.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/stochastic_volatility/heston/plot_02_heston.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_heston_plot_02_heston.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_heston_plot_02_heston.py:


The Heston model
================

Steven Heston (1993) gave the variance a mean-reverting square-root
diffusion correlated with the stock,

.. math::

   dS = r S\,dt + \sqrt{v} S\,dW_1, \qquad dv = \kappa(\theta - v)\,dt + \xi\sqrt{v}\,dW_2, \qquad dW_1 dW_2 = \rho\,dt,

and found its characteristic function in closed form. The call is
:math:`S P_1 - K e^{-rT} P_2`, with each exercise probability a single
Fourier integral. A negative correlation, the stock falling as
volatility rises, produces the equity skew; the volatility of variance
:math:`\xi` produces the curvature.

.. GENERATED FROM PYTHON SOURCE LINES 20-35

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.stochastic_volatility import HestonModel, heston_paths, heston_price
    from financialkit.stochastic_volatility.visualizers import plot_heston_paths, plot_smile

    S, T, r = 100.0, 1.0, 0.02
    base = dict(v0=0.04, kappa=1.5, theta=0.04, xi=0.5)

    m = HestonModel(rho=-0.7, **base)
    print("Feller condition 2 kappa theta >= xi^2:", m.feller_satisfied)
    print(f"at-the-money call by Heston's two probabilities: {heston_price(m, S, 100.0, T, r):.4f}")

    plot_heston_paths(heston_paths(m, S, T, r, n_steps=250, n_paths=10, rng=0))




.. image-sg:: /api/gallery/stochastic_volatility/heston/images/sphx_glr_plot_02_heston_001.png
   :alt: Heston paths (qe)
   :srcset: /api/gallery/stochastic_volatility/heston/images/sphx_glr_plot_02_heston_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Feller condition 2 kappa theta >= xi^2: False
    at-the-money call by Heston's two probabilities: 8.1950

    array([<Axes: title={'center': 'Heston paths (qe)'}, xlabel='t', ylabel='spot S'>,
           <Axes: xlabel='t', ylabel='volatility sqrt(v)'>], dtype=object)



.. GENERATED FROM PYTHON SOURCE LINES 36-38

Correlation sets the skew, vol of vol the curvature
---------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 38-49

.. code-block:: Python


    K = np.linspace(70, 135, 27)
    F = S * np.exp(r * T)
    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4), sharey=True)
    for rho in (-0.7, 0.0, 0.7):
        plot_smile(K, HestonModel(rho=rho, **base).implied_vol(S, K, T, r), forward=F, ax=ax1, label=f"rho = {rho}")
    for xi in (0.2, 0.5, 0.9):
        plot_smile(K, HestonModel(0.04, 1.5, 0.04, xi, 0.0).implied_vol(S, K, T, r), forward=F, ax=ax2, label=f"xi = {xi}, rho = 0")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/stochastic_volatility/heston/images/sphx_glr_plot_02_heston_002.png
   :alt: plot 02 heston
   :srcset: /api/gallery/stochastic_volatility/heston/images/sphx_glr_plot_02_heston_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.193 seconds)


.. _sphx_glr_download_api_gallery_stochastic_volatility_heston_plot_02_heston.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/heston/plot_02_heston.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_heston.ipynb <plot_02_heston.ipynb>`

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

      :download:`Download Python source code: plot_02_heston.py <plot_02_heston.py>`

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

      :download:`Download zipped: plot_02_heston.zip <plot_02_heston.zip>`


.. only:: html

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

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