.. 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_02_laasonen_implicit.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_02_laasonen_implicit.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_02_laasonen_implicit.py:


Laasonen's implicit scheme
==========================

Pentti Laasonen (1949) evaluated the space derivatives of the heat
equation at the *new* time level,
:math:`(I - \Delta\tau A) V^{n+1} = V^n`. Every step is then a linear
solve, but no step is too large: the scheme is stable and monotone for
any :math:`\Delta\tau`, at the price of first-order accuracy in time.
:class:`~financialkit.pde_methods.ImplicitScheme` shows both sides: a
large step that would wreck the explicit scheme gives a sensible answer,
and halving the step halves the error.

.. GENERATED FROM PYTHON SOURCE LINES 16-26

.. 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, ImplicitScheme, explicit_stability
    from financialkit.pde_methods.visualizers import plot_grid_convergence

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








.. GENERATED FROM PYTHON SOURCE LINES 27-29

A time step 400 times the explicit limit
----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 29-35

.. code-block:: Python


    res = ImplicitScheme(400, 10).price(S, K, T, r, sigma, option_type="put")
    limit = explicit_stability(res.S, 1.0, sigma, r).limit
    print(f"explicit limit {limit:.2e}, implicit step {T / 10:.2e}")
    print(f"implicit put with 10 steps {res.price:.4f}, Black-Scholes {exact:.4f}")





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

 .. code-block:: none

    explicit limit 1.63e-04, implicit step 1.00e-01
    implicit put with 10 steps 5.4785, Black-Scholes 5.5735




.. GENERATED FROM PYTHON SOURCE LINES 36-38

First order in time, against Crank-Nicolson's second
----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 38-48

.. code-block:: Python


    ns = np.array([10, 20, 40, 80, 160])
    imp = [abs(ImplicitScheme(800, n, s_max=400.0).price(S, K, T, r, sigma, option_type="put").price - exact) for n in ns]
    cn = [abs(CrankNicolson(800, n, rannacher_steps=2, s_max=400.0).price(S, K, T, r, sigma, option_type="put").price - exact) for n in ns]
    fig, ax = plt.subplots(figsize=(6, 4))
    plot_grid_convergence(ns, imp, ax=ax, label="Laasonen (implicit)", rates=(1.0,))
    ax.loglog(ns, cn, "s-", label="Crank-Nicolson")
    ax.set_xlabel("time steps N")
    ax.legend()
    plt.show()



.. image-sg:: /api/gallery/pde_methods/schemes/images/sphx_glr_plot_02_laasonen_implicit_001.png
   :alt: plot 02 laasonen implicit
   :srcset: /api/gallery/pde_methods/schemes/images/sphx_glr_plot_02_laasonen_implicit_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.101 seconds)


.. _sphx_glr_download_api_gallery_pde_methods_schemes_plot_02_laasonen_implicit.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_02_laasonen_implicit.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_laasonen_implicit.ipynb <plot_02_laasonen_implicit.ipynb>`

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

      :download:`Download Python source code: plot_02_laasonen_implicit.py <plot_02_laasonen_implicit.py>`

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

      :download:`Download zipped: plot_02_laasonen_implicit.zip <plot_02_laasonen_implicit.zip>`


.. only:: html

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

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