.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/pde_methods/schemes/plot_05_schwartz_explicit.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_pde_methods_schemes_plot_05_schwartz_explicit.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_pde_methods_schemes_plot_05_schwartz_explicit.py:


Schwartz: explicit finite differences
=====================================

Eduardo Schwartz (1977) valued warrants, which no formula covered, by
replacing the derivatives in the Black-Scholes equation with differences
and stepping back from expiry one explicit step at a time:

.. math:: V_i^{n+1} = V_i^n + \Delta\tau\,(a_i V_{i-1}^n + b_i V_i^n + c_i V_{i+1}^n).

No linear system is solved, so each step is cheap, but the time step
must stay below a limit set by the finest spot spacing.
:class:`~financialkit.pde_methods.ExplicitScheme` reproduces Hull's
Example 15.6 call, and its whole grid matches Black-Scholes.

.. GENERATED FROM PYTHON SOURCE LINES 18-30

.. code-block:: Python

    import matplotlib.pyplot as plt

    from financialkit.options import black_scholes_price
    from financialkit.pde_methods import ExplicitScheme
    from financialkit.pde_methods.visualizers import plot_value_profile

    # Hull, 10th ed., Example 15.6
    S, K, T, r, sigma = 42.0, 40.0, 0.5, 0.10, 0.20
    res = ExplicitScheme(320, 2500, s_max=160.0).price(S, K, T, r, sigma)
    print(f"explicit {res.price:.4f}, Black-Scholes {black_scholes_price(S, K, T, r, sigma):.4f}")
    print(f"time step {res.stability.number:.2e} below the limit {res.stability.limit:.2e}: {res.stability.stable}")





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

 .. code-block:: none

    explicit 4.7576, Black-Scholes 4.7594
    time step 2.00e-04 below the limit 2.57e-04: True




.. GENERATED FROM PYTHON SOURCE LINES 31-33

The whole grid at once
----------------------

.. GENERATED FROM PYTHON SOURCE LINES 33-43

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    plot_value_profile(res, exact=black_scholes_price(res.S, K, T, r, sigma), ax=axes[0], xlim=(20, 70))
    err = res.V - black_scholes_price(res.S, K, T, r, sigma)
    axes[1].plot(res.S, err)
    axes[1].set_xlim(0, 100)
    axes[1].set_xlabel("spot S")
    axes[1].set_ylabel("explicit - Black-Scholes")
    fig.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/pde_methods/schemes/images/sphx_glr_plot_05_schwartz_explicit_001.png
   :alt: plot 05 schwartz explicit
   :srcset: /api/gallery/pde_methods/schemes/images/sphx_glr_plot_05_schwartz_explicit_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.103 seconds)


.. _sphx_glr_download_api_gallery_pde_methods_schemes_plot_05_schwartz_explicit.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/pde_methods/schemes/plot_05_schwartz_explicit.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_05_schwartz_explicit.ipynb <plot_05_schwartz_explicit.ipynb>`

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

      :download:`Download Python source code: plot_05_schwartz_explicit.py <plot_05_schwartz_explicit.py>`

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

      :download:`Download zipped: plot_05_schwartz_explicit.zip <plot_05_schwartz_explicit.zip>`


.. only:: html

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

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