.. 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_01_bachelier.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_01_bachelier.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_01_bachelier.py:


Bachelier's normal model
========================

In his 1900 thesis *Théorie de la spéculation* Louis Bachelier modelled
the price of a French government bond as a Brownian motion and priced
options on it, five years before Einstein's paper on Brownian motion.
In his model price changes are normal with an absolute volatility
:math:`\sigma_N`, so an at-the-money option is worth
:math:`\sigma_N\sqrt{T/(2\pi)}`: the square-root-of-time law. Below, the
formula is checked against simulated arithmetic Brownian paths and
compared with Black-Scholes at a matched volatility.

.. GENERATED FROM PYTHON SOURCE LINES 16-21

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.options import bachelier_price, black_scholes_price








.. GENERATED FROM PYTHON SOURCE LINES 22-24

Monte Carlo under arithmetic Brownian motion
--------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 24-33

.. code-block:: Python


    S0, T, sigma_n = 100.0, 1.0, 20.0
    rng = np.random.default_rng(0)
    S_T = S0 + sigma_n * np.sqrt(T) * rng.standard_normal(200_000)
    for K in (90.0, 100.0, 110.0):
        payoff = np.maximum(S_T - K, 0.0)
        mc, se = payoff.mean(), payoff.std() / np.sqrt(payoff.size)
        print(f"K = {K:5.1f}: Bachelier {bachelier_price(S0, K, T, 0.0, sigma_n):.4f}, Monte Carlo {mc:.4f} +/- {1.96 * se:.4f}")





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

 .. code-block:: none

    K =  90.0: Bachelier 13.9559, Monte Carlo 13.9691 +/- 0.0652
    K = 100.0: Bachelier 7.9788, Monte Carlo 7.9905 +/- 0.0512
    K = 110.0: Bachelier 3.9559, Monte Carlo 3.9597 +/- 0.0362




.. GENERATED FROM PYTHON SOURCE LINES 34-40

The square-root-of-time law, and the comparison with Black-Scholes
------------------------------------------------------------------

With :math:`\sigma_N = \sigma S_0` the two models nearly agree near the
money for short maturities; Bachelier's prices go negative only in the
far tail of the normal, which lognormal Black-Scholes never does.

.. GENERATED FROM PYTHON SOURCE LINES 40-57

.. code-block:: Python


    maturities = np.linspace(0.01, 4.0, 100)
    K = np.linspace(40, 160, 121)
    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    axes[0].plot(maturities, bachelier_price(S0, S0, maturities, 0.0, sigma_n), label="Bachelier ATM")
    axes[0].plot(maturities, sigma_n * np.sqrt(maturities / (2 * np.pi)), "k:", label=r"$\sigma_N\sqrt{T/2\pi}$")
    axes[0].set_xlabel("maturity T")
    axes[0].set_ylabel("at-the-money call")
    axes[0].legend()
    axes[1].plot(K, bachelier_price(S0, K, T, 0.0, sigma_n), label="Bachelier, $\\sigma_N = 20$")
    axes[1].plot(K, black_scholes_price(S0, K, T, 0.0, 0.2), "--", label="Black-Scholes, $\\sigma = 20\\%$")
    axes[1].set_xlabel("strike K")
    axes[1].set_ylabel("call price")
    axes[1].legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/options/closed_form/images/sphx_glr_plot_01_bachelier_001.png
   :alt: plot 01 bachelier
   :srcset: /api/gallery/options/closed_form/images/sphx_glr_plot_01_bachelier_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.603 seconds)


.. _sphx_glr_download_api_gallery_options_closed_form_plot_01_bachelier.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_01_bachelier.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_bachelier.ipynb <plot_01_bachelier.ipynb>`

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

      :download:`Download Python source code: plot_01_bachelier.py <plot_01_bachelier.py>`

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

      :download:`Download zipped: plot_01_bachelier.zip <plot_01_bachelier.zip>`


.. only:: html

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

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