.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/stochastic_volatility/smile/plot_02_svi.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_smile_plot_02_svi.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_smile_plot_02_svi.py:


Gatheral's SVI
==============

Jim Gatheral presented the stochastic volatility inspired (SVI)
parametrisation at Global Derivatives in 2004: the total implied
variance :math:`w(k) = \sigma_{BS}^2 T` in log-moneyness :math:`k`,

.. math:: w(k) = a + b\left[\rho(k - m) + \sqrt{(k - m)^2 + \sigma^2}\right],

a hyperbola whose asymptotes are the linear wings Lee's theorem
requires. It is the large-maturity limit of Heston's smile, and five
parameters fit equity smiles remarkably well. Durrleman's function
:math:`g(k)` must stay non-negative for the implied density to be a
density; here SVI is fitted to a Heston smile and checked.

.. GENERATED FROM PYTHON SOURCE LINES 19-46



.. image-sg:: /api/gallery/stochastic_volatility/smile/images/sphx_glr_plot_02_svi_001.png
   :alt: plot 02 svi
   :srcset: /api/gallery/stochastic_volatility/smile/images/sphx_glr_plot_02_svi_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    T = 0.5: a = 0.0078, b = 0.0468, rho = -0.656, m = 0.096, sigma = 0.045; rms vol error 10.6 bp
    T = 1.0: a = 0.0106, b = 0.0779, rho = -0.613, m = 0.115, sigma = 0.115; rms vol error 1.8 bp
    T = 2.0: a = 0.0099, b = 0.1167, rho = -0.508, m = 0.192, sigma = 0.240; rms vol error 0.8 bp






|

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.stochastic_volatility import HestonModel, durrleman_g, svi_fit
    from financialkit.stochastic_volatility.visualizers import plot_smile

    S, r = 100.0, 0.0
    m = HestonModel(0.04, 1.5, 0.04, 0.6, -0.7)
    k = np.linspace(-0.6, 0.4, 21)

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4))
    for T in (0.5, 1.0, 2.0):
        iv = m.implied_vol(S, S * np.exp(k), T, r)
        fit = svi_fit(k, iv**2 * T, T)
        rms = np.sqrt(np.mean((fit.implied_vol(k) - iv) ** 2))
        print(f"T = {T}: a = {fit.a:.4f}, b = {fit.b:.4f}, rho = {fit.rho:.3f}, m = {fit.m:.3f}, sigma = {fit.sigma:.3f}; rms vol error {rms * 1e4:.1f} bp")
        line = ax1.plot(k, iv, "o", ms=3)[0]
        kk = np.linspace(-1.0, 0.8, 200)
        plot_smile(S * np.exp(kk), fit.implied_vol(kk), forward=S, ax=ax1, label=f"SVI, T = {T}", color=line.get_color())
        ax2.plot(kk, durrleman_g(kk, fit.a, fit.b, fit.rho, fit.m, fit.sigma), label=f"T = {T}")
    ax2.axhline(0.0, color="k", lw=0.8)
    ax2.set_xlabel("log-moneyness k")
    ax2.set_ylabel("Durrleman g(k)")
    ax2.legend()
    fig.tight_layout()

    plt.show()


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

   **Total running time of the script:** (0 minutes 0.178 seconds)


.. _sphx_glr_download_api_gallery_stochastic_volatility_smile_plot_02_svi.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/smile/plot_02_svi.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_svi.ipynb <plot_02_svi.ipynb>`

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

      :download:`Download Python source code: plot_02_svi.py <plot_02_svi.py>`

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

      :download:`Download zipped: plot_02_svi.zip <plot_02_svi.zip>`


.. only:: html

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

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