.. 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_02_lewis_formula.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_02_lewis_formula.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_02_lewis_formula.py:


Lewis's single-integral formula
===============================

Alan Lewis (2001) moved the Fourier inversion contour to
:math:`\operatorname{Im} u = -\tfrac12`, half way between the strips
where the call and the put transforms live. There the price is one real
integral,

.. math::

   C = S e^{-qT} - \frac{\sqrt{FK}\,e^{-rT}}{\pi}\int_0^\infty
   \operatorname{Re}\!\left[e^{iu\ln(F/K)}\,\phi(u - \tfrac i2)\right]\frac{du}{u^2 + \tfrac14},

instead of Heston's two, with an integrand that decays like
:math:`u^{-2}` whatever the model. Here it is compared with Heston's
two-probability formula: the integrands, and the prices they give.

.. GENERATED FROM PYTHON SOURCE LINES 21-45



.. image-sg:: /api/gallery/stochastic_volatility/fourier/images/sphx_glr_plot_02_lewis_formula_001.png
   :alt: Heston, variance gamma
   :srcset: /api/gallery/stochastic_volatility/fourier/images/sphx_glr_plot_02_lewis_formula_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Heston: Lewis 0.41986248, Heston's formula 0.41986248
    variance gamma: Lewis 0.73063766, Heston's formula 0.73063765






|

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.stochastic_volatility import HestonModel, VarianceGammaModel, heston_price, lewis_price

    S, K, T, r = 100.0, 110.0, 0.25, 0.0
    models = {"Heston": HestonModel(0.04, 1.5, 0.04, 0.5, -0.7), "variance gamma": VarianceGammaModel(0.2, 0.2, -0.2)}

    fig, axes = plt.subplots(1, 2, figsize=(10, 4))
    u = np.linspace(1e-3, 60, 600)
    kap = np.log(S / K)
    for ax, (name, m) in zip(axes, models.items(), strict=True):
        lewis = np.real(np.exp(1j * u * kap) * m.char_func(u - 0.5j, T)) / (u**2 + 0.25)
        p2 = np.real(np.exp(1j * u * kap) * m.char_func(u, T) / (1j * u))
        ax.plot(u, np.abs(lewis), label="Lewis integrand")
        ax.plot(u, np.abs(p2), label="Heston P2 integrand")
        ax.set_yscale("log")
        ax.set_xlabel("u")
        ax.set_title(name)
        ax.legend()
        print(f"{name}: Lewis {lewis_price(m, S, K, T, r):.8f}, Heston's formula {heston_price(m, S, K, T, r):.8f}")
    fig.tight_layout()

    plt.show()


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

   **Total running time of the script:** (0 minutes 0.124 seconds)


.. _sphx_glr_download_api_gallery_stochastic_volatility_fourier_plot_02_lewis_formula.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_02_lewis_formula.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_lewis_formula.ipynb <plot_02_lewis_formula.ipynb>`

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

      :download:`Download Python source code: plot_02_lewis_formula.py <plot_02_lewis_formula.py>`

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

      :download:`Download zipped: plot_02_lewis_formula.zip <plot_02_lewis_formula.zip>`


.. only:: html

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

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