.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/options/trees/plot_04_boyle_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_options_trees_plot_04_boyle_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_options_trees_plot_04_boyle_trinomial.py:


Boyle's trinomial tree
======================

Phelim Boyle (1986) added a third, flat branch to the binomial step.
With :math:`u = e^{\lambda\sigma\sqrt{\Delta t}}`, :math:`d = 1/u` and a
stretch :math:`\lambda > 1`, the three probabilities are chosen to
match the risk-neutral mean and variance of the next price exactly. The
extra branch makes the lattice smoother: for the same number of steps
the trinomial tree is usually more accurate than the binomial one.

.. GENERATED FROM PYTHON SOURCE LINES 14-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.options import CRRTree, TrinomialTree, black_scholes_price
    from financialkit.options.visualizers import plot_tree_convergence

    S, K, T, r, sigma = 100.0, 100.0, 1.0, 0.05, 0.2








.. GENERATED FROM PYTHON SOURCE LINES 23-25

Moment matching
---------------

.. GENERATED FROM PYTHON SOURCE LINES 25-33

.. code-block:: Python


    tree = TrinomialTree(50)
    u, pu, pm, pd = tree.probabilities(T, r, sigma)
    dt = T / 50
    print(f"p_u = {pu:.4f}, p_m = {pm:.4f}, p_d = {pd:.4f}")
    print(f"mean: {pu * u + pm + pd / u:.8f} vs e^(r dt) = {np.exp(r * dt):.8f}")
    print(f"second moment: {pu * u**2 + pm + pd / u**2:.8f} vs {np.exp((2 * r + sigma**2) * dt):.8f}")





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

 .. code-block:: none

    p_u = 0.3427, p_m = 0.3320, p_d = 0.3254
    mean: 1.00100050 vs e^(r dt) = 1.00100050
    second moment: 1.00280392 vs 1.00280392




.. GENERATED FROM PYTHON SOURCE LINES 34-36

Trinomial vs binomial convergence
---------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 36-44

.. code-block:: Python


    exact = black_scholes_price(S, K, T, r, sigma)
    steps = np.arange(10, 300, 5)
    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    plot_tree_convergence(steps, [CRRTree(n).price(S, K, T, r, sigma).price for n in steps], exact, ax=axes[0], label="CRR binomial")
    plot_tree_convergence(steps, [TrinomialTree(n).price(S, K, T, r, sigma).price for n in steps], exact, ax=axes[0], label="Boyle trinomial")
    axes[0].set_title("European call")




.. image-sg:: /api/gallery/options/trees/images/sphx_glr_plot_04_boyle_trinomial_001.png
   :alt: European call
   :srcset: /api/gallery/options/trees/images/sphx_glr_plot_04_boyle_trinomial_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'European call')



.. GENERATED FROM PYTHON SOURCE LINES 45-47

The stretch parameter
---------------------

.. GENERATED FROM PYTHON SOURCE LINES 47-58

.. code-block:: Python


    stretches = np.linspace(1.05, 2.0, 20)
    errors = [abs(TrinomialTree(100, stretch=lam).price(S, K, T, r, sigma).price - exact) for lam in stretches]
    axes[1].semilogy(stretches, errors, "o-")
    axes[1].axvline(np.sqrt(1.5), color="k", ls=":", label=r"default $\lambda = \sqrt{3/2}$")
    axes[1].set_xlabel(r"stretch $\lambda$")
    axes[1].set_ylabel("error with 100 steps")
    axes[1].legend()
    fig.tight_layout()

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.168 seconds)


.. _sphx_glr_download_api_gallery_options_trees_plot_04_boyle_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/options/trees/plot_04_boyle_trinomial.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_boyle_trinomial.ipynb <plot_04_boyle_trinomial.ipynb>`

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

      :download:`Download Python source code: plot_04_boyle_trinomial.py <plot_04_boyle_trinomial.py>`

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

      :download:`Download zipped: plot_04_boyle_trinomial.zip <plot_04_boyle_trinomial.zip>`


.. only:: html

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

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