.. 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_05_black76.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_05_black76.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_05_black76.py:


Black-76: options on futures
============================

Fischer Black (1976) priced options on commodity futures. A futures
contract costs nothing to enter, so its risk-neutral drift is zero and
the Black-Scholes argument gives

.. math:: c = e^{-rT}[F N(d_1) - K N(d_2)],

with the futures price :math:`F` in place of the spot. The same formula
prices caps, floors and swaptions in
:mod:`financialkit.interest_rate_models`.

.. GENERATED FROM PYTHON SOURCE LINES 17-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.options import black76_price, black_scholes_price








.. GENERATED FROM PYTHON SOURCE LINES 23-25

Hull's futures put (Example 18.7)
---------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 25-28

.. code-block:: Python


    print(f"four-month put on futures at 20, strike 20: {black76_price(20.0, 20.0, 4 / 12, 0.09, 0.25, 'put'):.4f} (Hull: 1.12)")





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

 .. code-block:: none

    four-month put on futures at 20, strike 20: 1.1166 (Hull: 1.12)




.. GENERATED FROM PYTHON SOURCE LINES 29-34

Black-76 is Black-Scholes on the forward
----------------------------------------

For a stock with dividend yield :math:`q` the forward is
:math:`F = S e^{(r-q)T}`, and both formulas give the same price.

.. GENERATED FROM PYTHON SOURCE LINES 34-40

.. code-block:: Python


    S, T, r, q, sigma = 100.0, 1.0, 0.05, 0.02, 0.25
    F = S * np.exp((r - q) * T)
    K = np.linspace(70, 140, 8)
    print("max difference:", np.max(np.abs(black76_price(F, K, T, r, sigma) - black_scholes_price(S, K, T, r, sigma, q))))





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

 .. code-block:: none

    max difference: 0.0




.. GENERATED FROM PYTHON SOURCE LINES 41-43

Calls and puts on futures across maturities
-------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 43-56

.. code-block:: Python


    K = np.linspace(60, 140, 161)
    fig, ax = plt.subplots(figsize=(7, 4))
    for T in (0.25, 1.0, 3.0):
        ax.plot(K, black76_price(100.0, K, T, 0.05, 0.3), label=f"call, T = {T}")
        ax.plot(K, black76_price(100.0, K, T, 0.05, 0.3, "put"), "--", label=f"put, T = {T}")
    ax.axvline(100.0, color="k", lw=0.5)
    ax.set_xlabel("strike K (futures at 100)")
    ax.set_ylabel("option price")
    ax.legend(ncol=2)
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/options/closed_form/images/sphx_glr_plot_05_black76_001.png
   :alt: plot 05 black76
   :srcset: /api/gallery/options/closed_form/images/sphx_glr_plot_05_black76_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.045 seconds)


.. _sphx_glr_download_api_gallery_options_closed_form_plot_05_black76.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_05_black76.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_05_black76.ipynb <plot_05_black76.ipynb>`

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

      :download:`Download Python source code: plot_05_black76.py <plot_05_black76.py>`

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

      :download:`Download zipped: plot_05_black76.zip <plot_05_black76.zip>`


.. only:: html

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

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