.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/options/closed_form/plot_06_garman_kohlhagen.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_options_closed_form_plot_06_garman_kohlhagen.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_options_closed_form_plot_06_garman_kohlhagen.py:


Garman-Kohlhagen currency options
=================================

Mark Garman and Steven Kohlhagen (1983) noticed that a unit of foreign
currency is an asset that pays the foreign interest rate :math:`r_f`,
just as a stock pays a dividend yield. Merton's dividend formula then
prices FX options:

.. math:: c = S e^{-r_f T} N(d_1) - K e^{-r_d T} N(d_2).

A call on one currency is a put on the other, which gives a symmetry
check.

.. GENERATED FROM PYTHON SOURCE LINES 17-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.options import garman_kohlhagen_price








.. GENERATED FROM PYTHON SOURCE LINES 23-25

Haug's example
--------------

.. GENERATED FROM PYTHON SOURCE LINES 25-29

.. code-block:: Python


    price = garman_kohlhagen_price(1.56, 1.60, 0.5, 0.06, 0.08, 0.12)
    print(f"six-month call, S = 1.56, K = 1.60, r_d = 6%, r_f = 8%: {price:.4f} (Haug: 0.0291)")





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

 .. code-block:: none

    six-month call, S = 1.56, K = 1.60, r_d = 6%, r_f = 8%: 0.0291 (Haug: 0.0291)




.. GENERATED FROM PYTHON SOURCE LINES 30-36

Foreign-domestic symmetry
-------------------------

A call to buy one unit of foreign currency at :math:`K` equals
:math:`S K` times a put, in foreign units, to sell one unit of domestic
currency at :math:`1/K`, with the rates swapped and the spot inverted.

.. GENERATED FROM PYTHON SOURCE LINES 36-42

.. code-block:: Python


    S, K, T, rd, rf, sigma = 1.56, 1.60, 0.5, 0.06, 0.08, 0.12
    call = garman_kohlhagen_price(S, K, T, rd, rf, sigma)
    put_inverse = garman_kohlhagen_price(1 / S, 1 / K, T, rf, rd, sigma, "put")
    print(f"call {call:.6f} = S K x inverse put {S * K * put_inverse:.6f}")





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

 .. code-block:: none

    call 0.029099 = S K x inverse put 0.029099




.. GENERATED FROM PYTHON SOURCE LINES 43-45

The interest-rate differential shifts the forward
-------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 45-56

.. code-block:: Python


    spots = np.linspace(1.3, 1.9, 121)
    fig, ax = plt.subplots(figsize=(7, 4))
    for rf in (0.0, 0.04, 0.08):
        ax.plot(spots, garman_kohlhagen_price(spots, K, 1.0, 0.06, rf, sigma), label=f"r_f = {rf:.0%}, forward = {1.56 * np.exp(0.06 - rf):.3f}")
    ax.set_xlabel("spot exchange rate")
    ax.set_ylabel("call value (domestic per foreign)")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/options/closed_form/images/sphx_glr_plot_06_garman_kohlhagen_001.png
   :alt: plot 06 garman kohlhagen
   :srcset: /api/gallery/options/closed_form/images/sphx_glr_plot_06_garman_kohlhagen_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.042 seconds)


.. _sphx_glr_download_api_gallery_options_closed_form_plot_06_garman_kohlhagen.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/options/closed_form/plot_06_garman_kohlhagen.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_06_garman_kohlhagen.ipynb <plot_06_garman_kohlhagen.ipynb>`

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

      :download:`Download Python source code: plot_06_garman_kohlhagen.py <plot_06_garman_kohlhagen.py>`

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

      :download:`Download zipped: plot_06_garman_kohlhagen.zip <plot_06_garman_kohlhagen.zip>`


.. only:: html

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

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