Black_Scholes.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "Black-Scholes.ipynb",
      "provenance": [],
      "authorship_tag": "ABX9TyNh475Cfzu9wUulasUle81h"
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "# Options Pricing with Black Scholes\n",
        "\n",
        "**Defining the components of the Black Scholes equation.**"
      ],
      "metadata": {
        "id": "UjjbheNziVTu"
      }
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "XoLTqp42hs6C"
      },
      "outputs": [],
      "source": [
        "def covariance(sigma1,sigma2):\n",
        "    return sqrt(sigma1**2-2*sigma1*sigma2+sigma2**2)\n",
        "\n",
        "def d1(S1,S2,T,r,sigma1,sigma2):\n",
        "    return (log(S1/S2)+(r+(covariance(sigma1,sigma2)**2)/2.)*T)/(covariance(sigma1,sigma2)*sqrt(T))\n",
        "\n",
        "def d2(S1,S2,T,r,sigma1,sigma2):\n",
        "    return (log(S1/S2)+(r-(covariance(sigma1,sigma2)**2)/2.)*T)/(covariance(sigma1,sigma2)*sqrt(T))\n",
        "\n",
        "def bs_call(S1,S2,T,r,sigma1,sigma2):\n",
        "    return S1 * norm.cdf(d1(S1,S2,T,r,sigma1,sigma2)) - S2 * exp(-r * T) * norm.cdf(d2(S1,S2,T,r,sigma1,sigma2))\n",
        "\n",
        "def bs_put(S1,S2,T,r,sigma1,sigma2):\n",
        "    return S2 * exp(-r * T) - S1 + bs_call(S1,S2,T,r,sigma1,sigma2)"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "**Extract share price data from Yahoo Finance**"
      ],
      "metadata": {
        "id": "pljp3z-nEOEY"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "today = datetime.now()\n",
        "one_year_ago = today.replace(year=today.year-1)\n",
        "expiry = '02-25-2021'"
      ],
      "metadata": {
        "id": "hUClonogh4fB"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "stock1 = 'AAPL'"
      ],
      "metadata": {
        "id": "aSQONfSuh5xg"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "dfone = web.DataReader(stock1, 'yahoo', one_year_ago, today)"
      ],
      "metadata": {
        "id": "JAE2egjeh7Xx"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "dfone = dfone.sort_values(by=\"Date\")\n",
        "dfone = dfone.dropna()\n",
        "dfone = dfone.assign(close_day_before=dfone.Close.shift(1))\n",
        "dfone['returns'] = ((dfone.Close - dfone.close_day_before)/dfone.close_day_before)"
      ],
      "metadata": {
        "id": "CBQlIA9Gh8Zt"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "sigma1 = np.sqrt(252) * dfone['returns'].std()\n",
        "lcp1 = dfone['Close'].iloc[-1]"
      ],
      "metadata": {
        "id": "7JxDhtO9iB0Y"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "stock2 = 'GE'"
      ],
      "metadata": {
        "id": "rhKLUBLRiE1G"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "dftwo = web.DataReader(stock2, 'yahoo', one_year_ago, today)"
      ],
      "metadata": {
        "id": "ZWNTrs-JiGDx"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "dftwo = dftwo.sort_values(by=\"Date\")\n",
        "dftwo = dftwo.dropna()\n",
        "dftwo = dftwo.assign(close_day_before=dftwo.Close.shift(1))\n",
        "dftwo['returns'] = ((dftwo.Close - dftwo.close_day_before)/dftwo.close_day_before)"
      ],
      "metadata": {
        "id": "UesYuxpeiH6y"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "sigma2 = np.sqrt(252) * dftwo['returns'].std()\n",
        "lcp2 = dftwo['Close'].iloc[-1]"
      ],
      "metadata": {
        "id": "XIOEny4piJbg"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "uty = web.DataReader(\n",
        "    \"^TNX\", 'yahoo', today.replace(day=today.day-1), today)['Close'].iloc[-1]\n",
        "t = (datetime.strptime(expiry, \"%m-%d-%Y\") - datetime.utcnow()).days / 365"
      ],
      "metadata": {
        "id": "UH7DGOYciSST"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "print('The Option Price is: ', bs_call(lcp1, lcp2, t, uty, sigma1, sigma2))"
      ],
      "metadata": {
        "id": "h1uQloxuiTsx"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "!pip install opstrat"
      ],
      "metadata": {
        "id": "kA_2BdDaa61e"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "import opstrat as op"
      ],
      "metadata": {
        "id": "Ldrd_UFZa9qr"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "op_1 = {'op_type': 'c', 'strike':340, 'tr_type': 'b'}\n",
        "op_2 = {'op_type': 'p', 'strike':340, 'tr_type': 'b'}\n",
        "\n",
        "op.yf_plotter(ticker='GS', exp='2022-04-08', \n",
        "              op_list=[op_1, op_2])"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 441
        },
        "id": "GdKtyc3sbBcm",
        "outputId": "b1d0549b-3a5f-4899-a191-a6e87b22b327"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 720x432 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Portfolio with Options / Dynamic Hedging\n",
        "\n",
        "As we saw from the derivation and the actual Black Scholes model, the partial derivative can tell us how the option value changes when one of these exposures changes holding the others constant.\n",
        "\n",
        "In this example we have a short position in 1000 stock calls, we want to hedge our exposure to changes in volatility, movements in the underlying asset, and the speed of movements in the underlying asset."
