.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/risk/copulas/plot_02_t_copula_tail_dependence.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_risk_copulas_plot_02_t_copula_tail_dependence.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_risk_copulas_plot_02_t_copula_tail_dependence.py:


Demarta-McNeil: the t copula and tail dependence
================================================

Under a Gaussian copula, joint extremes become independent as one looks
further into the tail: the coefficient of tail dependence is zero for
any correlation below one. Markets do not behave that way; in a crash
everything falls together. Demarta and McNeil (2005) studied the t
copula, whose shared random scale makes extremes coincide with
probability

.. math::

   \lambda = 2\, t_{\nu + 1}\left(-\sqrt{\frac{(\nu + 1)(1 - \rho)}{1 + \rho}}\right) > 0,

even for zero or negative correlation.

.. GENERATED FROM PYTHON SOURCE LINES 20-31

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.risk import empirical_tail_dependence, gaussian_copula_sample, t_copula_sample, t_tail_dependence
    from financialkit.risk.visualizers import plot_copula

    corr = np.array([[1.0, 0.5], [0.5, 1.0]])
    n = 100_000
    g = gaussian_copula_sample(corr, n, rng=0)
    t = t_copula_sample(corr, 4, n, rng=1)








.. GENERATED FROM PYTHON SOURCE LINES 32-34

Same correlation, different corners
-----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 34-40

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(9, 4.5))
    plot_copula(g[:3000], ax=axes[0], title="Gaussian copula, rho = 0.5")
    plot_copula(t[:3000], ax=axes[1], title="t copula, rho = 0.5, nu = 4")
    fig.tight_layout()




.. image-sg:: /api/gallery/risk/copulas/images/sphx_glr_plot_02_t_copula_tail_dependence_001.png
   :alt: Gaussian copula, rho = 0.5, t copula, rho = 0.5, nu = 4
   :srcset: /api/gallery/risk/copulas/images/sphx_glr_plot_02_t_copula_tail_dependence_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 41-46

Tail dependence as the level shrinks
------------------------------------

:math:`P(U_2 < q \mid U_1 < q)` falls towards zero for the Gaussian
copula and levels off at :math:`\lambda` for the t copula.

.. GENERATED FROM PYTHON SOURCE LINES 46-57

.. code-block:: Python


    q = np.geomspace(0.002, 0.2, 25)
    fig, ax = plt.subplots(figsize=(7, 4))
    ax.semilogx(q, empirical_tail_dependence(*g.T, q), "o-", ms=3, label="Gaussian")
    ax.semilogx(q, empirical_tail_dependence(*t.T, q), "o-", ms=3, label="t, nu = 4")
    ax.axhline(t_tail_dependence(0.5, 4), color="C1", ls="--", label="t copula lambda")
    ax.set_xlabel("tail level q")
    ax.set_ylabel("P(U2 < q | U1 < q)")
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/risk/copulas/images/sphx_glr_plot_02_t_copula_tail_dependence_002.png
   :alt: plot 02 t copula tail dependence
   :srcset: /api/gallery/risk/copulas/images/sphx_glr_plot_02_t_copula_tail_dependence_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 58-60

The coefficient across correlation and degrees of freedom
---------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 60-72

.. code-block:: Python


    rho = np.linspace(-0.9, 0.99, 100)
    fig, ax = plt.subplots(figsize=(7, 4))
    for nu in (2, 4, 10, 30):
        ax.plot(rho, t_tail_dependence(rho, nu), label=f"nu = {nu}")
    ax.plot(rho, t_tail_dependence(rho, np.inf), "k--", label="Gaussian")
    ax.set_xlabel("correlation rho")
    ax.set_ylabel("tail dependence lambda")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/risk/copulas/images/sphx_glr_plot_02_t_copula_tail_dependence_003.png
   :alt: plot 02 t copula tail dependence
   :srcset: /api/gallery/risk/copulas/images/sphx_glr_plot_02_t_copula_tail_dependence_003.png
   :class: sphx-glr-single-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.254 seconds)


.. _sphx_glr_download_api_gallery_risk_copulas_plot_02_t_copula_tail_dependence.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/risk/copulas/plot_02_t_copula_tail_dependence.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_t_copula_tail_dependence.ipynb <plot_02_t_copula_tail_dependence.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_t_copula_tail_dependence.py <plot_02_t_copula_tail_dependence.py>`

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      :download:`Download zipped: plot_02_t_copula_tail_dependence.zip <plot_02_t_copula_tail_dependence.zip>`


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