.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/options/trees/plot_03_rendleman_bartter.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_03_rendleman_bartter.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_03_rendleman_bartter.py:


The Rendleman-Bartter tree
==========================

Richard Rendleman and Brit Bartter (1979) published the two-state model
in the same year as Cox, Ross and Rubinstein, but centred each step on
the drift instead of on today's price:

.. math:: \ln u, \ln d = (r - \tfrac12\sigma^2)\Delta t \pm \sigma\sqrt{\Delta t}.

The up and down moves are then (nearly) equally likely, and the tree
drifts with the stock instead of staying centred on the spot. Both
trees converge to Black-Scholes at the same :math:`1/n` rate.

.. GENERATED FROM PYTHON SOURCE LINES 17-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

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

    S, K, T, r, sigma = 100.0, 100.0, 1.0, 0.08, 0.3








.. GENERATED FROM PYTHON SOURCE LINES 26-28

Nearly equal probabilities, a drifting lattice
----------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-45

.. code-block:: Python


    for tree in (CRRTree(10), RendlemanBartterTree(10)):
        u, d, p, _ = tree._parameters(S, K, T, r, sigma, 0.0, 10)
        print(f"{tree.method:>17}: u = {u:.4f}, d = {d:.4f}, p = {p:.4f}, centre after 10 steps = {S * u**5 * d**5:.2f}")

    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    for ax, tree in zip(axes, (CRRTree(6), RendlemanBartterTree(6)), strict=True):
        u, d, _, n = tree._parameters(S, K, T, r, sigma, 0.0, 6)
        for i in range(n + 1):
            j = np.arange(i + 1)
            ax.plot(np.full(i + 1, i * T / n), S * u**j * d ** (i - j), "o", color="C0", ms=4)
        ax.axhline(S, color="k", lw=0.5)
        ax.set_title(f"{tree.method} nodes")
        ax.set_xlabel("time")
    axes[0].set_ylabel("stock price")
    fig.tight_layout()




.. image-sg:: /api/gallery/options/trees/images/sphx_glr_plot_03_rendleman_bartter_001.png
   :alt: crr nodes, rendleman_bartter nodes
   :srcset: /api/gallery/options/trees/images/sphx_glr_plot_03_rendleman_bartter_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

                  crr: u = 1.0995, d = 0.9095, p = 0.5186, centre after 10 steps = 100.00
    rendleman_bartter: u = 1.1034, d = 0.9127, p = 0.5000, centre after 10 steps = 103.56




.. GENERATED FROM PYTHON SOURCE LINES 46-48

Convergence of both trees
-------------------------

.. GENERATED FROM PYTHON SOURCE LINES 48-60

.. code-block:: Python


    exact = black_scholes_price(S, K, T, r, sigma)
    steps = np.arange(10, 400, 3)
    fig, ax = plt.subplots(figsize=(7, 4))
    for tree in (CRRTree, RendlemanBartterTree):
        prices = [tree(n).price(S, K, T, r, sigma).price for n in steps]
        plot_tree_convergence(steps, prices, exact, ax=ax, label=tree.method)
    ax.loglog(steps, 2.0 / steps, "k:", label="2/n")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/options/trees/images/sphx_glr_plot_03_rendleman_bartter_002.png
   :alt: plot 03 rendleman bartter
   :srcset: /api/gallery/options/trees/images/sphx_glr_plot_03_rendleman_bartter_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.270 seconds)


.. _sphx_glr_download_api_gallery_options_trees_plot_03_rendleman_bartter.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_03_rendleman_bartter.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_rendleman_bartter.ipynb <plot_03_rendleman_bartter.ipynb>`

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

      :download:`Download Python source code: plot_03_rendleman_bartter.py <plot_03_rendleman_bartter.py>`

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

      :download:`Download zipped: plot_03_rendleman_bartter.zip <plot_03_rendleman_bartter.zip>`


.. only:: html

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

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