.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/exotic_options/asian/plot_02_turnbull_wakeman.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_exotic_options_asian_plot_02_turnbull_wakeman.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_exotic_options_asian_plot_02_turnbull_wakeman.py:


Turnbull-Wakeman Asian approximation
====================================

Stuart Turnbull and Lee Wakeman (1991) priced the arithmetic average
:math:`A` as if it were lognormal with the right first two moments:
Black's formula on the forward :math:`E[A]` with volatility

.. math:: \sigma_A^2 = \frac{1}{T}\ln\frac{E[A^2]}{E[A]^2}.

The moments are exact (Levy's sums for discrete fixings), only the shape
of the distribution is approximated. The approximation is excellent at
moderate volatility and drifts as the average's distribution grows
more skewed than a lognormal.

.. GENERATED FROM PYTHON SOURCE LINES 18-29

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.exotic_options import arithmetic_asian_mc, geometric_asian_price, turnbull_wakeman_price

    S, K, T, r, n = 100.0, 100.0, 1.0, 0.05, 12
    sigmas = np.array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
    mc = [arithmetic_asian_mc(S, K, T, r, s, n_fixings=n, n_paths=20_000, rng=3) for s in sigmas]
    tw = np.array([turnbull_wakeman_price(S, K, T, r, s, n_fixings=n) for s in sigmas])
    geo = geometric_asian_price(S, K, T, r, sigmas, n_fixings=n)








.. GENERATED FROM PYTHON SOURCE LINES 30-32

Approximation against controlled Monte Carlo
--------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 32-48

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(12, 4))
    axes[0].plot(sigmas, tw, "s-", label="Turnbull-Wakeman")
    axes[0].errorbar(sigmas, [m.estimate for m in mc], yerr=[2 * m.std_error for m in mc], fmt="o", label="Monte Carlo")
    axes[0].plot(sigmas, geo, ":", label="geometric (lower bound)")
    axes[0].set_xlabel(r"volatility $\sigma$")
    axes[0].set_ylabel("at-the-money arithmetic Asian call")
    axes[0].legend()
    rel = (tw - np.array([m.estimate for m in mc])) / np.array([m.estimate for m in mc])
    axes[1].bar(sigmas, 100 * rel, width=0.06)
    axes[1].set_xlabel(r"volatility $\sigma$")
    axes[1].set_ylabel("relative error (%)")
    fig.tight_layout()
    for s, t, m in zip(sigmas, tw, mc, strict=True):
        print(f"sigma = {s:.1f}: TW {t:.4f}, MC {m.estimate:.4f} +/- {m.std_error:.4f}")
    plt.show()



.. image-sg:: /api/gallery/exotic_options/asian/images/sphx_glr_plot_02_turnbull_wakeman_001.png
   :alt: plot 02 turnbull wakeman
   :srcset: /api/gallery/exotic_options/asian/images/sphx_glr_plot_02_turnbull_wakeman_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    sigma = 0.1: TW 3.9105, MC 3.9056 +/- 0.0005
    sigma = 0.2: TW 6.1742, MC 6.1587 +/- 0.0017
    sigma = 0.3: TW 8.5178, MC 8.4808 +/- 0.0038
    sigma = 0.4: TW 10.8892, MC 10.8133 +/- 0.0069
    sigma = 0.5: TW 13.2800, MC 13.1401 +/- 0.0113
    sigma = 0.6: TW 15.6890, MC 15.4559 +/- 0.0171





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

   **Total running time of the script:** (0 minutes 0.173 seconds)


.. _sphx_glr_download_api_gallery_exotic_options_asian_plot_02_turnbull_wakeman.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/exotic_options/asian/plot_02_turnbull_wakeman.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_turnbull_wakeman.ipynb <plot_02_turnbull_wakeman.ipynb>`

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

      :download:`Download Python source code: plot_02_turnbull_wakeman.py <plot_02_turnbull_wakeman.py>`

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

      :download:`Download zipped: plot_02_turnbull_wakeman.zip <plot_02_turnbull_wakeman.zip>`


.. only:: html

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

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