.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/pde_methods/accuracy/plot_01_richardson_extrapolation.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_accuracy_plot_01_richardson_extrapolation.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_accuracy_plot_01_richardson_extrapolation.py:


Richardson extrapolation
========================

Lewis Fry Richardson (1911), computing stresses in masonry dams by
finite differences, noticed that when the error behaves like
:math:`C h^p`, two solutions on grids :math:`h` and :math:`h/2` determine
both the constant and the limit:
:math:`V \approx V_{h/2} + (V_{h/2} - V_h)/(2^p - 1)`.
:func:`~financialkit.pde_methods.richardson_extrapolation` applies this to
Crank-Nicolson prices; three grids also confirm that the observed order
is 2, which needs the strike on a node and Rannacher's start so that the
error is a clean power of :math:`h`.

.. GENERATED FROM PYTHON SOURCE LINES 17-35

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.options import black_scholes_price
    from financialkit.pde_methods import CrankNicolson, richardson_extrapolation

    S, K, T, r, sigma = 100.0, 100.0, 1.0, 0.05, 0.2
    exact = black_scholes_price(S, K, T, r, sigma)

    ms = np.array([20, 40, 80, 160, 320])
    prices = np.array([CrankNicolson(m, m, rannacher_steps=2, s_max=400.0).price(S, K, T, r, sigma).price for m in ms])
    extrapolated = [richardson_extrapolation(prices[: k + 1]).extrapolated for k in range(1, len(ms))]
    res = richardson_extrapolation(prices)
    print(f"observed order {res.observed_order:.3f}")
    for m, p in zip(ms, prices, strict=True):
        print(f"M = N = {m:3d}: {p:.6f} (error {p - exact:+.2e})")
    print(f"extrapolated:  {res.extrapolated:.6f} (error {res.extrapolated - exact:+.2e})")





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

 .. code-block:: none

    observed order 2.016
    M = N =  20: 9.214957 (error -1.24e+00)
    M = N =  40: 10.189260 (error -2.61e-01)
    M = N =  80: 10.387971 (error -6.26e-02)
    M = N = 160: 10.435070 (error -1.55e-02)
    M = N = 320: 10.446714 (error -3.87e-03)
    extrapolated:  10.450537 (error -4.62e-05)




.. GENERATED FROM PYTHON SOURCE LINES 36-38

The extrapolated sequence converges much faster
-----------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 38-47

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(6, 4))
    ax.loglog(ms, np.abs(prices - exact), "o-", label="Crank-Nicolson")
    ax.loglog(ms[1:], np.abs(np.array(extrapolated) - exact), "s-", label="Richardson from the last two")
    ax.loglog(ms, np.abs(prices[0] - exact) * (ms / ms[0]) ** -2.0, ":", color="0.4", label="$M^{-2}$")
    ax.set_xlabel("grid size M = N")
    ax.set_ylabel("absolute error")
    ax.legend()
    plt.show()



.. image-sg:: /api/gallery/pde_methods/accuracy/images/sphx_glr_plot_01_richardson_extrapolation_001.png
   :alt: plot 01 richardson extrapolation
   :srcset: /api/gallery/pde_methods/accuracy/images/sphx_glr_plot_01_richardson_extrapolation_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.067 seconds)


.. _sphx_glr_download_api_gallery_pde_methods_accuracy_plot_01_richardson_extrapolation.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

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        :alt: Launch JupyterLite
        :width: 150 px

    .. 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_richardson_extrapolation.py <plot_01_richardson_extrapolation.py>`

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