.. 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_01_crank_nicolson.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_01_crank_nicolson.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_01_crank_nicolson.py:


Crank-Nicolson
==============

John Crank and Phyllis Nicolson (1947) averaged the explicit and implicit
differences of the heat equation around the half step,

.. math:: \frac{V^{n+1} - V^n}{\Delta\tau} = \tfrac12 \bigl(A V^{n+1} + A V^n\bigr),

which makes the scheme unconditionally stable *and* second-order accurate
in time. Applied to the Black-Scholes equation it is the workhorse of
finite-difference option pricing. Here
:class:`~financialkit.pde_methods.CrankNicolson` prices a call and its
error falls by four each time both grids are halved.

.. GENERATED FROM PYTHON SOURCE LINES 18-31

.. 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
    from financialkit.pde_methods.visualizers import plot_grid_convergence, plot_value_surface

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

    res = CrankNicolson(200, 100, rannacher_steps=2).price(S, K, T, r, sigma)
    print(f"Crank-Nicolson {res.price:.5f}, Black-Scholes {exact:.5f}")





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

 .. code-block:: none

    Crank-Nicolson 10.44060, Black-Scholes 10.45058




.. GENERATED FROM PYTHON SOURCE LINES 32-34

The value surface, from the payoff at expiry back to today
----------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 34-39

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    plot_value_surface(res, ax=axes[0], s_range=(40, 180))
    axes[0].set_title("call value V(S, t)")




.. image-sg:: /api/gallery/pde_methods/schemes/images/sphx_glr_plot_01_crank_nicolson_001.png
   :alt: call value V(S, t)
   :srcset: /api/gallery/pde_methods/schemes/images/sphx_glr_plot_01_crank_nicolson_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'call value V(S, t)')



.. GENERATED FROM PYTHON SOURCE LINES 40-42

Second order in space and time together
---------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 42-49

.. code-block:: Python


    ns = np.array([25, 50, 100, 200, 400])
    errors = [abs(CrankNicolson(n, n, rannacher_steps=2, s_max=400.0).price(S, K, T, r, sigma).price - exact) for n in ns]
    plot_grid_convergence(ns, errors, ax=axes[1], label="Crank-Nicolson, M = N = n", rates=(2.0,))
    axes[1].set_title("error against grid size")
    fig.tight_layout()
    plt.show()








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

   **Total running time of the script:** (0 minutes 0.140 seconds)


.. _sphx_glr_download_api_gallery_pde_methods_schemes_plot_01_crank_nicolson.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_01_crank_nicolson.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_crank_nicolson.ipynb <plot_01_crank_nicolson.ipynb>`

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

      :download:`Download Python source code: plot_01_crank_nicolson.py <plot_01_crank_nicolson.py>`

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

      :download:`Download zipped: plot_01_crank_nicolson.zip <plot_01_crank_nicolson.zip>`


.. only:: html

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

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