Lecture 07_Introduction to Statistical Dependence and Copulas for Finance.ipynb

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    "# <center><font color=navy>Data Science for Quantitative Finance</font></center>\n",
    "## <center><font color=navy>Introduction to Statistical Dependence and Copulas for Finance</font> </center>\n",
    "### <center> Ali Habibnia</center>\n",
    "\n",
    "    \n",
    "<center> Assistant Professor, Department of Economics, </center>\n",
    "<center> and Division of Computational Modeling & Data Analytics at Virginia Tech</center>\n",
    "<center> habibnia@vt.edu </center> "
   ]
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    "### Readings:\n",
    "\n",
    "1. ***Chapter 8***, [Financial Signal Processing and Machine Learning](https://ieeexplore.ieee.org/document/7471040) pdf file is available on Canvas\n",
    "\n",
    "2. [Statistical Dependence: Beyond Pearson’s ρ ](https://projecteuclid.org/journals/statistical-science/volume-37/issue-1/Statistical-Dependence-Beyond-Pearsons-%CF%81/10.1214/21-STS823.full)"
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    "To many scientists and researchers, measuring dependence is synonymous with correlation and regression originated by Galton in 1888. However, in the 18th century over 100 years before Sir Francis Galton and Karl Pearson’s preoccupation with co-relation and correlation, various authors such as Thomas Bayes (in his 1763 paper) used verbal definitions of (in)dependence on occasion in their writings. "
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    "\n",
    "###  $$\\text{ “Anything that relies on correlation is charlatanism” (Nassim Taleb)}$$\n",
    "\n",
    "\n",
    "#### Pearson’s correlation coefficient\n",
    "\n",
    "The most common measure of correlation in stats is the Pearson Correlation. The full name is the Pearson Product Moment Correlation (PPMC). It shows the linear relationship between two sets of data. In simple terms, it answers the question, Can I draw a line graph to represent the data? Two letters are used to represent the Pearson correlation: Greek letter rho (ρ) for a population and the letter “r” for a sample.\n",
    "\n",
    "$$r_{xy}=\\frac{s_{xy}}{s_xs_y}$$\n",
    "\n",
    "$s_{x}$ and $s_{y}$ are the sample standard deviations, and $s_{xy}$ is the sample covariance.\n",
    "\n",
    "The explicit formula is\n",
    "\n",
    "$$r_{xy}=\\frac{n\\Sigma x_iy_i-\\Sigma x_i\\Sigma y_i}{\\sqrt{n\\Sigma x_i^2-(\\Sigma x_i)^2}\\sqrt{n\\Sigma y_i^2-(\\Sigma y_i)^2})}$$\n",
    "\n",
    "\n",
    "\n",
    "#### Limitations of the Pearson Correlation Coefficient:\n",
    "\n",
    "1. Linearity\n",
    "\n",
    "    a zero correlation does not necessarily imply independence of X and Y. Example: $Y = X^2$\n",
    "\n",
    "\n",
    "2. Non-robustness with respect to data distribution\n",
    "\n",
    "    only the first two moments are incorporated in the Pearson correlation coefficient and uncorrelatedness does not imply independence for some elliptical distributions or non-elliptical distributions. Pearson uncorrelatedness only implies independence if the joint distribution of the variables is normal.\n",
    "\n",
    "\n",
    "3. Stationarity and Heteroskedasticity\n",
    "    \n",
    "    compute correlation on return series not on prices\n",
    "\n",
    "\n",
    "4. Non-robustness with respect to outliers\n",
    "\n",
    "    Pearson coefficient applies the mean as a measure for the central tendency.\n",
    "\n",
    "\n",
    "5. Time frame sensitivity\n",
    "\n",
    "    Pearson is static and Short-term correlation can be totally different from the long-term correlation. Correlations typically increase in economic recessions.\n",
    "\n",
    "\n",
    "6. The correlation-causality fallacy\n",
    "\n",
    "7. symmetric"
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.         -0.00952015]\n",
      " [-0.00952015  1.        ]]\n"
     ]
    }
   ],
   "source": [
    "x = pd.Series(np.random.randn(10000))\n",
    "y = x**2\n",
    "plt.scatter(x, y)\n",
