.. 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_01_margrabe.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_01_margrabe.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_01_margrabe.py:


Margrabe exchange options
=========================

William Margrabe (1978) priced the right to exchange one asset for
another, :math:`(S_1 - S_2)^+`. Measured in units of the second asset,
it is a call on the ratio :math:`S_1/S_2` struck at 1, so the interest
rate drops out and only the volatility of the ratio,
:math:`\sigma^2 = \sigma_1^2 + \sigma_2^2 - 2\rho\sigma_1\sigma_2`,
matters. Change of numeraire, the idea behind this one-line proof,
became one of the central tools of derivatives pricing.

.. GENERATED FROM PYTHON SOURCE LINES 15-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.exotic_options import margrabe_price
    from financialkit.integrators import correlated_increments

    S1, S2, T, s1, s2 = 100.0, 100.0, 1.0, 0.25, 0.3
    rho = np.linspace(-0.95, 0.95, 50)








.. GENERATED FROM PYTHON SOURCE LINES 25-27

Correlation drives the price
----------------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-40

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(12, 4))
    axes[0].plot(rho, margrabe_price(S1, S2, T, s1, s2, rho), label="closed form")
    for c in (-0.6, 0.0, 0.6):
        W = correlated_increments([[1.0, c], [c, 1.0]], T, 1, 20_000, rng=int(10 * c) + 10)[:, 0, :]
        A = S1 * np.exp(-0.5 * s1**2 * T + s1 * W[:, 0])
        B = S2 * np.exp(-0.5 * s2**2 * T + s2 * W[:, 1])
        y = np.maximum(A - B, 0.0)  # r drops out: simulate with zero rate
        axes[0].errorbar(c, y.mean(), yerr=2 * y.std() / np.sqrt(y.size), fmt="k.", capsize=3)
    axes[0].set_xlabel(r"correlation $\rho$")
    axes[0].set_ylabel("exchange option")
    axes[0].legend()




.. image-sg:: /api/gallery/exotic_options/multi_asset/images/sphx_glr_plot_01_margrabe_001.png
   :alt: plot 01 margrabe
   :srcset: /api/gallery/exotic_options/multi_asset/images/sphx_glr_plot_01_margrabe_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x116f11a90>



.. GENERATED FROM PYTHON SOURCE LINES 41-44

Only the ratio's volatility matters
-----------------------------------
Pairs of volatilities with the same ratio volatility give the same price.

.. GENERATED FROM PYTHON SOURCE LINES 44-57

.. code-block:: Python


    target = 0.3
    for s_1 in (0.2, 0.3, 0.4):
        s_2 = np.linspace(0.05, 0.6, 200)
        c = (s_1**2 + s_2**2 - target**2) / (2 * s_1 * s_2)
        ok = np.abs(c) < 1
        axes[1].plot(s_2[ok], c[ok], label=rf"$\sigma_1$ = {s_1}")
    axes[1].set_xlabel(r"$\sigma_2$")
    axes[1].set_ylabel(r"$\rho$ giving a ratio volatility of 30%")
    axes[1].set_title(f"all priced at {margrabe_price(100.0, 100.0, 1.0, 0.3, 0.0, 0.0):.4f}")
    axes[1].legend()
    fig.tight_layout()
    plt.show()








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

   **Total running time of the script:** (0 minutes 0.074 seconds)


.. _sphx_glr_download_api_gallery_exotic_options_multi_asset_plot_01_margrabe.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_01_margrabe.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_margrabe.ipynb <plot_01_margrabe.ipynb>`

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

      :download:`Download Python source code: plot_01_margrabe.py <plot_01_margrabe.py>`

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

      :download:`Download zipped: plot_01_margrabe.zip <plot_01_margrabe.zip>`


.. only:: html

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

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