.. 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_04_madan_carr_chang.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_04_madan_carr_chang.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_04_madan_carr_chang.py:


Madan, Carr and Chang's skewed variance gamma
=============================================

Madan, Carr and Chang (1998) added a drift :math:`\theta` in business
time: :math:`X = \theta G_t + \sigma W_{G_t}`. A negative :math:`\theta`
moves mass into the left tail, turning the symmetric Madan-Seneta smile
into the equity skew with one extra parameter, and they priced options
in closed form. Fang and Oosterlee (2008) later used one of their
parameter sets as a benchmark; it is reproduced below.

.. GENERATED FROM PYTHON SOURCE LINES 14-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

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

    m = VarianceGammaModel(sigma=0.12, nu=0.2, theta=-0.14)
    for T, ref in ((0.1, 10.993703187), (1.0, 19.099354724)):
        print(f"T = {T}: call {lewis_price(m, 100.0, 90.0, T, 0.1):.9f} (Fang-Oosterlee reference {ref})")





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

 .. code-block:: none

    T = 0.1: call 10.993703187 (Fang-Oosterlee reference 10.993703187)
    T = 1.0: call 19.099354724 (Fang-Oosterlee reference 19.099354724)




.. GENERATED FROM PYTHON SOURCE LINES 25-27

The skew comes from theta
-------------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-40

.. code-block:: Python


    S, r, T = 100.0, 0.0, 0.5
    x = np.linspace(-0.7, 0.5, 400)
    K = S * np.exp(np.linspace(-0.4, 0.3, 22))
    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4))
    for theta in (0.0, -0.15, -0.3):
        v = VarianceGammaModel(0.2, 0.3, theta)
        plot_density(x, v.pdf(x, T), ax=ax1, label=f"theta = {theta}", log=True)
        plot_smile(K, v.implied_vol(S, K, T, r), forward=S, ax=ax2, label=f"theta = {theta}")
    ax1.set_ylim(1e-3, 10)
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/stochastic_volatility/jumps/images/sphx_glr_plot_04_madan_carr_chang_001.png
   :alt: plot 04 madan carr chang
   :srcset: /api/gallery/stochastic_volatility/jumps/images/sphx_glr_plot_04_madan_carr_chang_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.216 seconds)


.. _sphx_glr_download_api_gallery_stochastic_volatility_jumps_plot_04_madan_carr_chang.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_04_madan_carr_chang.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_madan_carr_chang.ipynb <plot_04_madan_carr_chang.ipynb>`

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

      :download:`Download Python source code: plot_04_madan_carr_chang.py <plot_04_madan_carr_chang.py>`

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

      :download:`Download zipped: plot_04_madan_carr_chang.zip <plot_04_madan_carr_chang.zip>`


.. only:: html

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

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