    "plt.show()\n",
    "Pearson_rho = np.corrcoef(x, y)\n",
    "print(Pearson_rho)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "26aea92c",
   "metadata": {},
   "outputs": [],
   "source": [
    "!pip install yfinance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "571a889f",
   "metadata": {},
   "outputs": [],
   "source": [
    "!pip install missingno"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "e792c8d3",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from pandas_datareader import data\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import missingno as msno\n",
    "import datetime\n",
    "import random\n",
    "import yfinance as yf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "5ba07a45",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[*********************100%***********************]  2 of 2 completed\n"
     ]
    },
    {
     "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>BTC-USD</th>\n",
       "      <th>^GSPC</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Date</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>2022-03-26</th>\n",
       "      <td>44500.828125</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2022-03-27</th>\n",
       "      <td>46820.492188</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2022-03-28</th>\n",
       "      <td>47128.003906</td>\n",
       "      <td>4575.520020</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2022-03-29</th>\n",
       "      <td>47465.730469</td>\n",
       "      <td>4631.600098</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2022-03-30</th>\n",
       "      <td>47062.664062</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                 BTC-USD        ^GSPC\n",
       "Date                                 \n",
       "2022-03-26  44500.828125          NaN\n",
       "2022-03-27  46820.492188          NaN\n",
       "2022-03-28  47128.003906  4575.520020\n",
       "2022-03-29  47465.730469  4631.600098\n",
       "2022-03-30  47062.664062          NaN"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 1800x720 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Correlation and roaling correlation between S&P 500 Index and Bitcoin\n",
    "sample = ['BTC-USD', '^GSPC']\n",
    "\n",
    "Pr = yf.download(sample, start='2018-01-01', end='2022-03-30')['Adj Close']\n",
    "msno.matrix(Pr)\n",
    "Pr.tail()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "91ce9628",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 1800x720 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Clean before calculating return values\n",
    "cleaned_Pr = Pr.copy()\n",
    "\n",
    "# Remove days (rows) with missing values for any of assets\n",
    "cleaned_Pr.dropna(axis='index', how='any', inplace=True)\n",
    "\n",
    "cleaned_Pr.tail()\n",
    "msno.matrix(cleaned_Pr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "4a0257f4",
   "metadata": {},
   "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>BTC-USD</th>\n",
       "      <th>^GSPC</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Date</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>2022-03-25</th>\n",
       "      <td>0.008783</td>\n",
       "      <td>0.005053</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2022-03-28</th>\n",
       "      <td>0.060783</td>\n",
       "      <td>0.007120</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2022-03-29</th>\n",
       "      <td>0.007141</td>\n",
       "      <td>0.012182</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "             BTC-USD     ^GSPC\n",
       "Date                          \n",
       "2022-03-25  0.008783  0.005053\n",
       "2022-03-28  0.060783  0.007120\n",
       "2022-03-29  0.007141  0.012182"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Re = np.log(cleaned_Pr) - np.log(cleaned_Pr.shift(1))\n",
    "Re = Re[1:]\n",
