.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/stochastic_volatility/jumps/plot_02_madan_seneta_variance_gamma.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_stochastic_volatility_jumps_plot_02_madan_seneta_variance_gamma.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_stochastic_volatility_jumps_plot_02_madan_seneta_variance_gamma.py:


Madan and Seneta's variance gamma
=================================

Madan and Seneta (1990) ran a Brownian motion on a random business clock:
gamma-distributed time :math:`G_t`, with variance rate :math:`\nu`. The
result, :math:`\sigma W_{G_t}`, is a pure-jump Lévy process with
infinitely many small jumps, symmetric, and with tails heavier than the
normal's but lighter than a power law. The density is closed form in
terms of a modified Bessel function. The clock's variance :math:`\nu`
sets the excess kurtosis (:math:`3\nu/T`).

.. GENERATED FROM PYTHON SOURCE LINES 15-33

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.stats import norm

    from financialkit.stochastic_volatility import VarianceGammaModel
    from financialkit.stochastic_volatility.visualizers import plot_density, plot_smile

    T, sigma = 0.5, 0.2
    x = np.linspace(-0.8, 0.8, 401)

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4))
    for nu in (0.05, 0.2, 0.5):
        m = VarianceGammaModel(sigma=sigma, nu=nu, theta=0.0)
        plot_density(x, m.pdf(x, T), ax=ax1, label=f"nu = {nu}", log=True)
    ax1.plot(x, norm.pdf(x, -0.5 * sigma**2 * T, sigma * np.sqrt(T)), "k--", lw=1, label="normal")
    ax1.set_ylim(1e-4, 20)
    ax1.legend()




.. image-sg:: /api/gallery/stochastic_volatility/jumps/images/sphx_glr_plot_02_madan_seneta_variance_gamma_001.png
   :alt: plot 02 madan seneta variance gamma
   :srcset: /api/gallery/stochastic_volatility/jumps/images/sphx_glr_plot_02_madan_seneta_variance_gamma_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x115224ad0>



.. GENERATED FROM PYTHON SOURCE LINES 34-36

Subordination: sampling and the density agree
---------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 36-41

.. code-block:: Python


    m = VarianceGammaModel(sigma=sigma, nu=0.2, theta=0.0)
    draws = m.sample_terminal(T, 100_000, rng=0)
    print(f"sample excess kurtosis {np.mean((draws - draws.mean()) ** 4) / draws.var() ** 2 - 3:.2f}, theory 3 nu / T = {3 * 0.2 / T:.2f}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    sample excess kurtosis 1.19, theory 3 nu / T = 1.20




.. GENERATED FROM PYTHON SOURCE LINES 42-44

A symmetric smile
-----------------

.. GENERATED FROM PYTHON SOURCE LINES 44-52

.. code-block:: Python


    S, r = 100.0, 0.0
    K = S * np.exp(np.linspace(-0.4, 0.4, 21))
    for nu in (0.05, 0.2, 0.5):
        plot_smile(K, VarianceGammaModel(sigma, nu).implied_vol(S, K, T, r), forward=S, ax=ax2, label=f"nu = {nu}")
    fig.tight_layout()

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.132 seconds)


.. _sphx_glr_download_api_gallery_stochastic_volatility_jumps_plot_02_madan_seneta_variance_gamma.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/stochastic_volatility/jumps/plot_02_madan_seneta_variance_gamma.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_madan_seneta_variance_gamma.ipynb <plot_02_madan_seneta_variance_gamma.ipynb>`

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

      :download:`Download Python source code: plot_02_madan_seneta_variance_gamma.py <plot_02_madan_seneta_variance_gamma.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_02_madan_seneta_variance_gamma.zip <plot_02_madan_seneta_variance_gamma.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_