.. 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_05_kou.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_05_kou.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_05_kou.py:


Kou's double-exponential jumps
==============================

Steven Kou (2002) took the log jump sizes to be exponential, with rate
:math:`\eta_1` upward (probability :math:`p`) and :math:`\eta_2`
downward. The two exponential tails give the asymmetric, leptokurtic
returns seen in data, and because the exponential distribution is
memoryless the overshoot over a barrier is again exponential, which
makes barrier and lookback options tractable. The tails also cap the
moments, :math:`E[S_T^{s}] < \infty` only for :math:`-\eta_2 < s < \eta_1`.

.. GENERATED FROM PYTHON SOURCE LINES 15-28

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.stochastic_volatility import KouModel, MertonJumpModel, cos_density
    from financialkit.stochastic_volatility.visualizers import plot_density

    m = KouModel(sigma=0.16, lam=1.0, p=0.4, eta1=10.0, eta2=5.0)
    print(f"call S = 100, K = 98, T = 0.5, r = 5%: {m.price(100.0, 98.0, 0.5, 0.05):.5f} (Kou 2002: 9.14732)")

    x = m.sample_terminal(0.5, 200_000, rng=0)
    pay = np.exp(-0.025) * np.maximum(100 * np.exp(0.025) * np.exp(x) - 98.0, 0.0)
    print(f"Monte Carlo: {pay.mean():.4f} +/- {2 * pay.std() / np.sqrt(pay.size):.4f}")





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

 .. code-block:: none

    call S = 100, K = 98, T = 0.5, r = 5%: 9.14732 (Kou 2002: 9.14732)
    Monte Carlo: 9.1638 +/- 0.0539




.. GENERATED FROM PYTHON SOURCE LINES 29-35

Exponential against Gaussian jump tails
---------------------------------------

A Merton model with the same jump intensity and the same jump mean and
variance has Gaussian jump tails; on a log scale Kou's tails are
straight lines, Merton's parabolas.

.. GENERATED FROM PYTHON SOURCE LINES 35-47

.. code-block:: Python


    p, e1, e2 = m.p, m.eta1, m.eta2
    mean = p / e1 - (1 - p) / e2
    var = 2 * p / e1**2 + 2 * (1 - p) / e2**2 - mean**2
    merton = MertonJumpModel(0.16, 1.0, mean, np.sqrt(var))
    grid = np.linspace(-2.5, 1.5, 400)
    fig, ax = plt.subplots(figsize=(7, 4))
    plot_density(grid, np.maximum(cos_density(m, grid, 1.0, N=1024), 1e-12), ax=ax, label="Kou", log=True)
    plot_density(grid, np.maximum(cos_density(merton, grid, 1.0, N=1024), 1e-12), ax=ax, label="Merton, matched jumps", log=True)
    ax.set_ylim(1e-8, 5)

    plt.show()



.. image-sg:: /api/gallery/stochastic_volatility/jumps/images/sphx_glr_plot_05_kou_001.png
   :alt: plot 05 kou
   :srcset: /api/gallery/stochastic_volatility/jumps/images/sphx_glr_plot_05_kou_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.073 seconds)


.. _sphx_glr_download_api_gallery_stochastic_volatility_jumps_plot_05_kou.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_05_kou.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_05_kou.ipynb <plot_05_kou.ipynb>`

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

      :download:`Download Python source code: plot_05_kou.py <plot_05_kou.py>`

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

      :download:`Download zipped: plot_05_kou.zip <plot_05_kou.zip>`


.. only:: html

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

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