.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/exotic_options/barriers/plot_01_merton_down_and_out.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_barriers_plot_01_merton_down_and_out.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_barriers_plot_01_merton_down_and_out.py:


Merton's down-and-out call
==========================

In the paper that put the Black-Scholes model on a rigorous footing,
Robert Merton (1973) also priced the first barrier option: a call that
dies if the stock falls to :math:`H`. The barrier is an absorbing boundary
for the log-price, and the method of images from the heat equation
cancels the probability of crossing it by subtracting a call started
from the mirror image :math:`H^2/S` of the spot:

.. math:: C_{do} = c(S) - \left(\frac{H}{S}\right)^{2b/\sigma^2 - 1} c\!\left(\frac{H^2}{S}\right).

The knock-out is cheaper than the call, and the discount grows as the
spot approaches the barrier, where the option is worth nothing.

.. GENERATED FROM PYTHON SOURCE LINES 19-28

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.exotic_options import barrier_option_mc, merton_down_and_out_call, plot_price_profile
    from financialkit.options import black_scholes_price

    K, T, r, sigma = 100.0, 1.0, 0.05, 0.25
    S = np.linspace(70.0, 140.0, 200)








.. GENERATED FROM PYTHON SOURCE LINES 29-31

The call, its image and their difference
----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 31-42

.. code-block:: Python


    H = 90.0
    alive = S > H
    call = black_scholes_price(S, K, T, r, sigma)
    knock_out = merton_down_and_out_call(S, K, H, T, r, sigma)
    fig, axes = plt.subplots(1, 2, figsize=(12, 4))
    plot_price_profile(
        S, {"vanilla call": call, "down-and-out call": knock_out, "image term": np.where(alive, call - knock_out, np.nan)}, ax=axes[0], markers=(H, K)
    )
    axes[0].set_title(f"barrier H = {H:g}")




.. image-sg:: /api/gallery/exotic_options/barriers/images/sphx_glr_plot_01_merton_down_and_out_001.png
   :alt: barrier H = 90
   :srcset: /api/gallery/exotic_options/barriers/images/sphx_glr_plot_01_merton_down_and_out_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'barrier H = 90')



.. GENERATED FROM PYTHON SOURCE LINES 43-47

Lower barriers cost less
------------------------
The ratio to the vanilla call rises to one as the barrier moves away.
Bridge-corrected Monte Carlo confirms one point on each curve.

.. GENERATED FROM PYTHON SOURCE LINES 47-64

.. code-block:: Python


    for H in (60.0, 75.0, 90.0, 95.0):
        ratio = np.where(S > H, merton_down_and_out_call(S, K, H, T, r, sigma) / call, 0.0)
        axes[1].plot(S, ratio, label=f"H = {H:g}")
        res = barrier_option_mc(110.0, K, H, T, r, sigma, n_steps=25, n_paths=20_000, rng=0)
        axes[1].errorbar(
            110.0,
            res.estimate / black_scholes_price(110.0, K, T, r, sigma),
            yerr=2 * res.std_error / black_scholes_price(110.0, K, T, r, sigma),
            fmt="k.",
            capsize=3,
        )
    axes[1].set_xlabel("spot S")
    axes[1].set_ylabel("knock-out / vanilla")
    axes[1].legend()
    fig.tight_layout()
    plt.show()








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

   **Total running time of the script:** (0 minutes 0.109 seconds)


.. _sphx_glr_download_api_gallery_exotic_options_barriers_plot_01_merton_down_and_out.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/barriers/plot_01_merton_down_and_out.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_merton_down_and_out.ipynb <plot_01_merton_down_and_out.ipynb>`

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

      :download:`Download Python source code: plot_01_merton_down_and_out.py <plot_01_merton_down_and_out.py>`

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

      :download:`Download zipped: plot_01_merton_down_and_out.zip <plot_01_merton_down_and_out.zip>`


.. only:: html

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

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