.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/exotic_options/compound/plot_02_forward_start.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_compound_plot_02_forward_start.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_compound_plot_02_forward_start.py:


Rubinstein forward-start options
================================

A forward-start option is struck at :math:`\alpha S_t` on a future date
:math:`t`, the way executive options are granted at the money. Mark
Rubinstein (1990) noticed that on the start date it is worth
:math:`S_t` times the option on a unit spot, so today it is worth

.. math:: V = S e^{-qt}\, \mathrm{BS}(1, \alpha, T - t),

independent of where the price goes before :math:`t`. Its value is
proportional to the spot, so it has no gamma until it starts, and a
string of them is a cliquet.

.. GENERATED FROM PYTHON SOURCE LINES 18-27

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.exotic_options import forward_start_price
    from financialkit.options import black_scholes_price

    S, T, r, sigma, q = 100.0, 1.0, 0.05, 0.25, 0.02
    print(f"Haug (2007), 4.3: {forward_start_price(60.0, 1.1, 0.25, 1.0, 0.08, 0.30, q=0.04):.4f}")





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

 .. code-block:: none

    Haug (2007), 4.3: 4.4065




.. GENERATED FROM PYTHON SOURCE LINES 28-30

Value against the start date, with a Monte Carlo check
------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 30-46

.. code-block:: Python


    t = np.linspace(0.0, 0.95, 40)
    fig, axes = plt.subplots(1, 2, figsize=(12, 4))
    for alpha in (0.9, 1.0, 1.1):
        axes[0].plot(t, forward_start_price(S, alpha, t, T, r, sigma, q), label=rf"$\alpha$ = {alpha}")
    rng = np.random.default_rng(0)
    for ts in (0.25, 0.5, 0.75):
        z = rng.standard_normal((2, 20_000))
        St = S * np.exp((r - q - 0.5 * sigma**2) * ts + sigma * np.sqrt(ts) * z[0])
        ST = St * np.exp((r - q - 0.5 * sigma**2) * (T - ts) + sigma * np.sqrt(T - ts) * z[1])
        y = np.exp(-r * T) * np.maximum(ST - St, 0.0)
        axes[0].errorbar(ts, y.mean(), yerr=2 * y.std() / np.sqrt(y.size), fmt="k.", capsize=3)
    axes[0].set_xlabel("start date t")
    axes[0].set_ylabel("forward-start call")
    axes[0].legend()




.. image-sg:: /api/gallery/exotic_options/compound/images/sphx_glr_plot_02_forward_start_001.png
   :alt: plot 02 forward start
   :srcset: /api/gallery/exotic_options/compound/images/sphx_glr_plot_02_forward_start_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x116c66900>



.. GENERATED FROM PYTHON SOURCE LINES 47-49

Homogeneous in the spot: no gamma before the start
--------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 49-57

.. code-block:: Python


    Ss = np.linspace(50.0, 150.0, 100)
    axes[1].plot(Ss, forward_start_price(Ss, 1.0, 0.5, T, r, sigma, q), label="forward start (t = 0.5)")
    axes[1].plot(Ss, black_scholes_price(Ss, S, T, r, sigma, q), "--", label="vanilla call, K = 100")
    axes[1].set_xlabel("spot S")
    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_compound_plot_02_forward_start.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/compound/plot_02_forward_start.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_forward_start.ipynb <plot_02_forward_start.ipynb>`

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

      :download:`Download Python source code: plot_02_forward_start.py <plot_02_forward_start.py>`

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

      :download:`Download zipped: plot_02_forward_start.zip <plot_02_forward_start.zip>`


.. only:: html

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

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