      ],
      "metadata": {
        "id": "od97Pa-Qf05-"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import math\n",
        "from scipy.stats import norm"
      ],
      "metadata": {
        "id": "H_BP0ecHfvmv"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Delta: the first-order partial-derivative with respect to the underlying asset of the Black-Scholes equation is known as delta. Delta refers to how the option value changes when there is a change in the underlying asset price. \n",
        "\n",
        "Gamma: the second-order partial-derivative with respect to the underlying asset of the Black-Scholes equation is known as gamma. Gamma refers to how the option’s delta changes when there is a change in the underlying asset price.\n",
        "\n",
        "Vega: The first-order partial-derivative with respect to the underlying asset volatility of the Black-Scholes equation is known as vega. Vega refers to how the option value changes when there is a change in the underlying asset volatility."
      ],
      "metadata": {
        "id": "_EpEBjPLj2U5"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "class EuropeanCall:\n",
        "\n",
        "    def call_delta(\n",
        "        self, asset_price, asset_volatility, strike_price,\n",
        "        time_to_expiration, risk_free_rate\n",
        "            ):\n",
        "        b = math.exp(-risk_free_rate*time_to_expiration)\n",
        "        x1 = math.log(asset_price/(b*strike_price)) + .5*(asset_volatility*asset_volatility)*time_to_expiration\n",
        "        x1 = x1/(asset_volatility*(time_to_expiration**.5))\n",
        "        z1 = norm.cdf(x1)\n",
        "        return z1\n",
        "\n",
        "    def call_gamma(\n",
        "        self, asset_price, asset_volatility, strike_price,\n",
        "        time_to_expiration, risk_free_rate\n",
        "            ):\n",
        "        b = math.exp(-risk_free_rate*time_to_expiration)\n",
        "        x1 = math.log(asset_price/(b*strike_price)) + .5*(asset_volatility*asset_volatility)*time_to_expiration\n",
        "        x1 = x1/(asset_volatility*(time_to_expiration**.5))\n",
        "        z1 = norm.cdf(x1)\n",
        "        z2 = z1/(asset_price*asset_volatility*math.sqrt(time_to_expiration))\n",
        "        return z2\n",
        "\n",
        "    def call_vega(\n",
        "        self, asset_price, asset_volatility, strike_price,\n",
        "        time_to_expiration, risk_free_rate\n",
        "            ):\n",
        "        b = math.exp(-risk_free_rate*time_to_expiration)\n",
        "        x1 = math.log(asset_price/(b*strike_price)) + .5*(asset_volatility*asset_volatility)*time_to_expiration\n",
        "        x1 = x1/(asset_volatility*(time_to_expiration**.5))\n",
        "        z1 = norm.cdf(x1)\n",
        "        z2 = asset_price*z1*math.sqrt(time_to_expiration)\n",
        "        return z2/100\n",
        "\n",
        "    def call_price(\n",
        "        self, asset_price, asset_volatility, strike_price,\n",
        "        time_to_expiration, risk_free_rate\n",
        "            ):\n",
        "        b = math.exp(-risk_free_rate*time_to_expiration)\n",
        "        x1 = math.log(asset_price/(b*strike_price)) + .5*(asset_volatility*asset_volatility)*time_to_expiration\n",
        "        x1 = x1/(asset_volatility*(time_to_expiration**.5))\n",
        "        z1 = norm.cdf(x1)\n",
        "        z1 = z1*asset_price\n",
        "        x2 = math.log(asset_price/(b*strike_price)) - .5*(asset_volatility*asset_volatility)*time_to_expiration\n",
        "        x2 = x2/(asset_volatility*(time_to_expiration**.5))\n",
        "        z2 = norm.cdf(x2)\n",
        "        z2 = b*strike_price*z2\n",
        "        return z1 - z2\n",
        "\n",
