.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/exotic_options/path_dependent/plot_01_floating_lookback.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_path_dependent_plot_01_floating_lookback.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_path_dependent_plot_01_floating_lookback.py:


Goldman-Sosin-Gatto floating lookbacks
======================================

"Buy at the low, sell at the high": Barry Goldman, Howard Sosin and
Mary Ann Gatto (1979) priced the option whose strike is the lowest
price over its life, :math:`S_T - \min_t S_t` (or the highest, for the
put). It is never out of the money, so it costs much more than a
vanilla call. The joint law of a Brownian motion and its minimum gives
the closed form; Monte Carlo can match it exactly only by sampling each
step's minimum from the Brownian bridge, because checking the price at
discrete dates misses the true low.

.. GENERATED FROM PYTHON SOURCE LINES 16-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.exotic_options import floating_lookback_price, lookback_option_mc
    from financialkit.options import black_scholes_price

    S, T, r, sigma = 100.0, 1.0, 0.05, 0.25
    print(f"Haug (2007), 4.15.1: {floating_lookback_price(120.0, 0.5, 0.10, 0.30, q=0.06, S_extreme=100.0):.4f}")





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

 .. code-block:: none

    Haug (2007), 4.15.1: 25.3534




.. GENERATED FROM PYTHON SOURCE LINES 26-28

Lookback against at-the-money vanilla options
---------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-38

.. code-block:: Python


    vols = np.linspace(0.05, 0.6, 40)
    fig, axes = plt.subplots(1, 2, figsize=(12, 4))
    axes[0].plot(vols, floating_lookback_price(S, T, r, vols), label="floating lookback call")
    axes[0].plot(vols, floating_lookback_price(S, T, r, vols, option_type="put"), label="floating lookback put")
    axes[0].plot(vols, black_scholes_price(S, S, T, r, vols), "k--", label="ATM call")
    axes[0].set_xlabel(r"volatility $\sigma$")
    axes[0].set_ylabel("value")
    axes[0].legend()




.. image-sg:: /api/gallery/exotic_options/path_dependent/images/sphx_glr_plot_01_floating_lookback_001.png
   :alt: plot 01 floating lookback
   :srcset: /api/gallery/exotic_options/path_dependent/images/sphx_glr_plot_01_floating_lookback_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x116c67cb0>



.. GENERATED FROM PYTHON SOURCE LINES 39-41

Discrete monitoring misses the minimum; the bridge does not
-----------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 41-54

.. code-block:: Python


    steps = np.array([2, 4, 8, 16, 32, 64])
    exact = floating_lookback_price(S, T, r, sigma)
    for continuous, label in ((False, "minimum over the grid"), (True, "bridge-sampled minimum")):
        res = [lookback_option_mc(S, T, r, sigma, n_steps=int(m), n_paths=20_000, continuous=continuous, rng=int(m)) for m in steps]
        axes[1].errorbar(steps, [x.estimate for x in res], yerr=[2 * x.std_error for x in res], fmt="o-", capsize=3, label=label)
    axes[1].axhline(exact, color="k", lw=1, label="closed form")
    axes[1].set_xscale("log", base=2)
    axes[1].set_xlabel("time steps")
    axes[1].set_ylabel("floating lookback call")
    axes[1].legend()
    fig.tight_layout()
    plt.show()








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

   **Total running time of the script:** (0 minutes 0.159 seconds)


.. _sphx_glr_download_api_gallery_exotic_options_path_dependent_plot_01_floating_lookback.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/path_dependent/plot_01_floating_lookback.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_floating_lookback.ipynb <plot_01_floating_lookback.ipynb>`

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

      :download:`Download Python source code: plot_01_floating_lookback.py <plot_01_floating_lookback.py>`

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

      :download:`Download zipped: plot_01_floating_lookback.zip <plot_01_floating_lookback.zip>`


.. only:: html

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

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