.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/options/trees/plot_05_barone_adesi_whaley.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_05_barone_adesi_whaley.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_05_barone_adesi_whaley.py:


The Barone-Adesi-Whaley approximation
=====================================

Giovanni Barone-Adesi and Robert Whaley (1987), building on MacMillan
(1986), found an almost closed-form American price. The early-exercise
premium solves the Black-Scholes PDE; dropping one small time-derivative
term leaves an ordinary differential equation with power-law solutions,
and smooth pasting at a critical price :math:`S^*` pins down the
coefficient. One scalar root replaces a whole lattice.

.. GENERATED FROM PYTHON SOURCE LINES 14-23

.. code-block:: Python

    import time

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.options import CRRTree, barone_adesi_whaley

    K, T, r, sigma = 100.0, 0.5, 0.08, 0.3








.. GENERATED FROM PYTHON SOURCE LINES 24-26

Accuracy and speed against a 2000-step tree
-------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 26-46

.. code-block:: Python


    spots = np.linspace(70, 130, 13)
    t0 = time.perf_counter()
    baw = np.array([barone_adesi_whaley(s, K, T, r, sigma, option_type="put").price for s in spots])
    t_baw = time.perf_counter() - t0
    t0 = time.perf_counter()
    tree = np.array([CRRTree(2000).price(s, K, T, r, sigma, option_type="put", american=True).price for s in spots])
    t_tree = time.perf_counter() - t0
    print(f"max |BAW - tree| = {np.max(np.abs(baw - tree)):.4f}; time {t_baw * 1e3:.1f} ms vs {t_tree * 1e3:.0f} ms")

    res = barone_adesi_whaley(100.0, K, T, r, sigma, option_type="put")
    print(f"at S = 100: American {res.price:.4f}, European {res.european_price:.4f}, critical price {res.critical_price:.2f}")

    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    axes[0].plot(spots, baw - tree, "o-")
    axes[0].axhline(0, color="k", lw=0.5)
    axes[0].set_xlabel("spot S")
    axes[0].set_ylabel("BAW - tree")
    axes[0].set_title("American put, 6 months")




.. image-sg:: /api/gallery/options/trees/images/sphx_glr_plot_05_barone_adesi_whaley_001.png
   :alt: American put, 6 months
   :srcset: /api/gallery/options/trees/images/sphx_glr_plot_05_barone_adesi_whaley_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    max |BAW - tree| = 0.0863; time 8.3 ms vs 424 ms
    at S = 100: American 6.8670, European 6.4671, critical price 78.65

    Text(0.5, 1.0, 'American put, 6 months')



.. GENERATED FROM PYTHON SOURCE LINES 47-49

The critical price across maturities
------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 49-62

.. code-block:: Python


    maturities = np.linspace(0.02, 3.0, 40)
    put_crit = [barone_adesi_whaley(100.0, K, t, r, sigma, option_type="put").critical_price for t in maturities]
    call_crit = [barone_adesi_whaley(100.0, K, t, r, sigma, q=0.10).critical_price for t in maturities]
    axes[1].plot(maturities, put_crit, label="put: exercise below")
    axes[1].plot(maturities, call_crit, label="call with q = 10%: exercise above")
    axes[1].axhline(K, color="k", lw=0.5)
    axes[1].set_xlabel("time to expiry")
    axes[1].set_ylabel("critical price")
    axes[1].legend()
    fig.tight_layout()

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.539 seconds)


.. _sphx_glr_download_api_gallery_options_trees_plot_05_barone_adesi_whaley.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_05_barone_adesi_whaley.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_05_barone_adesi_whaley.ipynb <plot_05_barone_adesi_whaley.ipynb>`

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

      :download:`Download Python source code: plot_05_barone_adesi_whaley.py <plot_05_barone_adesi_whaley.py>`

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

      :download:`Download zipped: plot_05_barone_adesi_whaley.zip <plot_05_barone_adesi_whaley.zip>`


.. only:: html

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

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