Part IV - Dynamic Hedging.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "qoXhfECLXpxy"
   },
   "source": [
    "# QF620 Project Part IV - Dynamic Hedging"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Group 8**\\\n",
    "Ee Jing Michelle\\\n",
    "Ishani Maitra\\\n",
    "Jermayne Lim Jie Min\\\n",
    "Lim Fang Yi\\\n",
    "Muhammad Saqif Bin Juhaimee\\\n",
    "Rohen S/O Veera Kumaran"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "id": "4emKrv4V429h"
   },
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "from matplotlib.ticker import PercentFormatter\n",
    "\n",
    "from scipy.stats import norm\n",
    "\n",
    "import warnings\n",
    "warnings.filterwarnings(\"ignore\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "wtk-rwpaWYXf"
   },
   "source": [
    "### Initialize Parameters\n",
    "\n",
    "\n",
    "*   Set $S_0 = 100$, $\\sigma = 0.2$, $r = 0.05$, $T = \\frac{1}{12}$, $K = 100$, $N = 21$, $N = 84$\n",
    "*   Simulate 50,000 paths for $S_t$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "id": "tXoanYGlmKpQ"
   },
   "outputs": [],
   "source": [
    "# Parameters\n",
    "np.random.seed(620)\n",
    "\n",
    "S0 = 100\n",
    "sigma = 0.2\n",
    "r = 0.05\n",
    "T = 1/12     # 1 month\n",
    "K = 100\n",
    "\n",
    "N1 = 21      # Daily Hedging\n",
    "N2 = 84      # 4x Daily Hedging\n",
    "\n",
    "paths = 50000"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "1XOQkMFOY2Tl"
   },
   "source": [
    "### Simulate Stock Price Paths\n",
    "\n",
    "Using Brownian Motion to simulate stock prices:\n",
    "\n",
    "  \\begin{equation*}\n",
    "    S_{t+\\Delta t} = S_t e^{(r-\\frac{1}{2}σ^2)Δt+σ \\sqrt{Δt}Z}\n",
    "  \\end{equation*}\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "id": "wQtKxbBPFwE9"
   },
   "outputs": [],
   "source": [
    "# Brownian Motion\n",
    "def simulate_Brownian_Motion(paths, steps, T):\n",
    "    deltaT = T/steps\n",
    "    t = np.linspace(0, T, steps+1)\n",
    "    X = np.c_[np.zeros((paths, 1)),\n",
    "              np.random.randn(paths, steps)]\n",
    "    return t, np.cumsum(np.sqrt(deltaT) * X, axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "id": "fw5R6aedlZTs"
   },
   "outputs": [],
   "source": [
    "# Black-Scholes Model\n",
    "def BlackScholesCall(S, K, r, sigma, T):\n",
    "    d1 = (np.log(S/K) + (r+sigma**2 /2)*T) / (sigma*np.sqrt(T))\n",
    "    d2 = d1 - sigma*np.sqrt(T)\n",
    "    return S*norm.cdf(d1) - K*np.exp(-r*T)*norm.cdf(d2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "id": "birvCOWFliHG"
   },
   "outputs": [],
   "source": [
    "# Stock Price Simulation\n",
    "def simulate_Stock_Price(steps):\n",
    "    t, Wt = simulate_Brownian_Motion(paths, steps, 1/12)\n",
    "    t = np.linspace(0,T,steps+1)\n",
    "    S = S0*np.exp((r-(sigma**2)/2)*t+sigma*Wt)\n",
    "    return S"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "id": "9oTrQmxZCZjm"
   },
   "outputs": [],
   "source": [
    "# Simulate Stock Paths\n",
    "stockpaths21 = simulate_Stock_Price(N1)\n",
    "stockpaths84 = simulate_Stock_Price(N2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "isIhmDRcafBR"
   },
   "source": [
    "### Delta Hedging Strategy\n",
    "\n",
    "The hedged portfolio is $V_t=\\phi_tS_t+\\psi_tB_t$, where\n",
    "\n",
    "\\begin{equation*}\n",
    "\\phi_t=\\Delta_t=\\frac{\\partial C}{\\partial S}=\\Phi\\left(\\frac{\\log\\frac{S}{K}+\\left(r+\\frac{\\sigma^2}{2}\\right)T}{\\sigma\\sqrt{T}}\\right)\n",
    "\\end{equation*}\n",
    "\n",
    "and\n",
    "\n",
    "\\begin{equation*}\n",
    "\\psi_tB_t=-Ke^{-rT}\\Phi\\left(\\frac{\\log\\frac{S}{K}+\\left(r-\\frac{\\sigma^2}{2}\\right)T}{\\sigma\\sqrt{T}}\\right)\n",
    "\\end{equation*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "id": "56XIqZjzWdTp"
   },
   "outputs": [],
   "source": [
    "# Stock and bond holding functions\n",
    "def phi(S, K, r, sigma, T):\n",
    "    d1 = (np.log(S / K) + (r + sigma**2 / 2) * T) / (sigma * np.sqrt(T))\n",
    "    return norm.cdf(d1)\n",
    "\n",
    "def psi_Bt(S, K, r, sigma, T):\n",
    "    d1 = (np.log(S / K) + (r + sigma**2 / 2) * T) / (sigma * np.sqrt(T))\n",
    "    d2 = d1 - sigma * np.sqrt(T)\n",
    "    return -K * np.exp(-r * T) * norm.cdf(d2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "id": "xSu6vNHvcSRt"
   },
   "outputs": [],
   "source": [
