.. 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_03_black_scholes.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_03_black_scholes.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_03_black_scholes.py:


The Black-Scholes formula
=========================

Fischer Black and Myron Scholes (1973) showed that a continuously
rebalanced position in the stock and a bond replicates a European
option when the stock follows geometric Brownian motion, so the option
price cannot depend on investors' expected return:

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

The formula equals the discounted expected payoff under the
risk-neutral drift :math:`r`, which a Monte Carlo over exact GBM paths
confirms.

.. GENERATED FROM PYTHON SOURCE LINES 18-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.integrators import gbm_paths
    from financialkit.options import black_scholes_price
    from financialkit.options.visualizers import plot_option_value








.. GENERATED FROM PYTHON SOURCE LINES 26-28

Risk-neutral Monte Carlo agrees with the formula
------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-35

.. code-block:: Python


    S0, K, T, r, sigma = 100.0, 105.0, 1.0, 0.05, 0.2
    _, paths = gbm_paths(S0, r, sigma, T, n_steps=1, n_paths=400_000, rng=0)
    discounted = np.exp(-r * T) * np.maximum(paths[:, -1] - K, 0.0)
    print(f"Black-Scholes: {black_scholes_price(S0, K, T, r, sigma):.4f}")
    print(f"Monte Carlo:   {discounted.mean():.4f} +/- {1.96 * discounted.std() / np.sqrt(discounted.size):.4f}")





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

 .. code-block:: none

    Black-Scholes: 8.0214
    Monte Carlo:   8.0392 +/- 0.0409




.. GENERATED FROM PYTHON SOURCE LINES 36-38

The price as a function of spot, approaching the payoff at expiry
-----------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 38-53

.. code-block:: Python


    S = np.linspace(60, 150, 181)
    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    for tau in (1.0, 0.25, 0.02):
        plot_option_value(S, black_scholes_price(S, K, tau, r, sigma), K, ax=axes[0], label=f"T = {tau}")
    axes[0].set_title("call value against spot")
    for vol in (0.1, 0.2, 0.4):
        axes[1].plot(S, black_scholes_price(S, K, T, r, vol, option_type="put"), label=f"sigma = {vol}")
    axes[1].set_xlabel("spot S")
    axes[1].set_ylabel("put value")
    axes[1].set_title("higher volatility, higher price")
    axes[1].legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/options/closed_form/images/sphx_glr_plot_03_black_scholes_001.png
   :alt: call value against spot, higher volatility, higher price
   :srcset: /api/gallery/options/closed_form/images/sphx_glr_plot_03_black_scholes_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.092 seconds)


.. _sphx_glr_download_api_gallery_options_closed_form_plot_03_black_scholes.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_03_black_scholes.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_black_scholes.ipynb <plot_03_black_scholes.ipynb>`

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

      :download:`Download Python source code: plot_03_black_scholes.py <plot_03_black_scholes.py>`

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

      :download:`Download zipped: plot_03_black_scholes.zip <plot_03_black_scholes.zip>`


.. only:: html

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

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