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      "source": [
        "# Gaussian vs Non-Gaussian Copulas for Pairs Trading\n",
        "\n",
        "Compare simple dependence assumptions and explore why correlation can miss tail dependence.\n",
        "\n",
        "**How to use this notebook:** We give you the structure, the key ideas, starter code, and places where you need to make the important research decisions yourself."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## What you should have by the end\n",
        "\n",
        "- A working prototype\n",
        "- A few graphs/tables that explain what your model is doing\n",
        "- A short write-up of your assumptions\n",
        "- At least one extension that makes the project feel like yours"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Concept\n",
        "\n",
        "Pairs trading is often introduced with correlation or cointegration. Copulas are a more advanced way to model dependence separately from each asset's marginal distribution.\n",
        "\n",
        "The core idea: two assets can have the same correlation but very different joint tail behaviour."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Starter Code\n",
        "\n",
        "1. Generate or load two return series.\n",
        "2. Transform each marginal into ranks/uniform variables.\n",
        "3. Fit or compare Gaussian-style dependence against heavier-tail dependence.\n",
        "4. Create a simple mispricing score.\n",
        "5. Explain when the signal breaks."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import pandas as pd\n",
        "from scipy.stats import rankdata, norm, t\n",
        "\n",
        "rng = np.random.default_rng(202)\n",
        "\n",
        "# Toy data with dependence.\n",
        "n = 2000\n",
        "z = rng.normal(size=n)\n",
        "x = 0.01 * z + 0.01 * rng.normal(size=n)\n",
        "y = 0.008 * z + 0.012 * rng.normal(size=n)\n",
        "\n",
        "returns = pd.DataFrame({\"asset_x\": x, \"asset_y\": y})\n",
        "returns.corr()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
      "outputs": [],
      "source": [
        "def pseudo_observations(series):\n",
        "    # Convert data to approximately uniform ranks in (0, 1).\n",
        "    ranks = rankdata(series)\n",
        "    return ranks / (len(series) + 1)\n",
        "\n",
        "u = pseudo_observations(returns[\"asset_x\"])\n",
        "v = pseudo_observations(returns[\"asset_y\"])\n",
        "\n",
        "copula_data = pd.DataFrame({\"u\": u, \"v\": v})\n",
        "copula_data.head()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
      "outputs": [],
      "source": [
        "def gaussian_copula_score(u, v, rho):\n",
        "    # Very rough starter score for dependence under a Gaussian copula.\n",
        "    # TODO:\n",
        "    # - Replace this with a proper log-density.\n",
        "    # - Compare against t-copula assumptions.\n",
        "    # - Build a trading signal from extreme conditional probabilities.\n",
        "    x = norm.ppf(u)\n",
        "    y = norm.ppf(v)\n",
        "    return (x * y * rho)\n",
        "\n",
        "rho_hat = returns.corr().iloc[0, 1]\n",
        "copula_data[\"gaussian_score\"] = gaussian_copula_score(copula_data[\"u\"], copula_data[\"v\"], rho_hat)\n",
        "\n",
        "copula_data.sort_values(\"gaussian_score\").head()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Open-ended extensions\n",
        "\n",
        "- Implement a t-copula score\n",
        "- Compare Gaussian vs t-copula tail events\n",
        "- Build a pair trading rule based on conditional mispricing\n",
        "- Backtest with transaction costs\n",
        "- Explain why high correlation does not necessarily mean a good pair trade"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Final write-up prompts\n",
        "\n",
        "Answer these before putting the project on your resume:\n",
        "\n",
        "1. What problem were you trying to solve?\n",
        "2. What assumptions did your model make?\n",
        "3. Where did the model work well?\n",
        "4. Where did it fail?\n",
        "5. What would you improve with more time?\n",
        "6. What is one interesting thing you learned that was not obvious before starting?"
      ]
    }
  ],
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