        "    def __init__(\n",
        "        self, asset_price, asset_volatility, strike_price,\n",
        "        time_to_expiration, risk_free_rate\n",
        "            ):\n",
        "        self.asset_price = asset_price\n",
        "        self.asset_volatility = asset_volatility\n",
        "        self.strike_price = strike_price\n",
        "        self.time_to_expiration = time_to_expiration\n",
        "        self.risk_free_rate = risk_free_rate\n",
        "        self.price = self.call_price(asset_price, asset_volatility, strike_price, time_to_expiration, risk_free_rate)\n",
        "        self.delta = self.call_delta(asset_price, asset_volatility, strike_price, time_to_expiration, risk_free_rate)\n",
        "        self.gamma = self.call_gamma(asset_price, asset_volatility, strike_price, time_to_expiration, risk_free_rate)\n",
        "        self.vega = self.call_vega(asset_price, asset_volatility, strike_price, time_to_expiration, risk_free_rate)"
      ],
      "metadata": {
        "id": "3yB1FjRRgYjo"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "class EuropeanPut:\n",
        "\n",
        "    def put_delta(\n",
        "        self, asset_price, asset_volatility, strike_price,\n",
        "        time_to_expiration, risk_free_rate\n",
        "            ):\n",
        "        b = math.exp(-risk_free_rate*time_to_expiration)\n",
        "        x1 = math.log(asset_price/(b*strike_price)) + .5*(asset_volatility*asset_volatility)*time_to_expiration\n",
        "        x1 = x1/(asset_volatility*(time_to_expiration**.5))\n",
        "        z1 = norm.cdf(x1)\n",
        "        return z1 - 1\n",
        "\n",
        "    def put_gamma(\n",
        "        self, asset_price, asset_volatility, strike_price,\n",
        "        time_to_expiration, risk_free_rate\n",
        "            ):\n",
        "        b = math.exp(-risk_free_rate*time_to_expiration)\n",
        "        x1 = math.log(asset_price/(b*strike_price)) + .5*(asset_volatility*asset_volatility)*time_to_expiration\n",
        "        x1 = x1/(asset_volatility*(time_to_expiration**.5))\n",
        "        z1 = norm.cdf(x1)\n",
        "        z2 = z1/(asset_price*asset_volatility*math.sqrt(time_to_expiration))\n",
        "        return z2\n",
        "\n",
        "    def put_vega(\n",
        "        self, asset_price, asset_volatility, strike_price,\n",
        "        time_to_expiration, risk_free_rate\n",
        "            ):\n",
        "        b = math.exp(-risk_free_rate*time_to_expiration)\n",
        "        x1 = math.log(asset_price/(b*strike_price)) + .5*(asset_volatility*asset_volatility)*time_to_expiration\n",
        "        x1 = x1/(asset_volatility*(time_to_expiration**.5))\n",
        "        z1 = norm.cdf(x1)\n",
        "        z2 = asset_price*z1*math.sqrt(time_to_expiration)\n",
        "        return z2/100\n",
        "\n",
        "    def put_price(\n",
        "        self, asset_price, asset_volatility, strike_price,\n",
        "        time_to_expiration, risk_free_rate\n",
        "            ):\n",
        "        b = math.exp(-risk_free_rate*time_to_expiration)\n",
        "        x1 = math.log((b*strike_price)/asset_price) + .5*(asset_volatility*asset_volatility)*time_to_expiration\n",
        "        x1 = x1/(asset_volatility*(time_to_expiration**.5))\n",
        "        z1 = norm.cdf(x1)\n",
        "        z1 = b*strike_price*z1\n",
        "        x2 = math.log((b*strike_price)/asset_price) - .5*(asset_volatility*asset_volatility)*time_to_expiration\n",
        "        x2 = x2/(asset_volatility*(time_to_expiration**.5))\n",
        "        z2 = norm.cdf(x2)\n",
        "        z2 = asset_price*z2\n",
        "        return z1 - z2\n",
        "\n",
        "    def __init__(\n",
