.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/stochastic_volatility/local_vol/plot_02_dupire.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_local_vol_plot_02_dupire.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_local_vol_plot_02_dupire.py:


Dupire's local volatility
=========================

In 1994 Bruno Dupire showed that a full surface of call prices
:math:`C(K, T)` pins down a unique local volatility
:math:`\sigma(S, t)` that reproduces it,

.. math::

   \sigma^2(K, T) = \frac{\partial_T C + r K\,\partial_K C}{\tfrac12 K^2\,\partial_{KK} C},

the forward equation of the option price read backwards. Here the prices
come from Cox's CEV model, whose local volatility
:math:`\sigma S^{\beta - 1}` is known, so finite differences of the price
grid can be checked against it. Simulating the recovered diffusion then
reprices the options.

.. GENERATED FROM PYTHON SOURCE LINES 21-44

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.stochastic_volatility import cev_local_vol, cev_price, dupire_local_vol, local_vol_paths
    from financialkit.stochastic_volatility.visualizers import plot_local_vol

    S, r, beta, sigma = 100.0, 0.03, 0.5, 2.5
    K = np.linspace(50, 180, 131)
    T = np.linspace(0.25, 1.5, 51)
    C = cev_price(S, K[None, :], T[:, None], r, sigma, beta)

    lv = dupire_local_vol(K, T, C, r)
    true = cev_local_vol(K, sigma, beta)
    # The first maturity row uses one-sided differences in T, which are least accurate in the wings.
    inner = (K > 60) & (K < 160)
    print("max |Dupire - CEV| for 60 < K < 160, T > 0.25:", f"{np.nanmax(np.abs(lv.vol[1:, inner] - true[inner])):.1e}")

    fig, ax = plt.subplots(figsize=(7, 4))
    plot_local_vol(lv, maturities=[0.5, 1.0, 1.5], ax=ax)
    ax.plot(K, true, "k--", lw=1, label="CEV: sigma S^(beta - 1)")
    ax.set_ylim(0.1, 0.4)
    ax.legend()




.. image-sg:: /api/gallery/stochastic_volatility/local_vol/images/sphx_glr_plot_02_dupire_001.png
   :alt: plot 02 dupire
   :srcset: /api/gallery/stochastic_volatility/local_vol/images/sphx_glr_plot_02_dupire_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    max |Dupire - CEV| for 60 < K < 160, T > 0.25: 3.7e-03

    <matplotlib.legend.Legend object at 0x1172b4050>



.. GENERATED FROM PYTHON SOURCE LINES 45-47

Repricing by simulation
-----------------------

.. GENERATED FROM PYTHON SOURCE LINES 47-54

.. code-block:: Python


    t, paths = local_vol_paths(lv, S, 1.0, r, n_steps=100, n_paths=20_000, rng=1)
    for k in (80.0, 100.0, 120.0):
        pay = np.exp(-r) * np.maximum(paths[:, -1] - k, 0.0)
        print(f"K = {k:5.0f}: CEV {cev_price(S, k, 1.0, r, sigma, beta):.4f}, local-vol MC {pay.mean():.4f} +/- {2 * pay.std() / np.sqrt(pay.size):.4f}")

    plt.show()




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

 .. code-block:: none

    K =    80: CEV 24.4555, local-vol MC 24.6751 +/- 0.3131
    K =   100: CEV 11.3548, local-vol MC 11.5660 +/- 0.2367
    K =   120: CEV 4.0744, local-vol MC 4.2337 +/- 0.1463





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

   **Total running time of the script:** (0 minutes 2.561 seconds)


.. _sphx_glr_download_api_gallery_stochastic_volatility_local_vol_plot_02_dupire.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/local_vol/plot_02_dupire.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_dupire.ipynb <plot_02_dupire.ipynb>`

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

      :download:`Download Python source code: plot_02_dupire.py <plot_02_dupire.py>`

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

      :download:`Download zipped: plot_02_dupire.zip <plot_02_dupire.zip>`


.. only:: html

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

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