    "Re.tail(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "87b2e635",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:xlabel='Date'>"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "cleaned_Pr.plot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "1d69268f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:xlabel='Date'>"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "Re.plot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "id": "d8fbf836",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1.         0.24164016]\n",
      " [0.24164016 1.        ]]\n"
     ]
    }
   ],
   "source": [
    "# Pearson correlation for the entire data\n",
    "\n",
    "s1 = pd.Series(Re['BTC-USD'].values)\n",
    "s2 = pd.Series(Re['^GSPC'].values)\n",
    "Pearson_rho = np.corrcoef(s1, s2)\n",
    "print(Pearson_rho)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "id": "92f39683",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.16271877709550786\n",
      "0.2207956763495446\n"
     ]
    },
    {
     "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": [
    "#Rolling correlations \n",
    "\n",
    "s1 = pd.Series(Re['BTC-USD'].values)\n",
    "s2 = pd.Series(Re['^GSPC'].values)\n",
    "n = 150\n",
    "s1.rolling(n).corr(s2).plot()\n",
    "print(s1.rolling(n).corr(s2).mean())\n",
    "print(s1.rolling(n).corr(s2).std())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "f5cbec3e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Confidence Interval:  [0.12691264 0.35713291]\n",
      "Standard Error:  0.06\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYwAAAEGCAYAAAB2EqL0AAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjQuMywgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/MnkTPAAAACXBIWXMAAAsTAAALEwEAmpwYAAAbwUlEQVR4nO3de5QU5Z3/8ffXYUBhWLkZHC4yKCCim7Q6GsVLNP4Myq5glBhjVFQUE0nEy+8kXjYxK+svJPGCoqshMRvdXIyrMXJcNVGQlawxiNigoOiADBdHVLzgiOIw8/390TXYM8zl6Zmu7hr4vM6ZM9XP81T3t+rQ8+Gpqq42d0dERKQ9uxW7ABER6RoUGCIiEkSBISIiQRQYIiISRIEhIiJBuhW7gM4YMGCAV1RUFLuM+K1cmfm9//7FrUOkmPQ+yJvnn3/+HXffK9f1unRgVFRUsHjx4mKXEb/jjsv8XrCgmFWIFJfeB3ljZtUdWU+HpEREJIgCQ0REgigwREQkiAJDRESCKDBERCSIAkNERIIoMEREJIgCQ0REgigwREQkSJf+pLdIPs2tndtq34SyCQWsRCSZNMMQEZEgCgwREQmiQ1IigXTISnZ1mmGIiEgQBYaIiARRYIiISBAFhoiIBIktMMxsdzNbZGZLzWy5mf1r1D7czP5uZlVm9gcz6x6194geV0X9FXHVJiIiuYtzhrEV+LK7fwFIASeZ2RHAT4Bb3H0E8B4wJRo/BXgvar8lGiciIgkRW2B4Rm30sDT6ceDLwANR+z3AqdHyxOgxUf8JZmZx1SciIrmJ9XMYZlYCPA+MAO4AVgHvu/u2aMh6YHC0PBhYB+Du28zsA6A/8E6z55wKTAUoLy8nnU7HuQmJMKI2k7tVu8C2FpPXeat96dJ0u/0SL70Pii/WwHD3eiBlZn2Ah4DReXjOOcAcgMrKSk+lUp19yuQrKwNgl9jWIlpbu7bVvlRZqt1+iZneB0VXkE96u/v7ZvYUcCTQx8y6RbOMIcCGaNgGYCiw3sy6AXsCmwpRn3R97X0KW5/SFum8OK+S2iuaWWBmewAnAi8DTwGTomGTgYej5bnRY6L++e7e+jEAEREpqDhnGOXAPdF5jN2A+939ETNbAdxnZv8GvADcHY2/G/hPM6sC3gXOjLE2kbzTLEZ2drEFhrsvAw5uoX01cHgL7Z8AX4urHhER6RzdrVakQDQDka5OtwYREZEgCgwREQmiwBARkSAKDBERCaLAEBGRIAoMEREJostqJRF0yalI8ikwpEtQoIgUnw5JiYhIEAWGiIgEUWCIiEgQBYaIiARRYIiISBAFhoiIBFFgiIhIEAWGiIgEUWCIiEgQBYaIiARRYIiISBAFhoiIBFFgiIhIEAWGiIgEie325mY2FLgXGAg4MMfdbzWzHwEXAW9HQ69x90ejda4GpgD1wKXu/ue46hNJIt3GXZIszu/D2AZc6e5LzKw38LyZPRH13eLuN2YPNrMxwJnAgcAg4EkzG+Xu9THWKCIigWILDHevAWqi5Q/N7GVgcBurTATuc/etwOtmVgUcDvwtrhqlMPS/ZpGdQ0G+cc/MKoCDgb8DRwHfMbNzgcVkZiHvkQmTZ7NWW08LAWNmU4GpAOXl5aTT6VhrT4IRtbUAVHXRbfU6b7UvXZoOGhN3f1epYVfW1d8HO4PYA8PMyoAHgcvcfbOZ3QnMIHNeYwZwE3BB6PO5+xxgDkBlZaWnUqm815w4ZWUAdNVtXVu7ttW+VFkqaEzc/V2lhl1aF38f7AxivUrKzErJhMVv3f2PAO6+0d3r3b0B+AWZw04AG4ChWasPidpERCQB4rxKyoC7gZfd/eas9vLo/AbAV4GXouW5wO/M7GYyJ71HAoviqk+kK9L5ICmmOA9JHQWcA7xoZumo7RrgG2aWInNIag1wMYC7Lzez+4EVZK6wmqYrpEREkiPOq6T+ClgLXY+2sc4NwA1x1SQiIh2nT3qLiEgQBYaIiARRYIiISBAFhoiIBFFgiIhIEAWGiIgEUWCIiEgQBYaIiARRYIiISBAFhoiIBFFgiIhIEAWGiIgEUWCIiEgQBYaIiARRYIiISBAFhoiIBFFgiIhIEAWGiIgEUWCIiEgQBYaIiARRYIiISBAFhoiIBFFgiIhIkG7FLkCSb27t3Fb7JpRNaLdfRHYOsc0wzGyomT1lZivMbLmZTY/a+5nZE2b2WvS7b9RuZnabmVWZ2TIzOySu2kREJHdxHpLaBlzp7mOAI4BpZjYGuAqY5+4jgXnRY4CTgZHRz1TgzhhrExGRHMUWGO5e4+5LouUPgZeBwcBE4J5o2D3AqdHyROBez3gW6GNm5XHVJyIiuSnIOQwzqwAOBv4ODHT3mqjrTWBgtDwYWJe12vqorSarDTObSmYGQnl5Oel0Ora6k2JEbS0AVUXaVq/zVvvSpelO9xfiNXaVGnZmxX4fSAECw8zKgAeBy9x9s5lt73N3N7PW3wEtcPc5wByAyspKT6VSeaw2ocrKACjWtq6tXdtqX6os1en+QrzGrlLDTq3I7wOJ+bJaMyslExa/dfc/Rs0bGw81Rb/fito3AEOzVh8StYmISALEeZWUAXcDL7v7zVldc4HJ0fJk4OGs9nOjq6WOAD7IOnQlIiJFFuchqaOAc4AXzSwdtV0DzATuN7MpQDVwRtT3KDAeqAK2AOfHWJuIiOQotsBw978