    "# Delta Hedging Function\n",
    "def delta_hedging(stock_paths, steps, r, sigma, K, T):\n",
    "\n",
    "    dt = T / steps\n",
    "    t_grid = np.linspace(0, T, steps + 1)\n",
    "    paths = stock_paths.shape[0]\n",
    "\n",
    "    # Initialize cumulative hedging errors\n",
    "    errors = np.zeros(paths)\n",
    "\n",
    "    # Iterate over time steps\n",
    "    for i in range(steps):\n",
    "        St = stock_paths[:, i]               # Stock prices at time t\n",
    "        St_1 = stock_paths[:, i + 1]         # Stock prices at time t+1\n",
    "        deltaT = T - t_grid[i]               # Time to maturity from current step\n",
    "        changeT = t_grid[i + 1] - t_grid[i]  # Time difference between steps\n",
    "\n",
    "        # Delta (φ)\n",
    "        phi_t = phi(St, K, r, sigma, deltaT)\n",
    "\n",
    "        # Bond Position (ψ)\n",
    "        psi = psi_Bt(St, K, r, sigma, deltaT)\n",
    "\n",
    "        # Update Hedging Error\n",
    "        errors += (St_1 * phi_t + psi * np.exp(r * changeT)) - (St * phi_t + psi)\n",
    "\n",
    "    # Add Initial Option Price\n",
    "    C0 = phi(S0, K, r, sigma, T) * S0 + psi_Bt(S0, K, r, sigma, T)\n",
    "    errors += C0\n",
    "\n",
    "    # Subtract Terminal Payoff\n",
    "    errors -= np.maximum(stock_paths[:, -1] - K, 0)\n",
    "\n",
    "    return errors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "OAdSHqXFdFn1"
   },
   "source": [
    "### Hedging Errors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 564
    },
    "id": "Qs9RQulub-Nf",
    "outputId": "c09584f8-f5b7-4b8a-e386-2914cb3ea69c"
   },
   "outputs": [
    {
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",
      "text/plain": [
       "<Figure size 1200x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Calculate hedging errors\n",
    "errors21 = delta_hedging(stockpaths21, N1, r, sigma, K, T)\n",
    "errors84 = delta_hedging(stockpaths84, N2, r, sigma, K, T)\n",
    "\n",
    "# Plot histograms\n",
    "plt.figure(figsize=(12, 6))\n",
    "plt.hist(errors21, bins=100, alpha=0.6, label=\"N = 21\")\n",
    "plt.hist(errors84, bins=100, alpha=0.6, label=\"N = 84\")\n",
    "plt.xlabel(\"Hedging Error\")\n",
    "plt.ylabel(\"Frequency\")\n",
    "plt.title(\"Histogram of Hedging Errors\")\n",
    "plt.legend()\n",
    "plt.grid(True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 125
    },
    "id": "xMBrNL2Zveqo",
    "outputId": "173638a1-d307-40a6-d5ad-e15333cd6ec5"
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Hedging Frequency</th>\n",
       "      <th>Mean</th>\n",
       "      <th>Standard Deviation</th>\n",
       "      <th>Standard Deviation (% of Initial Option Price)</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>N=21</td>\n",
       "      <td>-0.002597</td>\n",
       "      <td>0.430180</td>\n",
       "      <td>17.124551</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>N=84</td>\n",
       "      <td>-0.001248</td>\n",
       "      <td>0.216545</td>\n",
       "      <td>8.620199</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "  Hedging Frequency      Mean  Standard Deviation  \\\n",
       "0              N=21 -0.002597            0.430180   \n",
       "1              N=84 -0.001248            0.216545   \n",
       "\n",
       "   Standard Deviation (% of Initial Option Price)  \n",
       "0                                       17.124551  \n",
       "1                                        8.620199  "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Compute initial option price\n",
    "C0 = BlackScholesCall(S0, K, r, sigma, T)\n",
    "\n",
    "# Compute statistics\n",
    "summary =\\\n",
    "(\n",
    "  pd\n",
    "  .DataFrame({\"Hedging Frequency\": [\"N=21\", \"N=84\"],\n",
    "              \"Mean\": [np.mean(errors21), np.mean(errors84)],\n",
    "              \"Standard Deviation\": [np.std(errors21), np.std(errors84)],\n",
    "              \"Standard Deviation (% of Initial Option Price)\": [np.std(errors21) / C0 * 100,\n",
    "                                                                 np.std(errors84) / C0 * 100\n",
    "                                                                 ]\n",
    "              })\n",
    ")\n",
    "\n",
    "summary"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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