.. 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_03_penalty_method.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_03_penalty_method.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_03_penalty_method.py:


Forsyth-Vetzal penalty method
=============================

Peter Forsyth and Kenneth Vetzal (2002) replaced the complementarity
constraint by a penalty: on nodes where the value falls below the
payoff, add :math:`\rho\,(g - x)` with a large :math:`\rho`. The resulting
nonlinear system is solved by a Newton (policy) iteration that only
changes which nodes are penalised, and converges in two or three
tridiagonal solves per time step, independently of the grid size --
unlike PSOR, whose sweep count grows as the grid is refined.
:func:`~financialkit.pde_methods.penalty_solve` implements it.

.. GENERATED FROM PYTHON SOURCE LINES 16-33

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.pde_methods import CrankNicolson

    S, K, T, r, sigma = 100.0, 100.0, 1.0, 0.06, 0.3
    ms = np.array([50, 100, 200, 400])
    iters = {"penalty": [], "psor": []}
    prices = {"penalty": [], "psor": []}
    for m in ms:
        for method in iters:
            res = CrankNicolson(m, 50, rannacher_steps=2, american_method=method).price(S, K, T, r, sigma, option_type="put", american=True)
            iters[method].append(res.lcp_iterations / 54)  # 50 steps, two of them as four half steps
            prices[method].append(res.price)
    for m, a, b in zip(ms, prices["penalty"], prices["psor"], strict=True):
        print(f"M = {m:3d}: penalty {a:.6f}, PSOR {b:.6f}")





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

 .. code-block:: none

    M =  50: penalty 9.432503, PSOR 9.432503
    M = 100: penalty 9.510534, PSOR 9.510534
    M = 200: penalty 9.523474, PSOR 9.523474
    M = 400: penalty 9.526267, PSOR 9.526266




.. GENERATED FROM PYTHON SOURCE LINES 34-36

Iterations per time step as the grid is refined
------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 36-44

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(6, 4))
    ax.semilogy(ms, iters["penalty"], "o-", label="penalty (Newton)")
    ax.semilogy(ms, iters["psor"], "s-", label="PSOR")
    ax.set_xlabel("spot intervals M")
    ax.set_ylabel("iterations per time step")
    ax.legend()
    plt.show()



.. image-sg:: /api/gallery/pde_methods/american/images/sphx_glr_plot_03_penalty_method_001.png
   :alt: plot 03 penalty method
   :srcset: /api/gallery/pde_methods/american/images/sphx_glr_plot_03_penalty_method_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.068 seconds)


.. _sphx_glr_download_api_gallery_pde_methods_american_plot_03_penalty_method.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_03_penalty_method.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_penalty_method.ipynb <plot_03_penalty_method.ipynb>`

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

      :download:`Download Python source code: plot_03_penalty_method.py <plot_03_penalty_method.py>`

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

      :download:`Download zipped: plot_03_penalty_method.zip <plot_03_penalty_method.zip>`


.. only:: html

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

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