C1kr3CS2Md2BaXPWIiEjn6NYgIiISRIEhIiJBFBgiIhJEgSEiIkEUGCIiEkS3NxfZiehW8xInzTBERCSIAkNERIIoMEREJEi75zDMrF9b/e7+bv7KERGRpAo56b2EzF1k3yNzq48+QOM9lh3YN5bKREQkUUIOST0BnOLuA9y9P/DPwF/cfbi7KyxERHYRIYFxhLs/2vjA3R8DxsZXkoiIJFHIIak3zOxfgN9Ej78JvBFfSSIikkQhM4xvAHsBDwF/jJa/EWdRIiKSPO3OMKKroKabWS93/6gANYmISAKFXFY7FvglUAbsY2ZfAC5290viLk5E8ku3DpHOCDkkdQswDtgE4O5LgWPjLEpERJIn6JPe7r6uWVN9DLWIiEiChVwltS46LOVmVgpMB16OtywREUmakBnGt4BpwGBgA5CKHouIyC6kzRmGmZUAt7r7NwtUj4iIJFSbMwx3rweGmVn3AtUjIiIJFXIOYzXwv2Y2F9j+OQx3vzm2qkREJHFaDQwz+093PweYQObS2t2A3qFPbGa/InOjwrfc/aCo7UfARcDb0bBrGu9TZWZXA1PIXIF1qbv/OeetkZy1dV0+6Np8EflMWzOMQ81sEJlbmc/uwHP/GrgduLdZ+y3ufmN2g5mNAc4EDgQGAU+a2ajokJiIiCRAW4FxFzAPGA4szmo3Ar4Hw92fNrOKwDomAve5+1bgdTOrAg4H/ha4voiIxKzVwHD324DbzOxOd/92Hl/zO2Z2LpkQutLd3yNzye6zWWPWR207MLOpwFSA8vJy0ul0HktLphG1tQBUxbCtXudt9qdL022OyUd/e3WohsLWkFRxvg8kTMjNB/MZFncCM8jMUGYANwEX5PIE7j4HmANQWVnpqVQqj+UlVFkZAHFs69ratW32p8pSbY7JR397daiGwtaQWDG+DyRMyFVSeePuGxuXzewXwCPRww1kvga20ZCoTUQKTDcolNYE3UsqX8ysPOvhV4GXouW5wJlm1sPMhgMjgUWFrE1ERNoW2wzDzH4PHAcMMLP1wHXAcWaWInNIag1wMYC7Lzez+4EVwDZgmq6QEhFJltgCw91b+la+u9sYfwNwQ1z1iIhI5xT0kJSIiHRdCgwREQmiwBARkSAKDBERCaLAEBGRIAoMEREJosAQEZEgCgwREQmiwBARkSAKDBERCaLAEBGRIAW9vbkUnm5VLSL5ohmGiIgEUWCIiEgQBYaIiARRYIiISBAFhoiIBNFVUiKSE115t+vSDENERIIoMEREJIgCQ0REgigwREQkiAJDRESCxBYYZvYrM3vLzF7KautnZk+Y2WvR775Ru5nZbWZWZWbLzOyQuOoSEZGOiXOG8WvgpGZtVwHz3H0kMC96DHAyMDL6mQrcGWNdIiLSAbEFhrs/DbzbrHkicE+0fA9walb7vZ7xLNDHzMrjqk1ERHJX6A/uDXT3mmj5TWBgtDwYWJc1bn3UVkMzZjaVzCyE8vJy0ul0bMUmxYjaWgCqOrCtXuet9qVL0232h4zJR39n61QNyaohLp15H0h+FO2T3u7uZtb2X6uW15sDzAGorKz0VCqV79KSp6wMgI5s69rata32pcpSbfaHjMlHf2frVA3JqiE2nXgfSH4UOjA2mlm5u9dEh5zeito3AEOzxg2J2kSki9GtQ3Zehb6sdi4wOVqeDDyc1X5udLXUEcAHWYeuREQkAWKbYZjZ74HjgAFmth64DpgJ3G9mU4Bq4Ixo+KPAeKAK2AKcH1ddIiLSMbEFhrt/o5WuE1oY68C0uGoREZHO0ye9RUQkiAJDRESCKDBERCSIAkNERILoK1q7OF3zLiKFohmGiIgEUWCIiEgQBYaIiARRYIiISBAFhoiIBFFgiIhIEAWGiIgEUWCIiEgQBYaIiARRYIiISBAFhoiIBFFgiIhIEAWGiIgEUWCIiEgQBYaIiARRYIiISBB9gVLCza2dy9j6TQA80+zLkvQFSdIV6Uu/ui7NMEREJIgCQ0REghTlkJSZrQE+BOqBbe5eaWb9gD8AFcAa4Ax3f68Y9YmIyI6KeQ7jeHd/J+vxVcA8d59pZldFj79fnNJEpJhaOs/ReC5vQKGLke2SdEhqInBPtHwPcGrxShERkeaKNcNw4C9m5sDP3X0OMNDda6L+N4GBLa1oZlOBqQDl5eWk0+kClFs8XuewxTPLL3uTvnRpOtPfis72F+I10qVpgNhfQzXsBDVE74Od/T2fZMUKjKPdfYOZfQ54wsxeye50d4/CZAdRuMwBqKys9FQqFXuxxbS2di30NADsAGvSlypLZfpb0dn+QrxGqiwFEPtrqIadoIbofbCzv+eTrCiHpNx9Q/T7LeAh4HBgo5mVA0S/3ypGbSIi0rKCB4aZ9TKz3o3LwFeAl4C5wORo2GTg4ULXJiIirSvGIamBwENm1vj6v3P3x83sOeB+M5sCVANnFKE2ERFpRcEDw91XA19ooX0TcEKh6ykm3SJBRLqSJF1WKyIiCabAEBGRIAoMEREJotubi8hOR+cH46EZhoiIBNEMQ0S6HM0gikMzDBERCaLAEBGRIAoMEREJonMYMdJxVhHZmWiGISIiQRQYIiISRIEhIiJBFBgiIhJEJ707QSe1RWRXohmGiIgE0QxDpMCWPLGEX3zvFzQ0NHDiuScy6cpJTfrfXv82s6bO4v233sfMGHf+OE655BQAPv3kU6456RrqttZRv62esaeO5axrz2L9q+tJXZBic8NmAN5c8yZnXXsWE6blNtNdv349C+ct5JjTj2mx/8ILL6R8XDmHnXxYh7b5Cr+CseeM3WGbGz3++ONMnz6dzds2N9k3te/Xcv7La3llyyd0O+AAzr/9fEZ/cXRONWQLOTqgIwg70gxDpIDq6+v5+ZU/57o/Xsftz93OwgcWsvaVtU3GlHQr4YL/dwF3LL6Dn87/KY/OeXT7mNIepcx4ZAa3/u1WZj0ziyVPLmHlopUMGTWEdDrNrGdmcdPCm+ixRw+OOOWInOubN28eq9OrW+1/4YUXGP754R3e5hUrVrS4zY3jpk2bxmOPPbbDvvnl937Jl/uW8bdDR7F06VKG7D8ktw2TvNAMQyTQtSdfy7dnfZsh+w9h86bNXHvytcxeNDun53ht8Wvsve/e7D18bwCOOf0YFj2yCCo/G9Nv737027sfAD1792TI/kN49413ATAz9ijbA4D6unrq6+rBmr7GsgXL2Hv43nxun8+1Wcv8387nkbseYVvdNnr27km/n/XjiiuuoPue3Xlh3gtc9durqN9Wz+xLZrNl8xaOnnQ0b775JgMGD+jwNnfv3n37Nu8zep8dxo0YMYJ9992Xl2pf2j6uf3l/lj+znD8M2QuA7t27U9anLKcaJD8UGCKBalbXMGjkIADWLF/DsAOHNem/+itX83Htxzusd94N5zHhlMwhjE01m5r8we0/uD+vLn611dfcWL2R1ctWM6py1Pa2+vp6rjzmSmpW1zD+ovHsf9j+TdZZ+MBCjv3asW1uy5YPt/DgLQ8y65lZlHYvpfb9Wo4ecjSHHXYY/3T9PzFszDDqttZx+dGXc+mdlzKqchR3XX4Xo0c3PQzUfJt/uNsP2dywmfNuOI/U8amctnlTzSaGDh26w7iN1RvZc8CefPe1DSz/6BO+eOGFfOWGr7B7r93b3EbJPwVGG3QMUxpVV1fTb1A/dtstcxS3+qVqKg6saDLmx3/5cV5f8+Paj/nJ2T/hwpkX0vMfem5vLykpYdYzs6h9v5Yfn/VjqldUM2xMJrzqPq1j0aOLOPdfz23zuUtKSvj0k0/5j2v+g+PPOp6Rh4wEYOXKlVw86mIAnn3kWUYcPGJ7WA09YCi9evdq8jzNt3lC2YQ23zcdUb+tnlXpVcw+qIJDe/dkRq9ePHjzg3zzB9/M6+tI+xQYIgGWLl3aJCCq0lUcc1rTE8MhM4z+5f15Z8M72/s2bdhE//L+O6yzrW4bM8+eyZfO+BJHTjyyxZrK+pTxj8f+I0ueWLI9MJb8ZQn7pfajz+f6tLk9PXr2YPbfZ7PosUX8+6X/zomTT2TsOWPZc889KelWAkD18mr2O3i/7eusemEV557YNIhCZhih29y/vD/pdekdxg0YPIABgwdwaO9MaE6aNInL