.. 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_01_kemna_vorst.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_01_kemna_vorst.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_01_kemna_vorst.py:


Kemna-Vorst Asian options
=========================

An average-rate option pays on the average price over its life, which
makes it cheaper than a vanilla option and harder to manipulate at
expiry. Angelien Kemna and Ton Vorst (1990) observed that a *geometric*
average of lognormal prices is lognormal, with volatility
:math:`\sigma/\sqrt3` under continuous averaging, so its option has a
Black-Scholes formula. The arithmetic average traded in practice has
none, but the geometric option is so tightly correlated with it that it
makes a near-perfect Monte Carlo control variate.

.. GENERATED FROM PYTHON SOURCE LINES 16-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.exotic_options import arithmetic_asian_mc, geometric_asian_price
    from financialkit.integrators import gbm_paths
    from financialkit.options import black_scholes_price

    S, T, r, sigma, n = 100.0, 1.0, 0.05, 0.25, 12








.. GENERATED FROM PYTHON SOURCE LINES 26-28

Arithmetic and geometric payoffs on the same paths
--------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-43

.. code-block:: Python


    _, paths = gbm_paths(S, r, sigma, T, n, 3000, rng=0)
    A = paths[:, 1:].mean(axis=1)
    G = np.exp(np.log(paths[:, 1:]).mean(axis=1))
    fig, axes = plt.subplots(1, 2, figsize=(12, 4.5))
    axes[0].scatter(np.maximum(G - 100, 0), np.maximum(A - 100, 0), s=3, alpha=0.4)
    axes[0].plot([0, 60], [0, 60], "k:", lw=1)
    axes[0].set_xlabel("geometric-average call payoff")
    axes[0].set_ylabel("arithmetic-average call payoff")

    res = arithmetic_asian_mc(S, 100.0, T, r, sigma, n_fixings=n, n_paths=20_000, rng=2)
    print(f"geometric (closed form) {geometric_asian_price(S, 100.0, T, r, sigma, n_fixings=n):.4f}")
    print(f"arithmetic: crude {res.crude_estimate:.4f} +/- {res.crude_std_error:.4f}, controlled {res.estimate:.4f} +/- {res.std_error:.4f}")
    print(f"R^2 = {res.r_squared:.5f}: variance reduced {1 / (1 - res.r_squared):.0f} times")




.. image-sg:: /api/gallery/exotic_options/asian/images/sphx_glr_plot_01_kemna_vorst_001.png
   :alt: plot 01 kemna vorst
   :srcset: /api/gallery/exotic_options/asian/images/sphx_glr_plot_01_kemna_vorst_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    geometric (closed form) 6.9907
    arithmetic: crude 7.3192 +/- 0.0755, controlled 7.3102 +/- 0.0027
    R^2 = 0.99875: variance reduced 801 times




.. GENERATED FROM PYTHON SOURCE LINES 44-46

Averaging lowers the price
--------------------------

.. GENERATED FROM PYTHON SOURCE LINES 46-57

.. code-block:: Python


    K = np.linspace(70.0, 130.0, 13)
    axes[1].plot(K, black_scholes_price(S, K, T, r, sigma), "k--", label="vanilla call")
    axes[1].plot(K, geometric_asian_price(S, K, T, r, sigma), label="geometric, continuous")
    arith = [arithmetic_asian_mc(S, k, T, r, sigma, n_fixings=n, n_paths=5_000, rng=2) for k in K]
    axes[1].errorbar(K, [a.estimate for a in arith], yerr=[2 * a.std_error for a in arith], fmt="o", ms=4, label=f"arithmetic, {n} fixings (MC)")
    axes[1].set_xlabel("strike K")
    axes[1].set_ylabel("value")
    axes[1].legend()
    fig.tight_layout()
    plt.show()








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

   **Total running time of the script:** (0 minutes 0.065 seconds)


.. _sphx_glr_download_api_gallery_exotic_options_asian_plot_01_kemna_vorst.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_01_kemna_vorst.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_kemna_vorst.ipynb <plot_01_kemna_vorst.ipynb>`

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

      :download:`Download Python source code: plot_01_kemna_vorst.py <plot_01_kemna_vorst.py>`

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

      :download:`Download zipped: plot_01_kemna_vorst.zip <plot_01_kemna_vorst.zip>`


.. only:: html

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

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