.. 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_02_fixed_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_02_fixed_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_02_fixed_lookback.py:


Conze-Viswanathan fixed lookbacks
=================================

Antoine Conze and Viswanathan (1991) priced the fixed-strike lookback:
a call on the maximum, :math:`(\max_t S_t - K)^+`, and a put on the
minimum, :math:`(K - \min_t S_t)^+`. Once the running maximum has passed
the strike, the call has locked in :math:`\max - K` and what remains is
a call on any further rise; that is how a seasoned option is priced,
with the maximum observed so far as an input.

.. GENERATED FROM PYTHON SOURCE LINES 14-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

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

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








.. GENERATED FROM PYTHON SOURCE LINES 23-25

New options across strikes
--------------------------

.. GENERATED FROM PYTHON SOURCE LINES 25-40

.. code-block:: Python


    K = np.linspace(70.0, 140.0, 50)
    fig, axes = plt.subplots(1, 2, figsize=(12, 4))
    axes[0].plot(K, fixed_lookback_price(S, K, T, r, sigma), label="call on the maximum")
    axes[0].plot(K, fixed_lookback_price(S, K, T, r, sigma, option_type="put"), label="put on the minimum")
    axes[0].plot(K, black_scholes_price(S, K, T, r, sigma), "k--", label="vanilla call")
    axes[0].plot(K, black_scholes_price(S, K, T, r, sigma, option_type="put"), "k:", label="vanilla put")
    for k in (80.0, 100.0, 120.0):
        for option_type in ("call", "put"):
            res = lookback_option_mc(S, T, r, sigma, option_type=option_type, K=k, n_steps=10, n_paths=10_000, rng=int(k))
            axes[0].errorbar(k, res.estimate, yerr=2 * res.std_error, fmt="k.", capsize=3)
    axes[0].set_xlabel("strike K")
    axes[0].set_ylabel("value")
    axes[0].legend()




.. image-sg:: /api/gallery/exotic_options/path_dependent/images/sphx_glr_plot_02_fixed_lookback_001.png
   :alt: plot 02 fixed lookback
   :srcset: /api/gallery/exotic_options/path_dependent/images/sphx_glr_plot_02_fixed_lookback_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x116f51fd0>



.. GENERATED FROM PYTHON SOURCE LINES 41-46

A seasoned call: the running maximum so far
-------------------------------------------
Halfway through, with the spot at 100 and a strike of 105, the value
rises with the maximum already reached, linearly once it passes the
strike.

.. GENERATED FROM PYTHON SOURCE LINES 46-56

.. code-block:: Python


    Smax = np.linspace(100.0, 140.0, 100)
    axes[1].plot(Smax, fixed_lookback_price(S, 105.0, 0.5, r, sigma, S_extreme=Smax), label="seasoned call on the maximum")
    axes[1].plot(Smax, np.exp(-r * 0.5) * np.maximum(Smax - 105.0, 0.0), "--", label=r"locked in $e^{-r\tau}(S_{\max} - K)^+$")
    axes[1].axvline(105.0, color="0.5", lw=1)
    axes[1].set_xlabel(r"running maximum $S_{\max}$")
    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 0.109 seconds)


.. _sphx_glr_download_api_gallery_exotic_options_path_dependent_plot_02_fixed_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_02_fixed_lookback.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_fixed_lookback.ipynb <plot_02_fixed_lookback.ipynb>`

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

      :download:`Download Python source code: plot_02_fixed_lookback.py <plot_02_fixed_lookback.py>`

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

      :download:`Download zipped: plot_02_fixed_lookback.zip <plot_02_fixed_lookback.zip>`


.. only:: html

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

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