/+3yNrdP4rHLBoZmD5KLdDpN3dY6AN6oeoNF/72Is39wdpMxITOMkYeOpGZVDRvXbKTfoH4sfHAhV/7qyiZj3J3Z02YzdP+hTPzuxCZ9H7z9ASWlJZT1KWPrx1tZOn8pp11+2vb+px94mmMmNQ2yH/zzD7hszmXw2VEt3qh6g0EjBnHspGNZ98o66j6pY82aNQwaNGj7mN79erN2Reakc9ULVSx8YCE3/d+bqKa61W1uaYaRvc2fjvq0xW1uHHfXa3fx+uuvU7dn3fZxfQf2ZcDgAVRt2cqInj2YN28eQ0cP3WF9id8uGxgiuVi6dCkN3RqYfuR0Kg6qYOj+Q5n/u/l8/ftfz+l5SrqVMPXGqfzo1B/R0NDACeecwD4HfHby9/rTr+f0y09nwe8XMOzAYVw29jIAzr7ubCacPoH3Nr7HrItn0VDfgDc4R5121PZLXD/66COWzl/KJbdesv35GhoaqFldQ1nfpieJ/+tn/8Uri15h9167s88B+/C12V9j9O6jeeedd/ju4d/lktsu4fgzj+f6Sddz2djLGDRyEL327MWYMWOo3lpNLrK3eabP5Kizj9phm6fdPo3+5f25/fbbGTduHB9u+7DJvrnoxov41rirqWtwRvZPc8bsM3KqIQ674n86ExcYZnYScCtQAvzS3WcWuSQRli1bxoynZ9Czd8/2B7ejclwlleMqW+z74YM/BODhDx9usb/ioApm/e+sFvt69erFb9b+pknbulfWceSEI+mxR48m7dN/Pn2H9cvKyli0aFGTP4Q3LrixyZjS0lLY2uLLt6lxm1uagTRuM8D48eMZP378DmP2/fy+PJkaAcCAP/0p7+dJiqW90ElaKCUqMMysBLgDOBFYDzxnZnPdfUVxK5Nd2ZYPt2BmeQmLQhs2ZhhTZk4pdhm7pKT9sc+HpH1w73Cgyt1Xu/unwH3AxHbWEYlVz949efXV1i99FdlVJGqGAQwG1mU9Xg98MXuAmU0FpkYPa81sZYFqCzEAeKfdUR3VO2/ZGW+d+dVValWd+dV6nWYtNhdJV9mf0LTWYW0NbE3SAqNd7j4HmFPsOlpiZovdveWD0wnSVeqErlOr6swv1Zl/+ag1aYekNgDZ18sNidpERKTIkhYYzwEjzWy4mXUHzgR2jsshRES6uEQdknL3bWb2HeDPZC6r/ZW7Ly9yWblI5KGyFnSVOqHr1Ko680t15l+nazV3z0chIiKyk0vaISkREUkoBYaIiARRYAQys5PMbKWZVZnZVS30H2tmS8xsm5lNatY32cxei34mJ7jOejNLRz+xXmwQUOcVZrbCzJaZ2TwzG5bVl6T92VadBdufgbV+y8xejOr5q5mNyeq7OlpvpZmNS2KdZlZhZh9n7dO7illn1rjTzczNrDKrLTH7s7U6O7Q/3V0/7fyQOQG/CtgX6A4sBcY0G1MBfB64F5iU1d4PWB397hst901anVFfbYL25/FAz2j528AfEro/W6yzkPszh1r/IWt5AvB4tDwmGt8DGB49T0kC66wAXkrK/ozG9QaeBp4FKpO4P9uoM+f9qRlGmHZvWeLua9x9GdDQbN1xwBPu/q67vwc8AZyUwDoLKaTOp9x9S/TwWTKfyYHk7c/W6iy0kFo3Zz3sBTRe8TIRuM/dt7r760BV9HxJq7OQQm9TNAP4CfBJVlui9mcbdeZMgRGmpVuWDC7Aurnq7GvtbmaLzexZMzs1r5U1lWudU4DHOrhuZ3SmTijc/oTAWs1smpmtAn4KXJrLugmoE2C4mb1gZv9jZsc0X6+QdZrZIcBQd//vXNfNo87UCTnuz0R9DkOKbpi7bzCzfYH5Zvaiu68qZkFmdjZQCXypmHW0p5U6E7c/3f0O4A4zOwv4FyDWc0Ad1UqdNcA+7r7JzA4F/mRmBzabkRSEme0G3AycV+jXzkU7dea8PzXDCNOZW5YU8nYnnXotd98Q/V4NLAAOzmdxWYLqNLP/A1wLTHD3rbmsm4A6C7k/g2vNch9wagfX7YwO1xkd4tkULT9P5tj9qNZX7ZT26uwNHAQsMLM1wBHA3OiEcpL2Z6t1dmh/xn3yaGf4ITMTW03mBFbjiaUDWxn7a3Y86f06mRO0faPlfgmssy/QI1oeALxGCyfPClUnmT+uq4CRzdoTtT/bqLNg+zOHWkdmLZ8CLI6WD6TpSdrVxHeStjN17tVYF5mTvBuS8F6Kxi/gs5PJidqfbdSZ8/6M5R/vzvgDjAdejf44XBu1XU/mf5UAh5E5fvgRsAlYnrXuBWROfFUB5yexTmAs8GL0D+5FYEqR63wS2Aiko5+5Cd2fLdZZ6P0ZWOutwPKozqey/7CQmSGtAlYCJyexTuD0rPYlwCnFrLPZ2AVEf4iTtj9bq7Mj+1O3BhERkSA6hyEiIkEUGCIiEkSBISIiQRQYIiISRIEhIiJBFBgiHWBme5vZfWa2ysyeN7NHzSyuD5GJJIICQyRHZmbAQ8ACd9/P3Q8FrgYGBqzbra3HIkmmwBDJ3fFAnbtv//4Ad18K/NXMfmZmL0Xf5/B1ADM7zswWRt+JsaL54+Jsgkju9L8bkdwdBDzfQvtpQAr4ApnbgTxnZk9HfYcAB7n762Z2XPbj2KsVyRPNMETy52jg9+5e7+4bgf8hcysWgEXNwqH5Y5HEU2CI5G45cGiO63zUzmORxFNgiORuPtDDzKY2NpjZ54H3ga+bWYmZ7QUcCywqToki+adzGCI5cnc3s68Cs8zs+2S+9nINcBlQRuYOtQ58z93fNLPRxapVJJ90t1oREQmiQ1IiIhJEgSEiIkEUGCIiEkSBISIiQRQYIiISRIEhIiJBFBgiIhLk/wM7MbdM7o9htgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Bootsrtaping (random resampling with replacement) to show the sensitivity of correlation to sample \n",
    "# and to test the significance of the correlation coefficient\n",
    "\n",
    "Corr = []\n",
    "for i in range(5000):\n",
    "    data = Re.sample(n = len(Re), replace = True)\n",
    "    Corr.append(data['BTC-USD'].corr(data['^GSPC'], method='pearson'))  # 'pearson' 'spearman' 'kendall'\n",
    "\n",
    "confidence_interval = np.percentile(Corr,[2.5,97.5])\n",
    "standard_error = np.round( np.std(Corr),3 )\n",
    "print(\"Confidence Interval: \",confidence_interval)\n",
    "print(\"Standard Error: \",standard_error)\n",
    "plt.hist(x= Corr,bins='auto', color = 'lightgreen',alpha=0.7, rwidth=0.85)\n",
    "plt.grid(axis='y', alpha=0.75)\n",
    "plt.xlabel('Corr')\n",
    "plt.ylabel('freq')\n",
    "plt.text(np.round(np.mean(Corr),2), 5,r'$\\mu={}, std={}$'.format(np.round( np.mean(Corr),3 ),np.round( np.std(Corr),3 ) ))\n",
    "plt.axvline(x=np.percentile(Corr,[2.5]), ymin=0, ymax=1,label=confidence_interval[0],c='r')\n",
    "plt.axvline(x=np.percentile(Corr,[97.5]), ymin=0, ymax=1,label=confidence_interval[1],c='r')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "korean-rocket",