        "        self, asset_price, asset_volatility, strike_price,\n",
        "        time_to_expiration, risk_free_rate\n",
        "            ):\n",
        "        self.asset_price = asset_price\n",
        "        self.asset_volatility = asset_volatility\n",
        "        self.strike_price = strike_price\n",
        "        self.time_to_expiration = time_to_expiration\n",
        "        self.risk_free_rate = risk_free_rate\n",
        "        self.price = self.put_price(asset_price, asset_volatility, strike_price, time_to_expiration, risk_free_rate)\n",
        "        self.delta = self.put_delta(asset_price, asset_volatility, strike_price, time_to_expiration, risk_free_rate)\n",
        "        self.gamma = self.put_gamma(asset_price, asset_volatility, strike_price, time_to_expiration, risk_free_rate)\n",
        "        self.vega = self.put_vega(asset_price, asset_volatility, strike_price, time_to_expiration, risk_free_rate)"
      ],
      "metadata": {
        "id": "eQvOYCWJgaxm"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "short = EuropeanCall(543, .53, 545, 30/365, .015)\n",
        "print(short.delta*(-1000))\n",
        "print(short.gamma*(-1000))\n",
        "print(short.vega*(-1000))"
      ],
      "metadata": {
        "id": "8HE0uAVqgiBb"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "call_a = EuropeanCall(543, .53, 550, 30/365, .015)\n",
        "print(call_a.delta)\n",
        "print(call_a.gamma)\n",
        "print(call_a.vega)\n",
        "\n",
        "call_b = EuropeanCall(543, .53, 555, 30/365, .015)\n",
        "print(call_b.delta)\n",
        "print(call_b.gamma)\n",
        "print(call_b.vega)"
      ],
      "metadata": {
        "id": "PI6fymQWgvvK"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "**Underlying Asset (NVDA Stock):\n",
        "Delta: 1\n",
        "Gamma: 0\n",
        "Vega: 0**\n",
        "\n",
        "Using a combination of these assets we can neutralize our portfolio’s exposure to delta, gamma, and vega."
      ],
      "metadata": {
        "id": "ZBVTp1UugzlK"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "greeks = np.array([[call_a.gamma, call_b.gamma], [call_a.vega, call_b.vega]])\n",
        "portfolio_greeks = [[short.gamma*1000], [short.vega*1000]]\n",
        "\n",
        "# We need to round otherwise we can end up with a non-invertible matrix\n",
        "inv = np.linalg.inv(np.round(greeks, 2))\n",
        "print(inv)"
      ],
      "metadata": {
        "id": "P8CCEhMvg1N9"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "w = np.dot(inv, portfolio_greeks)\n",
        "print(w)"
      ],
      "metadata": {
        "id": "-7cxfqVdkp0_"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Using these weights we will have effectively neutralized our exposure to gamma and vega…"
      ],
      "metadata": {
        "id": "yP8hFX72kt7L"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "print(np.round(np.dot(np.round(greeks, 2), w) - portfolio_greeks))"
      ],
      "metadata": {
        "id": "Y2GyYVMnkvBU"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "To find our new exposure, we take the sum-product of all option positions in our portfolio with their respective deltas."
      ],
      "metadata": {
        "id": "6_ZzOWaEkzeH"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Greeks including delta\n",
        "portfolio_greeks = [[short_nvda.delta*-1000], [short_nvda.gamma*-1000], [short_nvda.vega*-1000]]\n",
        "greeks = np.array([[call_a.delta, call_b.delta], [call_a.gamma, call_b.gamma], [call_a.vega, call_b.vega]])\n",
        "print(np.round(np.dot(np.round(greeks, 2), w) + portfolio_greeks))"
      ],
      "metadata": {
        "id": "MzSrvD48kwHn"
      },
      "execution_count": null,
      "outputs": []
    }
  ]
}