.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/exotic_options/multi_asset/plot_02_stulz_max_min.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_multi_asset_plot_02_stulz_max_min.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_multi_asset_plot_02_stulz_max_min.py:


Stulz options on the maximum and minimum
========================================

Rene Stulz (1982) priced calls and puts on the better or the worse of
two assets, :math:`(\max(S_1, S_2) - K)^+` and
:math:`(\min(S_1, S_2) - K)^+`, the simplest "rainbow" options. The
answer needs the bivariate normal distribution. Because
:math:`\max + \min = S_1 + S_2`, the two calls always add up to the two
vanilla calls, and correlation moves value between them: diversification
makes the maximum more valuable and the minimum less.

.. GENERATED FROM PYTHON SOURCE LINES 15-36

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.exotic_options import stulz_price
    from financialkit.options import black_scholes_price

    S1, S2, K, T, r, s1, s2 = 100.0, 100.0, 100.0, 1.0, 0.05, 0.25, 0.3
    rho = np.linspace(-0.95, 0.95, 60)

    fig, axes = plt.subplots(1, 2, figsize=(12, 4))
    cmax = stulz_price(S1, S2, K, T, r, s1, s2, rho, kind="max")
    cmin = stulz_price(S1, S2, K, T, r, s1, s2, rho, kind="min")
    vanilla = black_scholes_price(S1, K, T, r, s1) + black_scholes_price(S2, K, T, r, s2)
    axes[0].plot(rho, cmax, label="call on the max")
    axes[0].plot(rho, cmin, label="call on the min")
    axes[0].plot(rho, cmax + cmin, "k--", lw=2, label="sum")
    axes[0].axhline(vanilla, color="C3", ls=":", lw=3, label="two vanilla calls")
    axes[0].set_xlabel(r"correlation $\rho$")
    axes[0].set_ylabel("value")
    axes[0].legend()




.. image-sg:: /api/gallery/exotic_options/multi_asset/images/sphx_glr_plot_02_stulz_max_min_001.png
   :alt: plot 02 stulz max min
   :srcset: /api/gallery/exotic_options/multi_asset/images/sphx_glr_plot_02_stulz_max_min_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x11713d160>



.. GENERATED FROM PYTHON SOURCE LINES 37-39

Puts on the maximum and minimum across strikes
----------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 39-48

.. code-block:: Python


    Ks = np.linspace(70.0, 130.0, 60)
    for kind in ("max", "min"):
        axes[1].plot(Ks, stulz_price(S1, S2, Ks, T, r, s1, s2, 0.5, option_type="put", kind=kind), label=f"put on the {kind}")
    axes[1].plot(Ks, black_scholes_price(S1, Ks, T, r, s1, option_type="put"), "k:", label="put on asset 1")
    axes[1].set_xlabel("strike K")
    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_multi_asset_plot_02_stulz_max_min.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/multi_asset/plot_02_stulz_max_min.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_stulz_max_min.ipynb <plot_02_stulz_max_min.ipynb>`

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

      :download:`Download Python source code: plot_02_stulz_max_min.py <plot_02_stulz_max_min.py>`

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

      :download:`Download zipped: plot_02_stulz_max_min.zip <plot_02_stulz_max_min.zip>`


.. only:: html

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

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