.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/monte_carlo/basics/plot_03_boyle_option_pricing.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_monte_carlo_basics_plot_03_boyle_option_pricing.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_monte_carlo_basics_plot_03_boyle_option_pricing.py:


Boyle: options by Monte Carlo
=============================

Phelim Boyle (1977) observed that risk-neutral valuation turns an option
price into an expectation, and so into a simulation:

.. math:: V = e^{-rT} E^Q[\text{payoff}(S_T)], \qquad
          S_T = S e^{(r - \sigma^2/2)T + \sigma\sqrt T Z}.

He priced European options this way, checked them against Black-Scholes,
and noted that the method extends to payoffs no formula covers. Here the
terminal prices come from :func:`financialkit.integrators.gbm_paths`.

.. GENERATED FROM PYTHON SOURCE LINES 17-27

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.monte_carlo import european_option_mc
    from financialkit.monte_carlo.visualizers import plot_error_convergence
    from financialkit.options import black_scholes_price

    # Hull, 10th ed., Example 15.6.
    S, K, T, r, sigma = 42.0, 40.0, 0.5, 0.10, 0.20








.. GENERATED FROM PYTHON SOURCE LINES 28-30

Monte Carlo against Black-Scholes
---------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 30-36

.. code-block:: Python


    for option_type in ("call", "put"):
        res = european_option_mc(S, K, T, r, sigma, option_type=option_type, n_paths=200_000, rng=0)
        exact = black_scholes_price(S, K, T, r, sigma, option_type=option_type)
        print(f"{option_type}: MC {res.estimate:.4f} +/- {res.std_error:.4f}, Black-Scholes {exact:.4f}")





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

 .. code-block:: none

    call: MC 4.7636 +/- 0.0111, Black-Scholes 4.7594
    put: MC 0.8110 +/- 0.0041, Black-Scholes 0.8086




.. GENERATED FROM PYTHON SOURCE LINES 37-39

Prices across strikes, with 95% intervals
-----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 39-49

.. code-block:: Python


    strikes = np.linspace(30, 55, 11)
    mc = [european_option_mc(S, k, T, r, sigma, n_paths=5000, rng=1) for k in strikes]
    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    axes[0].plot(strikes, black_scholes_price(S, strikes, T, r, sigma), "k", label="Black-Scholes")
    axes[0].errorbar(strikes, [m.estimate for m in mc], yerr=[m.half_width for m in mc], fmt="o", capsize=3, label="MC, 5000 paths")
    axes[0].set_xlabel("strike K")
    axes[0].set_ylabel("call price")
    axes[0].legend()




.. image-sg:: /api/gallery/monte_carlo/basics/images/sphx_glr_plot_03_boyle_option_pricing_001.png
   :alt: plot 03 boyle option pricing
   :srcset: /api/gallery/monte_carlo/basics/images/sphx_glr_plot_03_boyle_option_pricing_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x11207dbe0>



.. GENERATED FROM PYTHON SOURCE LINES 50-52

The standard error halves when the paths quadruple
--------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 52-58

.. code-block:: Python


    ns = 1000 * 4 ** np.arange(5)
    se = [european_option_mc(S, K, T, r, sigma, n_paths=n, rng=2).std_error for n in ns]
    plot_error_convergence(ns, se, ax=axes[1], label="standard error")
    fig.tight_layout()
    plt.show()








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

   **Total running time of the script:** (0 minutes 0.079 seconds)


.. _sphx_glr_download_api_gallery_monte_carlo_basics_plot_03_boyle_option_pricing.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/monte_carlo/basics/plot_03_boyle_option_pricing.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_boyle_option_pricing.ipynb <plot_03_boyle_option_pricing.ipynb>`

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

      :download:`Download Python source code: plot_03_boyle_option_pricing.py <plot_03_boyle_option_pricing.py>`

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

      :download:`Download zipped: plot_03_boyle_option_pricing.zip <plot_03_boyle_option_pricing.zip>`


.. only:: html

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

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