.. 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_06_explicit_is_trinomial.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_06_explicit_is_trinomial.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_06_explicit_is_trinomial.py:


Brennan-Schwartz: explicit schemes are trinomial trees
======================================================

Brennan and Schwartz (1978) noticed that in :math:`x = \ln S` the explicit
scheme's weights on the three neighbouring nodes sum to one and match the
mean and variance of the log-price increment: the explicit scheme is a
trinomial tree, and its stability condition is just the requirement that
the weights be probabilities. With :math:`\Delta x = \sigma\sqrt{3\Delta t}`
the middle weight is 2/3, Hull's standard tree.
:func:`~financialkit.pde_methods.log_explicit_price` runs the scheme, and
:func:`~financialkit.pde_methods.explicit_trinomial_probabilities` shows the
weights turn negative exactly where the scheme becomes unstable.

.. GENERATED FROM PYTHON SOURCE LINES 17-28

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.options import CRRTree, black_scholes_price
    from financialkit.pde_methods import explicit_trinomial_probabilities, log_explicit_price

    S, K, T, r, sigma = 50.0, 50.0, 5 / 12, 0.10, 0.40
    dt = T / 200
    p = explicit_trinomial_probabilities(sigma, r, dt, sigma * np.sqrt(3 * dt))
    print(f"p_down {p.p_down:.4f}, p_mid {p.p_mid:.4f}, p_up {p.p_up:.4f}, total {p.total:.4f}")





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

 .. code-block:: none

    p_down 0.1660, p_mid 0.6667, p_up 0.1673, total 1.0000




.. GENERATED FROM PYTHON SOURCE LINES 29-31

The scheme prices like a tree: European and American puts
---------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 31-46

.. code-block:: Python


    ns = np.arange(50, 501, 50)
    eu = [log_explicit_price(S, K, T, r, sigma, option_type="put", n_time=n).price for n in ns]
    am = [log_explicit_price(S, K, T, r, sigma, option_type="put", american=True, n_time=n).price for n in ns]
    crr = CRRTree(2000).price(S, K, T, r, sigma, option_type="put", american=True).price

    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    axes[0].plot(ns, eu, "o-", label="explicit log-spot, European")
    axes[0].axhline(black_scholes_price(S, K, T, r, sigma, option_type="put"), color="k", lw=1, label="Black-Scholes")
    axes[0].plot(ns, am, "s-", label="explicit log-spot, American")
    axes[0].axhline(crr, color="C3", lw=1, ls="--", label="CRR tree, 2000 steps")
    axes[0].set_xlabel("time steps N")
    axes[0].set_ylabel("put value")
    axes[0].legend(fontsize=8)




.. image-sg:: /api/gallery/pde_methods/schemes/images/sphx_glr_plot_06_explicit_is_trinomial_001.png
   :alt: plot 06 explicit is trinomial
   :srcset: /api/gallery/pde_methods/schemes/images/sphx_glr_plot_06_explicit_is_trinomial_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x116d7e3c0>



.. GENERATED FROM PYTHON SOURCE LINES 47-49

Negative probabilities are instability
--------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 49-60

.. code-block:: Python


    lam = np.linspace(0.2, 1.6, 50)
    p_mid = [explicit_trinomial_probabilities(sigma, r, dt, sigma * np.sqrt(dt / l)).p_mid for l in lam]
    axes[1].plot(lam, p_mid)
    axes[1].axhline(0.0, color="k", lw=0.8)
    axes[1].axvline(1.0, color="C3", ls="--", label="stability limit")
    axes[1].set_xlabel(r"diffusion number $\sigma^2 \Delta t / \Delta x^2$")
    axes[1].set_ylabel("middle weight p_mid")
    axes[1].legend()
    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_06_explicit_is_trinomial.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_06_explicit_is_trinomial.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_06_explicit_is_trinomial.ipynb <plot_06_explicit_is_trinomial.ipynb>`

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

      :download:`Download Python source code: plot_06_explicit_is_trinomial.py <plot_06_explicit_is_trinomial.py>`

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

      :download:`Download zipped: plot_06_explicit_is_trinomial.zip <plot_06_explicit_is_trinomial.zip>`


.. only:: html

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

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