.. 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_01_carr_madan_fft.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_01_carr_madan_fft.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_01_carr_madan_fft.py:


Carr and Madan's FFT
====================

Carr and Madan (1999) noticed that the call price, as a function of the
log strike :math:`k`, is not integrable, but :math:`e^{\alpha k} C(k)`
is, and that its Fourier transform is a simple expression in the
characteristic function. One fast Fourier transform then prices calls
at thousands of strikes at once, for any model with a known
characteristic function. The grids are linked: integration spacing
:math:`\eta` and strike spacing :math:`\lambda` satisfy
:math:`\lambda\eta = 2\pi/N`.

.. GENERATED FROM PYTHON SOURCE LINES 16-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.options import black_scholes_price
    from financialkit.stochastic_volatility import BlackScholesModel, HestonModel, carr_madan_fft, carr_madan_price, lewis_price

    S, T, r = 100.0, 1.0, 0.03








.. GENERATED FROM PYTHON SOURCE LINES 25-27

A whole strike grid from one FFT
--------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-42

.. code-block:: Python


    bs = carr_madan_fft(BlackScholesModel(0.2), S, T, r)
    mid = (bs.strikes > 50) & (bs.strikes < 200)
    err = np.abs(bs.calls[mid] - black_scholes_price(S, bs.strikes[mid], T, r, 0.2))
    print(f"{mid.sum()} strikes between 50 and 200, max error against Black-Scholes {err.max():.1e}")

    heston = HestonModel(0.04, 1.5, 0.04, 0.5, -0.7)
    res = carr_madan_fft(heston, S, T, r)
    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4))
    ax1.plot(res.strikes[mid], res.calls[mid], label="Heston, FFT")
    ax1.plot(bs.strikes[mid], bs.calls[mid], "--", label="Black-Scholes 20%")
    ax1.set_xlabel("strike K")
    ax1.set_ylabel("call price")
    ax1.legend()




.. image-sg:: /api/gallery/stochastic_volatility/fourier/images/sphx_glr_plot_01_carr_madan_fft_001.png
   :alt: plot 01 carr madan fft
   :srcset: /api/gallery/stochastic_volatility/fourier/images/sphx_glr_plot_01_carr_madan_fft_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    226 strikes between 50 and 200, max error against Black-Scholes 2.2e-07

    <matplotlib.legend.Legend object at 0x116f57b60>



.. GENERATED FROM PYTHON SOURCE LINES 43-49

The damping exponent
--------------------

Too little damping and the integrand is large near :math:`v = 0`; too
much and the payoff's growth is amplified. Mid-range values of
:math:`\alpha` are accurate to many digits.

.. GENERATED FROM PYTHON SOURCE LINES 49-60

.. code-block:: Python


    K = np.array([80.0, 100.0, 120.0])
    ref = np.asarray(lewis_price(heston, S, K, T, r))
    alphas = np.linspace(0.1, 8.0, 40)
    errors = [np.max(np.abs(carr_madan_price(heston, S, K, T, r, alpha=a) - ref)) for a in alphas]
    ax2.semilogy(alphas, errors, marker="o", ms=3)
    ax2.set_xlabel("damping alpha")
    ax2.set_ylabel("max error at K = 80, 100, 120")
    fig.tight_layout()

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.749 seconds)


.. _sphx_glr_download_api_gallery_stochastic_volatility_fourier_plot_01_carr_madan_fft.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_01_carr_madan_fft.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_carr_madan_fft.ipynb <plot_01_carr_madan_fft.ipynb>`

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

      :download:`Download Python source code: plot_01_carr_madan_fft.py <plot_01_carr_madan_fft.py>`

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

      :download:`Download zipped: plot_01_carr_madan_fft.zip <plot_01_carr_madan_fft.zip>`


.. only:: html

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

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