.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/stochastic_volatility/fourier/plot_03_cos_method.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_fourier_plot_03_cos_method.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_fourier_plot_03_cos_method.py:


Fang and Oosterlee's COS method
===============================

Fang and Oosterlee (2008) truncated the density of the log return to an
interval :math:`[a, b]` set by its cumulants and expanded it in a cosine
series, whose coefficients are read directly off the characteristic
function. The payoff's cosine coefficients are closed form, so a price
is a dot product of two short vectors. For smooth densities the error
falls exponentially with the number of terms; for a density with a kink,
like short-dated variance gamma, only algebraically.

.. GENERATED FROM PYTHON SOURCE LINES 15-38

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.stochastic_volatility import BlackScholesModel, HestonModel, VarianceGammaModel, cos_density, cos_price, lewis_price

    N = 2 ** np.arange(3, 11)
    cases = {
        "Black-Scholes": (BlackScholesModel(0.25), 100.0, 1.0, 0.0),
        "Heston (Fang-Oosterlee)": (HestonModel(0.0175, 1.5768, 0.0398, 0.5751, -0.5711), 100.0, 1.0, 0.0),
        "VG, T = 1": (VarianceGammaModel(0.12, 0.2, -0.14), 90.0, 1.0, 0.1),
        "VG, T = 0.1": (VarianceGammaModel(0.12, 0.2, -0.14), 90.0, 0.1, 0.1),
    }

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4))
    for name, (m, K, T, r) in cases.items():
        ref = lewis_price(m, 100.0, K, T, r)
        err = [abs(cos_price(m, 100.0, K, T, r, N=n) - ref) for n in N]
        ax1.semilogy(N, np.maximum(err, 1e-16), marker="o", ms=3, label=name)
    ax1.set_xscale("log", base=2)
    ax1.set_xlabel("cosine terms N")
    ax1.set_ylabel("|COS - reference|")
    ax1.legend()




.. image-sg:: /api/gallery/stochastic_volatility/fourier/images/sphx_glr_plot_03_cos_method_001.png
   :alt: plot 03 cos method
   :srcset: /api/gallery/stochastic_volatility/fourier/images/sphx_glr_plot_03_cos_method_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x115467cb0>



.. GENERATED FROM PYTHON SOURCE LINES 39-41

Recovering the density
----------------------

.. GENERATED FROM PYTHON SOURCE LINES 41-53

.. code-block:: Python


    x = np.linspace(-0.6, 0.4, 400)
    vg = VarianceGammaModel(0.12, 0.2, -0.14)
    for n in (16, 64, 512):
        ax2.plot(x, cos_density(vg, x, 0.5, N=n), label=f"N = {n}")
    ax2.plot(x, vg.pdf(x, 0.5), "k--", lw=1, label="exact (Bessel)")
    ax2.set_xlabel("log return")
    ax2.set_title("variance gamma density, T = 0.5")
    ax2.legend()
    fig.tight_layout()

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.195 seconds)


.. _sphx_glr_download_api_gallery_stochastic_volatility_fourier_plot_03_cos_method.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/fourier/plot_03_cos_method.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_cos_method.ipynb <plot_03_cos_method.ipynb>`

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

      :download:`Download Python source code: plot_03_cos_method.py <plot_03_cos_method.py>`

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

      :download:`Download zipped: plot_03_cos_method.zip <plot_03_cos_method.zip>`


.. only:: html

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

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