.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/pde_methods/stability/plot_03_rannacher_smoothing.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_stability_plot_03_rannacher_smoothing.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_stability_plot_03_rannacher_smoothing.py:


Rannacher smoothing
===================

Crank-Nicolson is stable, but its amplification factor tends to
:math:`-1` for the stiffest modes, so the high-frequency error excited by
a payoff kink is never damped: it just flips sign every step. With large
time steps the price looks fine while delta and gamma oscillate around
the strike. Rolf Rannacher (1984) proposed starting with a few fully
implicit steps, which damp those modes, before switching to
Crank-Nicolson. :class:`~financialkit.pde_methods.CrankNicolson` with
``rannacher_steps=2`` replaces the first two steps by four implicit half
steps.

.. GENERATED FROM PYTHON SOURCE LINES 17-28

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.options import black_scholes_greeks
    from financialkit.pde_methods import CrankNicolson

    S, K, T, r, sigma = 100.0, 100.0, 0.25, 0.05, 0.2
    plain = CrankNicolson(400, 10, s_max=400.0).price(S, K, T, r, sigma)
    smooth = CrankNicolson(400, 10, rannacher_steps=2, s_max=400.0).price(S, K, T, r, sigma)
    print(f"prices: plain CN {plain.price:.4f}, Rannacher {smooth.price:.4f}")





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

 .. code-block:: none

    prices: plain CN 4.5980, Rannacher 4.6062




.. GENERATED FROM PYTHON SOURCE LINES 29-31

Gamma on the grid, from second differences
------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 31-46

.. code-block:: Python


    h = plain.S[1] - plain.S[0]
    s = plain.S[1:-1]
    near = (s > 75) & (s < 125)
    exact = black_scholes_greeks(s[near], K, T, r, sigma).gamma
    fig, axes = plt.subplots(1, 2, figsize=(11, 4), sharey=True)
    for ax, res, title in ((axes[0], plain, "Crank-Nicolson"), (axes[1], smooth, "Crank-Nicolson + Rannacher")):
        ax.plot(s[near], np.diff(res.V, 2)[near] / h**2, ".-", ms=3, label="grid gamma")
        ax.plot(s[near], exact, "k", lw=1, label="Black-Scholes")
        ax.set_title(title)
        ax.set_xlabel("spot S")
        ax.legend()
    axes[0].set_ylabel("gamma")
    fig.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/pde_methods/stability/images/sphx_glr_plot_03_rannacher_smoothing_001.png
   :alt: Crank-Nicolson, Crank-Nicolson + Rannacher
   :srcset: /api/gallery/pde_methods/stability/images/sphx_glr_plot_03_rannacher_smoothing_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.057 seconds)


.. _sphx_glr_download_api_gallery_pde_methods_stability_plot_03_rannacher_smoothing.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/stability/plot_03_rannacher_smoothing.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_rannacher_smoothing.ipynb <plot_03_rannacher_smoothing.ipynb>`

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

      :download:`Download Python source code: plot_03_rannacher_smoothing.py <plot_03_rannacher_smoothing.py>`

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

      :download:`Download zipped: plot_03_rannacher_smoothing.zip <plot_03_rannacher_smoothing.zip>`


.. only:: html

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

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