9 covered_call_strategy_gbm.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "Dk6jYTH3O0zn"
   },
   "source": [
    "# Covered Call Strategy Using GBM"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "gAT1LMewO2aX"
   },
   "source": [
    "Import the packages, functions, and parameters we need for this question."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "ACOCJOJrOjkb",
    "outputId": "d04a612a-0264-4003-be2d-755f702bce8d"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Collecting tabulate\n",
      "  Using cached tabulate-0.9.0-py3-none-any.whl (35 kB)\n",
      "Installing collected packages: tabulate\n",
      "Successfully installed tabulate-0.9.0\n",
      "Note: you may need to restart the kernel to use updated packages.\n"
     ]
    }
   ],
   "source": [
    "import math\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.stats import norm\n",
    "%pip install tabulate\n",
    "from tabulate import tabulate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "id": "Y36onwpuPACO"
   },
   "outputs": [],
   "source": [
    "# Adapted from Octave's financial toolkit\n",
    "from scipy.stats import norm\n",
    "import numpy as np\n",
    "    \n",
    "def blsprice(Price, Strike, Rate, Time, Volatility):\n",
    "    sigma_sqrtT = Volatility * np.sqrt (Time)\n",
    "\n",
    "    d1 = 1 / sigma_sqrtT * (np.log(Price / Strike) + (Rate + Volatility**2 / 2) * Time)\n",
    "    d2 = d1 - sigma_sqrtT\n",
    "\n",
    "    phi1 = norm.cdf(d1)\n",
    "    phi2 = norm.cdf(d2)\n",
    "    disc = np.exp (-Rate * Time)\n",
    "    F    = Price * np.exp ((Rate) * Time)\n",
    "\n",
    "    Call = disc * (F * phi1 - Strike * phi2)\n",
    "    Put  = disc * (Strike * (1 - phi2) + F * (phi1 - 1))\n",
    "    return Call, Put"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "id": "proNyxFgPFQ-"
   },
   "outputs": [],
   "source": [
    "sigma = 0.35    # Volatility\n",
    "r = 0.07        # Risk-free rate\n",
    "T = 1.5         # Time to expiry\n",
    "S0 = 100        # Initial asset price\n",
    "S_init = 100    # Initial investment\n",
    "mu = 0.09       # Real world drift\n",
    "N_sim = 80000   # Number of simulations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "zG1kzkzCP8bu"
   },
   "source": [
    "Write the covered call strategy function under the assumption of GBM."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "id": "2saUQzA4Q5WQ"
   },
   "outputs": [],
   "source": [
    "def Covered_Call(K):\n",
    "    # K is the strike price\n",
    "\n",
    "    S = S_init*np.ones(N_sim)\n",
    "    phi = np.random.normal(0, 1, N_sim)\n",
    "    # Stock price at time T\n",
    "    ST = S*np.exp((mu - (sigma**2/2))*T + sigma*phi*math.sqrt(T))\n",
    "\n",
    "    # Payoff for the call option\n",
    "    Payoff = np.maximum(ST - K, 0)\n",
    "\n",
    "    # Call option price at time 0\n",
    "    C = blsprice(S0, K, r, T, sigma)[0]\n",
    "    # Call option price at time T (value in the risk-free account)\n",
    "    B = C*np.exp(r*T)\n",
    "\n",
    "    # Final value at time T\n",
    "    V = ST + B - Payoff\n",
    "    \n",
    "    # Investment performance (log return)\n",
    "    R = np.log(V/S_init)\n",
    "    R_mean = np.mean(R)\n",
    "    R_std = np.std(R)\n",
    "    R_VAR = np.quantile(R, 0.05)\n",
    "    R_cVAR = R[R<=R_VAR].mean()\n",
    "\n",
    "    return R, R_mean, R_std, R_VAR, R_cVAR"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "6mOafmhCYMTA"
   },
   "source": [
    "Generate a table showing mean, standard deviation, $95\\%$ $VAR$, and $95\\%$ $cVAR$ corresponding to each of the three strike values $K$ of the call."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "YpaEbTpbYaxn",
    "outputId": "f3bd115c-a507-4699-d5b5-6cca5959dd40"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "+----------------+-----------+----------------------+-----------+------------+\n",
      "|   Strike Price |      Mean |   Standard Deviation |   95% VAR |   95% cVAR |\n",
      "+================+===========+======================+===========+============+\n",
      "|            101 | 0.100261  |             0.177812 | -0.28631  |  -0.401497 |\n",
      "+----------------+-----------+----------------------+-----------+------------+\n",
      "|            115 | 0.0939309 |             0.230095 | -0.375178 |  -0.503287 |\n",
      "+----------------+-----------+----------------------+-----------+------------+\n",
      "|            130 | 0.0876373 |             0.276532 | -0.444778 |  -0.585174 |\n",
      "+----------------+-----------+----------------------+-----------+------------+\n"
     ]
    }
   ],
   "source": [
    "# Strike prices\n",
    "K1 = 101\n",
    "K2 = 115\n",
    "K3 = 130\n",
    "\n",
    "# Investment Performances\n",
    "R1 = Covered_Call(K1)\n",
    "R2 = Covered_Call(K2)\n",
    "R3 = Covered_Call(K3)\n",
    "\n",
    "# Data for the table\n",
    "K = [K1, K2, K3]\n",
    "mean = [R1[1], R2[1], R3[1]]\n",
    "std = [R1[2], R2[2], R3[2]]\n",
    "VAR = [R1[3], R2[3], R3[3]]\n",
    "cVAR = [R1[4], R2[4], R3[4]]\n",
    "\n",
    "# Table for F, M, mean, std, VAR, and cVAR\n",
    "table = {\"Strike Price\": K,\n",
    "         \"Mean\": mean,\n",
    "         \"Standard Deviation\": std,\n",
    "         \"95% VAR\": VAR,\n",
    "         \"95% cVAR\": cVAR}\n",
    "print(tabulate(table, headers=\"keys\", tablefmt=\"grid\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "e5pwyWEUa_KY"
   },
   "source": [
    "Generate a plot of the (approximate) probability density function of the log return for the largest strike price."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 472
    },
    "id": "MU7HH15ybHtQ",
    "outputId": "6bc57630-b725-4d67-dc09-c9a72cbe3ddb"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 648x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot of probability function\n",
    "plt.rcParams['figure.figsize'] = [9, 6]\n",
    "plt.hist(R3[0], bins = 80, density = True)\n",
    "plt.title(\"Probability Density Function, K = 130\", fontsize = 12)\n",
    "plt.xlabel(\"Investment Performance\", fontsize = 12)\n",
    "plt.ylabel(\"Probability Density\", fontsize = 12)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "dxLRKVQ2bkQ8"
   },
   "source": [
    "Observation:\n",
    "\n",
    "> According to the table, as $K$ increases, the mean of the log return decreases, the standard devation of the log return increases, the $95\\%$ $VAR$ of the log return decreases, and the $95\\%$ $cVAR$ of the log return decreases.\n",
    "\n",
    "> According to the plot, the probability density function for $K=130$ is heavily left-skewed.\n",
    "\n"
   ]
  }
 ],
 "metadata": {
  "colab": {
   "provenance": []
  },
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.9.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}