   "metadata": {},
   "source": [
    "### Copulas\n",
    "\n",
    "In this section, we will discuss the basic framework of copula from the mathematical perspective and explain how to apply the approach in finance with an example of stock trading.\n",
    "\n",
    "- The word **copula** is derived from the Latin noun for a link or a tie (as is the English word “couple”) and its purpose is to describe the dependence structure between two variables. \n",
    "\n",
    "\n",
    "- Sklar’s theorem states that “Any multivariate joint distribution can be written in terms of univariate marginal distribution functions and a copula which describes the dependence structure between the two variables”.\n",
    "\n",
    "\n",
    "- Modeling multivariate probability distributions can be difficult when the marginal probability density functions of the component random variables are different. Copulas are a useful tool to model dependence between random variables with any marginal distributions.\n",
    "\n",
    "To understand how a copula can describe the dependence function between random variables its helpful to first review some simple statistics. We'll focus specifically on continuous random variables."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "innovative-color",
   "metadata": {},
   "source": [
    "### Probability Density Function (PDF)\n",
    "\n",
    "A probability density function $f(x)$ captures the likelihood that a random sample from the distribution is equal to $x$. For example, the probability density function for the standard normal distribution is given by\n",
    "\n",
    "\\begin{equation}\n",
    "f(x) = \\frac{1}{2 \\pi} e^{-x^2/2}\n",
    "\\end{equation}\n",
    "\n",
    "Note that the probability density function does **not** return a probability but rather a \"relative likelihood\" which can take on values in the interval $[0, \\infty)$; however, the integral over the probability density function from $-\\infty$ to $\\infty$ must be equal to one."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "complete-sleep",
   "metadata": {},
   "source": [
    "### Cumulative Distribution Function (CDF)\n",
    "\n",
    "In many cases, the probability density function can be hard to work with directly. Instead, we will use the cumulative distribution function $F(x)$ which is defined as the integral of the probability density function\n",
    "\n",
    "\\begin{equation}\n",
    "F(x) = \\int_{-\\infty}^x f(x)\n",
    "\\end{equation}\n",
    "\n",
    "The below figure shows the probability density function $f(x)$ and the cumulative distribution function $F(x)$ for a normal standard distribution with mean $0.0$ and variance $1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "thrown-supervisor",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "outputs": [],
   "source": [
    "import warnings\n",
    "\n",
    "warnings.filterwarnings('ignore')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "focused-roots",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "from scipy import stats\n",
    "\n",
    "def plot_cdf_pdf():\n",
    "    # Generate 10000 evenly distributed values from -4 to 4\n",
    "    x = np.linspace(-4.0, 4.0, 10000)\n",
    "\n",
    "    # Compute their Probability Densities and Cumulative Distributions\n",
    "    pdf = stats.norm.pdf(x)\n",
    "    cdf = stats.norm.cdf(x)\n",
    "\n",
    "    figure = plt.figure(figsize=(16, 4))\n",
    "    figure.add_subplot(1, 2, 1)\n",
    "    plt.plot(x, pdf)\n",
    "    plt.title(\"Probability Density Function\")\n",
    "    plt.xlabel(\"x\")\n",
    "    plt.ylabel(\"f(x)\")\n",
    "\n",
    "    figure.add_subplot(1, 2, 2)\n",
    "    plt.plot(x, cdf)\n",
    "    plt.title(\"Cumulative Density Function\")\n",
    "    plt.xlabel(\"x\")\n",
    "    plt.ylabel(\"F(x)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "essential-damages",
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 1152x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_cdf_pdf()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "completed-apollo",
