.. 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_01_hull_white_mixing.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_01_hull_white_mixing.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_01_hull_white_mixing.py:


Hull and White's stochastic volatility
======================================

Hull and White (1987) let the variance follow its own lognormal
diffusion, :math:`dV = \xi V\,dW`, independent of the stock. Conditional
on the variance path the stock is lognormal with the path's *average*
variance :math:`\bar V`, so the option price is the Black-Scholes price
averaged over the distribution of :math:`\bar V`: the mixing formula.
They expanded it in a series in :math:`\xi^2 T`; here it is also
computed by simulating variance paths. Because the variance is
uncorrelated with the stock the smile is symmetric, and the at-the-money
price falls below Black-Scholes, which is concave in the variance there.

.. GENERATED FROM PYTHON SOURCE LINES 17-26

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.options import black_scholes_price, implied_volatility
    from financialkit.stochastic_volatility import hull_white_mixing_mc, hull_white_price
    from financialkit.stochastic_volatility.visualizers import plot_smile

    S, T, r, sigma0 = 100.0, 1.0, 0.0, 0.2








.. GENERATED FROM PYTHON SOURCE LINES 27-32

Series against mixing Monte Carlo
---------------------------------

The series is accurate while :math:`\xi^2 T` is small and drifts away as
it grows; the mixing Monte Carlo has no such limit.

.. GENERATED FROM PYTHON SOURCE LINES 32-40

.. code-block:: Python


    bs = black_scholes_price(S, 100.0, T, r, sigma0)
    print(f"Black-Scholes at sigma0: {bs:.4f}")
    print("   xi   series   mixing MC  (+/- 2 se)")
    for xi in (0.2, 0.4, 0.8, 1.2):
        mc = hull_white_mixing_mc(S, 100.0, T, r, sigma0, xi, n_paths=20_000, rng=0)
        print(f"{xi:5.1f} {hull_white_price(S, 100.0, T, r, sigma0, xi):8.4f}   {mc.estimate:8.4f}   {2 * mc.std_error:.4f}")





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

 .. code-block:: none

    Black-Scholes at sigma0: 7.9656
       xi   series   mixing MC  (+/- 2 se)
      0.2   7.9524     7.9566   0.0065
      0.4   7.9156     7.9214   0.0130
      0.8   7.8466     7.7753   0.0256
      1.2   8.6741     7.5371   0.0377




.. GENERATED FROM PYTHON SOURCE LINES 41-43

The symmetric smile
-------------------

.. GENERATED FROM PYTHON SOURCE LINES 43-53

.. code-block:: Python


    F = S * np.exp(r * T)
    K = F * np.exp(np.linspace(-0.4, 0.4, 17))
    fig, ax = plt.subplots(figsize=(7, 4))
    for xi in (0.2, 0.3, 0.4):
        vols = implied_volatility(hull_white_price(S, K, T, r, sigma0, xi), S, K, T, r)
        plot_smile(K, vols, forward=F, ax=ax, label=f"xi = {xi}")
    ax.axhline(sigma0, color="k", lw=0.8, ls="--")

    plt.show()



.. image-sg:: /api/gallery/stochastic_volatility/heston/images/sphx_glr_plot_01_hull_white_mixing_001.png
   :alt: plot 01 hull white mixing
   :srcset: /api/gallery/stochastic_volatility/heston/images/sphx_glr_plot_01_hull_white_mixing_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.640 seconds)


.. _sphx_glr_download_api_gallery_stochastic_volatility_heston_plot_01_hull_white_mixing.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_01_hull_white_mixing.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_hull_white_mixing.ipynb <plot_01_hull_white_mixing.ipynb>`

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

      :download:`Download Python source code: plot_01_hull_white_mixing.py <plot_01_hull_white_mixing.py>`

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

      :download:`Download zipped: plot_01_hull_white_mixing.zip <plot_01_hull_white_mixing.zip>`


.. only:: html

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

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