.. 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_03_quanto.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_03_quanto.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_03_quanto.py:


Reiner quantos
==============

A quanto pays a foreign asset's option payoff in domestic currency at a
fixed exchange rate, so the investor keeps the asset's risk without the
currency's. Eric Reiner (1992) showed that, under the domestic measure,
the asset's drift picks up a correction :math:`-\rho\sigma_S\sigma_X`
from its correlation with the exchange rate:

.. math:: V = \bar X e^{-r_d T}[\hat F N(d_1) - K N(d_2)], \qquad
          \hat F = S e^{(r_f - q - \rho\sigma_S\sigma_X)T}.

A Monte Carlo under the *foreign* measure, which converts the payoff at
the floating rate, checks the drift adjustment independently.

.. GENERATED FROM PYTHON SOURCE LINES 19-42



.. image-sg:: /api/gallery/exotic_options/multi_asset/images/sphx_glr_plot_03_quanto_001.png
   :alt: plot 03 quanto
   :srcset: /api/gallery/exotic_options/multi_asset/images/sphx_glr_plot_03_quanto_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Haug (2007), 5.16: 5.3280






|

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.exotic_options import quanto_price
    from financialkit.integrators import correlated_increments

    S, K, T, rd, rf, sS, sX, q, Xbar, X0 = 100.0, 105.0, 0.5, 0.08, 0.05, 0.2, 0.1, 0.04, 1.5, 1.2
    print(f"Haug (2007), 5.16: {quanto_price(S, K, T, rd, rf, sS, sX, 0.3, q=q, fixed_rate=Xbar):.4f}")
    rho = np.linspace(-0.9, 0.9, 50)

    fig, ax = plt.subplots(figsize=(8, 4.5))
    for option_type in ("call", "put"):
        ax.plot(rho, quanto_price(S, K, T, rd, rf, sS, sX, rho, q, Xbar, option_type), label=f"quanto {option_type}")
        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) + 20)[:, 0, :]
            ST = S * np.exp((rf - q - 0.5 * sS**2) * T + sS * W[:, 0])
            YT = np.exp((rf - rd - 0.5 * sX**2) * T + sX * W[:, 1]) / X0  # foreign price of one domestic unit
            y = X0 * np.exp(-rf * T) * Xbar * np.maximum((1 if option_type == "call" else -1) * (ST - K), 0.0) * YT
            ax.errorbar(c, y.mean(), yerr=2 * y.std() / np.sqrt(y.size), fmt="k.", capsize=3)
    ax.set_xlabel(r"correlation $\rho$ between asset and exchange rate")
    ax.set_ylabel("value (domestic currency)")
    ax.legend()
    plt.show()


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

   **Total running time of the script:** (0 minutes 0.043 seconds)


.. _sphx_glr_download_api_gallery_exotic_options_multi_asset_plot_03_quanto.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_03_quanto.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_quanto.ipynb <plot_03_quanto.ipynb>`

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

      :download:`Download Python source code: plot_03_quanto.py <plot_03_quanto.py>`

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

      :download:`Download zipped: plot_03_quanto.zip <plot_03_quanto.zip>`


.. only:: html

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

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