   "metadata": {},
   "source": [
    "### Probability Integral Transform\n",
    "\n",
    "The probability integral transform is a key component in our toolkit for working with probability distributions. Suppose we have a random variable $X$ that comes from a distribution with cumulative density function $F(X)$. Then, we can define a random variable $U$ as\n",
    "\n",
    "\\begin{equation}\n",
    "U = F(X)\n",
    "\\end{equation}\n",
    "\n",
    "and prove that $U$ follows a uniform distribution over the interval $[0.0, 1.0]$.\n",
    "\n",
    "The figure below shows an example of this. We sample some data from a normal distribution and plot it on the left. Then, we use the CDF of the normal distribution to transform the data, plot it on the right, and observe that it resembles an uniform distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "id": "saving-college",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1152x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy import stats\n",
    "from matplotlib import pyplot as plt\n",
    "\n",
    "\n",
    "X = stats.norm.rvs(size=10000)\n",
    "X_pit = stats.norm.cdf(X)\n",
    "\n",
    "fig = plt.figure(figsize=(16, 4))\n",
    "fig.add_subplot(1, 2, 1)\n",
    "plt.hist(X, density=True, bins=10)\n",
    "plt.title(\"Samples\") \n",
    "plt.xlabel(\"x\")\n",
    "\n",
    "fig.add_subplot(1, 2, 2)\n",
    "plt.hist(X_pit, density=True, bins=10)\n",
    "plt.title(\"Transformed Samples\")\n",
    "plt.xlabel(\"U\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "separate-orlando",
   "metadata": {},
   "source": [
    "## Framework of Bivariate Copulas\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "higher-bahamas",
   "metadata": {},
   "source": [
    "## \\begin{equation}\n",
    "Let:\\\\\n",
    "X_1~ and ~X_2~ be ~RVs\\\\\n",
    "U_1 ~=~ F(X_1);~~~ U_2 ~=~ F(X_2)\\\\\n",
    "Then\\\\\n",
    "U_1 \\sim uniform\\\\\n",
    "U_2 \\sim uniform\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "architectural-mouth",
   "metadata": {},
   "source": [
    "The above statements can be summarized as saying that the values of the CDF of any marginal distribution are uniformly distributed on the interval [0,1] ie. if you make a random draw from any distribution, you have the same probability of drawing the largest value (U=1) of that distribution as the smallest possible value (U=0) or the median value (U=.5).\n",
    "\n",
    "So what does this have to do with copulas? A copula is actually a joint cumulative distribution function (CDFs) of the random variables $U$ it is modeling. Put formally:\n",
    "\n",
    "A k dimensional copula is a function $c:[0,1]^k$ $\\rightarrow [0,1]$ and is a CDF with uniform marginals $[1]$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "happy-slide",
   "metadata": {},
   "source": [
    "### \\begin{equation}\n",
    "Let:\\\\\n",
    "U_1 ~=~ F(X_1);~~~ U_2 ~=~ F(X_2)\\\\\n",
    "F(X_1,X_2) ~=~ P(X_1 \\leq x_1 \\cap X_2 \\leq x_2) \\\\\n",
    "~=~ P(F^-1(U_1) \\leq x_1 \\cap F^-1(U_2) \\leq x_2)\\\\\n",
    "~=~ P(U_1 \\leq F(x_1) \\cap U_2 \\leq F(x_2))\\\\\n",
    "~=~ P(U_1 \\leq u_1 \\cap U_2 \\leq u_2) ~=~ C(U_1,U_2)\n",
    "\\end{equation}\n",
    "\n",
    "\n",
    "\n",
    "Copulas function contains all the dependency characteristics of the marginal distributions and will better describe the linear and non-linear relationship between variables, using probability. They allow the marginal distributions to be modeled independently from each other, and no assumption on the joint behavior of the marginals is required.\n",
    "\n",
    "\n",
    "- Taking the partial derivative of the copula function over U_1 and U_2 would give the conditional distribution function.\n",
    "- There are many copula functions that enable us to describe dependence structures between variables, other than the Gaussian assumption. For instance the Clayton, Gumbel and Frank copula formulas from the Archimedean class. Archimedean copulas are based on the Laplace transforms φ of univariate distribution functions. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "distinct-contest",
