.. 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_03_bates.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_03_bates.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_03_bates.py:


Bates: jumps and stochastic volatility
======================================

David Bates (1996), studying Deutsche mark options, found that neither
jumps nor stochastic volatility alone fit the smile at every maturity.
Diffusive stochastic volatility needs time to build up a skew, so
Heston's short-dated smiles are too flat; jumps act at once but their
effect washes out with maturity. Putting Merton's lognormal jumps on
top of Heston gives both: the characteristic function is simply the
product of the two.

.. GENERATED FROM PYTHON SOURCE LINES 15-41



.. image-sg:: /api/gallery/stochastic_volatility/jumps/images/sphx_glr_plot_03_bates_001.png
   :alt: T = 0.08, T = 0.5, T = 3.0
   :srcset: /api/gallery/stochastic_volatility/jumps/images/sphx_glr_plot_03_bates_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    phi_Bates = phi_Heston * phi_jumps: True






|

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.stochastic_volatility import BatesModel, HestonModel, MertonJumpModel
    from financialkit.stochastic_volatility.visualizers import plot_smile

    S, r = 100.0, 0.02
    heston = HestonModel(0.03, 2.0, 0.03, 0.4, -0.6)
    bates = BatesModel(0.03, 2.0, 0.03, 0.4, -0.6, lam=0.3, mu_j=-0.12, delta=0.08)

    u = np.linspace(-10, 10, 5)
    print(
        "phi_Bates = phi_Heston * phi_jumps:",
        np.allclose(bates.char_func(u, 1.0), heston.char_func(u, 1.0) * MertonJumpModel(0.0, 0.3, -0.12, 0.08).char_func(u, 1.0)),
    )

    fig, axes = plt.subplots(1, 3, figsize=(12, 3.8), sharey=True)
    for ax, T in zip(axes, (0.08, 0.5, 3.0), strict=True):
        F = S * np.exp(r * T)
        K = F * np.exp(np.linspace(-0.3, 0.2, 21) * np.sqrt(max(T, 0.25)))
        plot_smile(K, heston.implied_vol(S, K, T, r), forward=F, ax=ax, label="Heston")
        plot_smile(K, bates.implied_vol(S, K, T, r), forward=F, ax=ax, label="Bates")
        ax.set_title(f"T = {T}")
    fig.tight_layout()

    plt.show()


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

   **Total running time of the script:** (0 minutes 0.142 seconds)


.. _sphx_glr_download_api_gallery_stochastic_volatility_jumps_plot_03_bates.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_03_bates.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_bates.ipynb <plot_03_bates.ipynb>`

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

      :download:`Download Python source code: plot_03_bates.py <plot_03_bates.py>`

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

      :download:`Download zipped: plot_03_bates.zip <plot_03_bates.zip>`


.. only:: html

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

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