.. 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_01_cryer_psor.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_01_cryer_psor.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_01_cryer_psor.py:


Cryer's projected SOR
=====================

An American option is either held, where it satisfies the Black-Scholes
equation, or exercised, where it equals its payoff; each implicit step
becomes a linear complementarity problem
:math:`Ax \ge b,\ x \ge g,\ (Ax - b)^\top(x - g) = 0`. Colin Cryer (1971)
solved it by successive over-relaxation, projecting each update back
onto :math:`x \ge g`. :func:`~financialkit.pde_methods.psor` does this
inside :class:`~financialkit.pde_methods.CrankNicolson`; the sweep count
depends strongly on the relaxation parameter :math:`\omega`.

.. GENERATED FROM PYTHON SOURCE LINES 16-30

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.options import CRRTree, black_scholes_price
    from financialkit.pde_methods import CrankNicolson
    from financialkit.pde_methods.visualizers import plot_exercise_boundary

    # Hull, 10th ed., Example 21.1
    S, K, T, r, sigma = 50.0, 50.0, 5 / 12, 0.10, 0.40
    res = CrankNicolson(200, 100, rannacher_steps=2, american_method="psor").price(S, K, T, r, sigma, option_type="put", american=True)
    tree = CRRTree(2000).price(S, K, T, r, sigma, option_type="put", american=True).price
    print(f"PSOR {res.price:.4f}, CRR tree {tree:.4f}, European {black_scholes_price(S, K, T, r, sigma, option_type='put'):.4f}")
    print(f"{res.lcp_iterations} PSOR sweeps over 100 time steps")





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

 .. code-block:: none

    PSOR 4.2793, CRR tree 4.2839, European 4.0760
    1156 PSOR sweeps over 100 time steps




.. GENERATED FROM PYTHON SOURCE LINES 31-33

The option and its early-exercise boundary
------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 33-42

.. code-block:: Python


    fig, axes = plt.subplots(1, 3, figsize=(14, 4))
    near = res.S < 90
    axes[0].plot(res.S[near], res.V[near], label="American put")
    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].legend()
    plot_exercise_boundary(res, ax=axes[1])




.. image-sg:: /api/gallery/pde_methods/american/images/sphx_glr_plot_01_cryer_psor_001.png
   :alt: plot 01 cryer psor
   :srcset: /api/gallery/pde_methods/american/images/sphx_glr_plot_01_cryer_psor_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <Axes: xlabel='time t', ylabel='critical spot S*(t)'>



.. GENERATED FROM PYTHON SOURCE LINES 43-45

Sweeps against the relaxation parameter
---------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 45-56

.. code-block:: Python


    omegas = np.linspace(1.0, 1.8, 9)
    sweeps = [
        CrankNicolson(100, 50, rannacher_steps=2, american_method="psor", omega=w).price(S, K, T, r, sigma, option_type="put", american=True).lcp_iterations
        for w in omegas
    ]
    axes[2].plot(omegas, sweeps, "o-")
    axes[2].set_xlabel(r"relaxation $\omega$")
    axes[2].set_ylabel("total sweeps (M = 100, N = 50)")
    fig.tight_layout()
    plt.show()








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

   **Total running time of the script:** (0 minutes 0.190 seconds)


.. _sphx_glr_download_api_gallery_pde_methods_american_plot_01_cryer_psor.py:

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        :alt: Launch JupyterLite
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    .. container:: sphx-glr-download sphx-glr-download-jupyter

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

      :download:`Download Python source code: plot_01_cryer_psor.py <plot_01_cryer_psor.py>`

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