   "metadata": {},
   "source": [
    "## Trading Strategy based on Copula"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "incorporate-novelty",
   "metadata": {},
   "source": [
    "In this section, we investigate using copula theory to identify trading opportunities. \n",
    "\n",
    "Stander Y, Marais D, Botha I. Trading strategies with copulas. Journal of Economic and Financial Sciences, 2013, 6(1): 83-107. https://journals.co.za/doi/pdf/10.10520/EJC135921\n",
    "\n",
    "Typically the asset price distribution is modeled by a  Gaussian distribution of return series but the joint normal distribution may fail to catch some key features of the dependence of stock pairs' price like tail dependence. \n",
    "\n",
    "To determine which stock pairs to include in the analysis, correlations between pairs are analyzed using daily log stock price returns.  \n",
    "\n",
    "- The Pearson correlation assumes that both variables should be normally distributed. Thus here we use Kendall rank as the correlation measure to choose the pairs."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "proof-belize",
   "metadata": {},
   "source": [
    "In order to construct the copula, we need to transform the log-return series R1 and R2 to two uniformly distributed values U1 and U2. This can be done by estimating the marginal distribution functions of Rx\n",
    "R1 and R2 and plugging the return values into a distribution function. As we make no assumptions about the distribution of the two log-return series, here we use the empirical distribution function to approach the marginal distribution F1(R1) and F2(R2). The Python ECDF function from the statsmodel library gives us the Empirical CDF as a step function.\n",
    "\n",
    "Then, we estimate the copula parameter (i.e. theta for Archimedean copulas) by the relationship between the copula and the dependence measure Kendall’s tau."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dressed-coast",
   "metadata": {},
   "source": [
    "The copula functions include all the information about the dependence structures of two return series. In Trading strategies with copulas, the fitted copula is used to derive the confidence bands for the conditional marginal distribution function of C(U1∣U2) and C(U2∣U1), that is the mispricing indexes. When the market observations fall outside the confidence band, it is an indication that pairs trading opportunity is available. \n",
    "\n",
    "Please also see \"Copula for Pairs Trading: A Detailed, But Practical Introduction\" for more details. \n",
    "https://hudsonthames.org/copula-for-pairs-trading-introduction/\n",
    "\n",
    "For python code see: https://www.quantconnect.com/tutorials/strategy-library/pairs-trading-copula-vs-cointegration"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "official-edgar",
   "metadata": {},
   "source": [
    "## The Gaussian Copula and the Financial Crisis"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "built-elevation",
   "metadata": {},
   "source": [
    "In fact, Gaussian copulas are said to have played a key role in the 2007-2008 Financial Crisis as tail-correlations were severely underestimated. If you've seen The Big Short, the default rates of individual mortgages (among other things) inside CDOs are correlated -- if one mortgage fails, the likelihood of another failing is increased. In the early 2000s, the banks only knew how to model the marginals of the default rates. But we could  use copulas to model the correlations between those marginals. Rating agencies relied on this model heavily, severly underestimating risk and giving false ratings. \n",
    "\n",
    "Read this paper http://samueldwatts.com/wp-content/uploads/2016/08/Watts-Gaussian-Copula_Financial_Crisis.pdf for an excellent description of Gaussian copulas and the Financial Crisis which argues that different copula choices would not have made a difference but instead the assumed correlation was way too low."
   ]
  }
 ],
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