.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/pde_methods/american/plot_02_brennan_schwartz.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_american_plot_02_brennan_schwartz.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_american_plot_02_brennan_schwartz.py:


Brennan-Schwartz American put
=============================

Michael Brennan and Eduardo Schwartz (1977) priced the American put with
an implicit scheme and a direct solve: eliminate the tridiagonal system
starting from the top of the grid, then substitute back upward from
:math:`S = 0`, taking the larger of the computed value and the payoff at
each node. Because the put is exercised on a single interval at the
bottom of the grid, this one pass solves the complementarity problem
exactly. :func:`~financialkit.pde_methods.brennan_schwartz` matches PSOR
to rounding at a fraction of the cost.

.. GENERATED FROM PYTHON SOURCE LINES 16-32

.. code-block:: Python

    import time

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.options import CRRTree
    from financialkit.pde_methods import CrankNicolson

    S, K, T, r, sigma = 50.0, 50.0, 5 / 12, 0.10, 0.40
    tree = CRRTree(2000).price(S, K, T, r, sigma, option_type="put", american=True).price
    for method in ("brennan_schwartz", "psor"):
        t0 = time.perf_counter()
        res = CrankNicolson(400, 200, rannacher_steps=2, american_method=method).price(S, K, T, r, sigma, option_type="put", american=True)
        print(f"{method:16s} {res.price:.6f} in {time.perf_counter() - t0:.3f} s, {res.lcp_iterations} iterations")
    print(f"CRR tree         {tree:.6f}")





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

 .. code-block:: none

    brennan_schwartz 4.282866 in 0.128 s, 202 iterations
    psor             4.282866 in 0.015 s, 3606 iterations
    CRR tree         4.283922




.. GENERATED FROM PYTHON SOURCE LINES 33-35

The put across spots and maturities
-----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 35-52

.. code-block:: Python


    maturities = [0.1, 0.25, 0.5, 1.0]
    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    for T_ in maturities:
        res = CrankNicolson(200, 100, rannacher_steps=2, american_method="brennan_schwartz").price(S, K, T_, r, sigma, option_type="put", american=True)
        near = res.S < 90
        axes[0].plot(res.S[near], res.V[near], label=f"T = {T_}")
        axes[1].plot(T_ - res.times, res.exercise_boundary, label=f"T = {T_}")
    axes[0].plot(res.S[near], np.maximum(K - res.S[near], 0.0), "k--", lw=1, label="payoff")
    axes[0].set_xlabel("spot S")
    axes[0].set_ylabel("American put")
    axes[0].legend()
    axes[1].set_xlabel("time to maturity")
    axes[1].set_ylabel("critical spot S*")
    axes[1].legend()
    fig.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/pde_methods/american/images/sphx_glr_plot_02_brennan_schwartz_001.png
   :alt: plot 02 brennan schwartz
   :srcset: /api/gallery/pde_methods/american/images/sphx_glr_plot_02_brennan_schwartz_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.259 seconds)


.. _sphx_glr_download_api_gallery_pde_methods_american_plot_02_brennan_schwartz.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/american/plot_02_brennan_schwartz.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_brennan_schwartz.ipynb <plot_02_brennan_schwartz.ipynb>`

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

      :download:`Download Python source code: plot_02_brennan_schwartz.py <plot_02_brennan_schwartz.py>`

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

      :download:`Download zipped: plot_02_brennan_schwartz.zip <plot_02_brennan_schwartz.zip>`


.. only:: html

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

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