sinh(beta*y) * scps.kv(1, alpha*np.abs(y) ) \n",
    "drift = quad(int_b, 0, 1, points=0, limit=2000)[0]\n",
    "\n",
    "print(\"The theoretical mean is theta: \", theta)\n",
    "print(\"The drift component is: \", drift )\n",
    "\n",
    "mean_greater1 = quad(int_m, -10, -1, limit=2000)[0] + quad(int_m, 1, 10, limit=2000)[0]\n",
    "\n",
    "print(\"The mean is the sum of the drift and the integration over |x>1| i.e. : \", drift + mean_greater1 )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### variance:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The variance obtained from integration of Lévy mesure is:  0.04363\n",
      "The theoretical variance is:  0.04363\n"
     ]
    }
   ],
   "source": [
    "int_s = lambda y: y**2 * NIG_measure(y)\n",
    "var_int = quad(int_s, -2, 2, points=0)[0]             # I inform that the point 0 has a singularity\n",
    "var_th = dev_X**2\n",
    "print(\"The variance obtained from integration of Lévy mesure is: \", round(var_int,6))\n",
    "print(\"The theoretical variance is: \", var_th.round(7))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Martingale correction:\n",
    "\n",
    "The martingale correction can be computed from the Lévy measure as well:\n",
    "\n",
    "$$ w = \\int_{\\mathbb{R} \\backslash [-\\epsilon,\\epsilon]} (e^z-1) \\nu(dz). $$\n",
    "\n",
    "we have to choose an $\\epsilon > 0$ very small, but it cannot be zero. The reason is that the NIG process has infinite variation (see **A.3** for the definition) and this means that the integral above does not converge for $\\epsilon = 0$. \n",
    "\n",
    "**Comment:**     \n",
    "For infinite variation processes we have that:\n",
    "\n",
    "$ \\int_{[-\\epsilon,\\epsilon]} (e^z-1) \\nu(dz) \\approx \\int_{[-\\epsilon,\\epsilon]} z \\nu(dz) = \\infty. $\n",
    "\n",
    "This is the main difference with the VG process, which instead has finite variation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Martingale correction:  -0.088817\n",
      "Martingale correction from integration:  -0.088817\n"
     ]
    }
   ],
   "source": [
    "epsilon = 0.0000001\n",
    "int_w = lambda y: (np.exp(y)-1) * NIG_measure(y)\n",
    "w2 = quad(int_w, -10, -epsilon)[0] + quad(int_w, epsilon, 10)[0]\n",
    "print(\"Martingale correction: \", w.round(6))\n",
    "print(\"Martingale correction from integration: \", round(w2,6) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec2.2'></a>\n",
    "## The NIG PIDE\n",
    "\n",
    "The NIG PIDE is \n",
    "\n",
    "$$\n",
    "\\frac{\\partial V(t,x)}{\\partial t} \n",
    "          + \\biggl( r - \\int_{\\mathbb{R}} \\bigl( e^z-1-z \\mathbb{1}_{|z|<\\epsilon} \\bigr) \\nu(dz) \\biggr) \\frac{\\partial V(t,x)}{\\partial x} $$\n",
    "$$          + \\int_{\\mathbb{R}} \\bigl( V(t,x+z)- V(t,x) - z \\frac{\\partial V(t,x) }{\\partial x} \\mathbb{1}_{|z|<\\epsilon} \\bigr) \\nu(dz)  = r V(t,x).\n",
    "$$\n",
    "\n",
    "for a small value of $\\epsilon$ we can use the Brownian approximation described in **3.2** and **A.3**.      \n",
    "The jump-diffusion type PIDE is:\n",
    "\n",
    "$$\n",
    "  \\frac{\\partial V(t,x)}{\\partial t} +\n",
    " \\bigl( r-\\frac{1}{2}\\sigma_{\\epsilon}^2 - w_{\\epsilon} \\bigr) \\frac{\\partial V(t,x)}{\\partial x} \n",
    " + \\frac{1}{2}\\sigma_{\\epsilon}^2 \\frac{\\partial^2 V(t,x)}{\\partial x^2}\n",
    " + \\int_{|z| \\geq \\epsilon} V(t,x+z) \\nu(dz) = (\\lambda_{\\epsilon} + r) V(t,x).\n",
    "$$\n",
    "\n",
    "with parameters   \n",
    "\n",
    "$$\n",
    "  \\sigma_{\\epsilon}^2 :=  \\int_{|z| < \\epsilon} z^2 \\nu(dz), \\quad \\quad w_{\\epsilon} := \\int_{|z| \\geq \\epsilon} (e^z-1) \\nu(dz), \\quad \\quad\n",
    " \\lambda_{\\epsilon} :=  \\int_{|z| \\geq \\epsilon} \\nu(dz) .\n",
    "$$\n",
    "\n",
    "The previous approximated equation is identical to the VG equation.     \n",
    "I will not repeat the calculation. I will just present some numerical values and some plots.\n",
    "\n",
    "**Comment:**      \n",
    "I would like to point out that the convergence to the right price is quite slow, in particular for small values of $\\kappa$.     \n",
    "We have seen a similar behavior for the VG PIDE, but in the NIG PIDE is a bit worse.      \n",
    "The algorithm works very well for PUT options, while for CALL options it is hard to obtain good results under our grid resolution.    \n",
    "In order to improve the performances I also used a big computational domain $[A_1,A_2]$, but the improvements are still not satisfactory. \n",
    "I expect that for a higher grid resolution (in particular for more time steps), we can achieve better performances."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from functions.Parameters import Option_param\n",
    "from functions.Processes import NIG_process\n",
    "import functions.NIG_pricer as NIG"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "opt_param = Option_param(S0=S0, K=K, T=T, exercise=\"European\", payoff=\"call\" )\n",
    "opt_param2 = Option_param(S0=S0, K=K, T=T, exercise=\"European\", payoff=\"put\" )\n",
    "NIG_param = NIG_process(r=r, sigma=sigma, theta=theta, kappa=kappa)\n",
    "\n",
    "NIG_c = NIG.NIG_pricer(opt_param, NIG_param)\n",
    "NIG_p = NIG.NIG_pricer(opt_param2, NIG_param)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### European Call"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(13.38723669349473, 251.3608090877533)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "NIG_c.PIDE_price((22000,50000), Time=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "NIG_c.plot([50,200,0,100])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### European Put"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3.9110477337157423, 152.98903918266296)"
      ]
     },
     "execution_count": 106,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "NIG_p.PIDE_price((20000,30000), Time=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "NIG_p.plot([20,150,0,80])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### American Put"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {},
   "outputs": [],
   "source": [
    "opt_param3 = Option_param(S0=S0, K=K, T=T, exercise=\"American\", payoff=\"put\" )\n",
    "NIG_am = NIG.NIG_pricer(opt_param3, NIG_param)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(5.06447605936324, 16.304216861724854)"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "NIG_am.PIDE_price((10000,10000), Time=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "NIG_am.plot([50,150,0,60])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## References\n",
    "\n",
    "[1] Rama Cont and Peter Tankov (2003) \"Financial Modelling with Jump Processes\", Chapman and Hall/CRC; 1 edition.  \n",
    "\n",
    "[2] Rydberg Tina (1997) \"The Normal Inverse Gaussian Lévy Process:\n",
    "Simulation and Approximation\",  Communications in Statistics. Stochastic Models. 13:4, 887-910\n",
    "\n",
    "[3] Barndorff-Nielsen, Ole E. (1998) \"Processes of Normal Inverse Gaussian type\" Finance and Stochastics 2, 41-68."
   ]
  }
 ],
 "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": "3.6.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}