.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/options/trees/plot_06_leisen_reimer.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_06_leisen_reimer.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_06_leisen_reimer.py:


The Leisen-Reimer tree
======================

Dietmar Leisen and Matthias Reimer (1996) removed the binomial tree's
oscillation. Instead of fixing :math:`u` and :math:`d` from the
volatility, they choose the probabilities so that the tree's
cumulative binomial distribution reproduces Black-Scholes's
:math:`N(d_2)` and :math:`N(d_1)`, using the Peizer-Pratt inversion of
the normal onto the binomial. The tree is centred on the strike, and
European prices converge smoothly at order :math:`1/n^2`.

.. GENERATED FROM PYTHON SOURCE LINES 15-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

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

    S, K, T, r, sigma = 100.0, 110.0, 1.0, 0.05, 0.25
    exact = black_scholes_price(S, K, T, r, sigma)








.. GENERATED FROM PYTHON SOURCE LINES 25-27

Second-order convergence
------------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-43

.. code-block:: Python


    steps = np.arange(11, 402, 10)
    crr = [CRRTree(n).price(S, K, T, r, sigma).price for n in steps]
    lr = [LeisenReimerTree(n).price(S, K, T, r, sigma).price for n in steps]
    order = np.polyfit(np.log(steps), np.log(np.abs(np.array(lr) - exact)), 1)[0]
    print(f"Leisen-Reimer empirical order: {-order:.2f}")
    err_crr = abs(CRRTree(51).price(S, K, T, r, sigma).price - exact)
    err_lr = abs(LeisenReimerTree(51).price(S, K, T, r, sigma).price - exact)
    print(f"error with 51 steps: CRR {err_crr:.1e}, Leisen-Reimer {err_lr:.1e}")

    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    plot_tree_convergence(steps, crr, exact, ax=axes[0], label="CRR")
    plot_tree_convergence(steps, lr, exact, ax=axes[0], label="Leisen-Reimer")
    axes[0].loglog(steps, 1.0 / steps**2, "k:", label="1/n^2")
    axes[0].legend()




.. image-sg:: /api/gallery/options/trees/images/sphx_glr_plot_06_leisen_reimer_001.png
   :alt: plot 06 leisen reimer
   :srcset: /api/gallery/options/trees/images/sphx_glr_plot_06_leisen_reimer_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Leisen-Reimer empirical order: 1.98
    error with 51 steps: CRR 8.7e-03, Leisen-Reimer 1.8e-04

    <matplotlib.legend.Legend object at 0x1166d6270>



.. GENERATED FROM PYTHON SOURCE LINES 44-46

American puts converge smoothly too
-----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 46-58

.. code-block:: Python


    reference = LeisenReimerTree(2001).price(S, K, T, r, sigma, option_type="put", american=True).price
    steps = np.arange(11, 302, 10)
    axes[1].plot(steps, [CRRTree(n).price(S, K, T, r, sigma, option_type="put", american=True).price for n in steps], label="CRR")
    axes[1].plot(steps, [LeisenReimerTree(n).price(S, K, T, r, sigma, option_type="put", american=True).price for n in steps], label="Leisen-Reimer")
    axes[1].axhline(reference, color="k", ls=":", label="LR, 2001 steps")
    axes[1].set_xlabel("time steps n")
    axes[1].set_ylabel("American put")
    axes[1].legend()
    fig.tight_layout()

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.264 seconds)


.. _sphx_glr_download_api_gallery_options_trees_plot_06_leisen_reimer.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

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        :target: ../../../../lite/lab/index.html?path=api/gallery/options/trees/plot_06_leisen_reimer.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_06_leisen_reimer.ipynb <plot_06_leisen_reimer.ipynb>`

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

      :download:`Download Python source code: plot_06_leisen_reimer.py <plot_06_leisen_reimer.py>`

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      :download:`Download zipped: plot_06_leisen_reimer.zip <plot_06_leisen_reimer.zip>`


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