.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/options/implied_vol/plot_02_breeden_litzenberger.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_implied_vol_plot_02_breeden_litzenberger.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_implied_vol_plot_02_breeden_litzenberger.py:


Breeden-Litzenberger risk-neutral density
=========================================

Douglas Breeden and Robert Litzenberger (1978) showed that the second
strike derivative of the call price is the discounted risk-neutral
density of the terminal price,

.. math:: f(K) = e^{rT} \frac{\partial^2 c}{\partial K^2}.

A butterfly spread around :math:`K` pays off only if :math:`S_T` lands
near :math:`K`, so its price is the market's state price for that
outcome. Black-Scholes prices give back the lognormal; prices from a
two-regime mixture reveal a bimodal market view.

.. GENERATED FROM PYTHON SOURCE LINES 18-26

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.integrate import trapezoid
    from scipy.stats import lognorm

    from financialkit.options import black_scholes_price, breeden_litzenberger_density
    from financialkit.options.visualizers import plot_density








.. GENERATED FROM PYTHON SOURCE LINES 27-29

Black-Scholes prices recover the lognormal density
--------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 29-41

.. code-block:: Python


    S, T, r, sigma = 100.0, 0.5, 0.03, 0.25
    K = np.linspace(30, 220, 381)
    res = breeden_litzenberger_density(K, black_scholes_price(S, K, T, r, sigma), T, r)
    exact = lognorm.pdf(res.strikes, s=sigma * np.sqrt(T), scale=S * np.exp((r - 0.5 * sigma**2) * T))
    print(f"max error against the lognormal: {np.max(np.abs(res.density - exact)):.1e}")

    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    plot_density(res, ax=axes[0])
    axes[0].plot(res.strikes, exact, "k:", label="lognormal")
    axes[0].legend()




.. image-sg:: /api/gallery/options/implied_vol/images/sphx_glr_plot_02_breeden_litzenberger_001.png
   :alt: plot 02 breeden litzenberger
   :srcset: /api/gallery/options/implied_vol/images/sphx_glr_plot_02_breeden_litzenberger_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    max error against the lognormal: 1.7e-06

    <matplotlib.legend.Legend object at 0x1167d17f0>



.. GENERATED FROM PYTHON SOURCE LINES 42-48

A bimodal market: an event with two outcomes
--------------------------------------------

Mix two lognormals with forwards 85 and 115, the same overall forward,
as before a binary event. The density from the mixed prices shows both
modes, which no single Black-Scholes volatility could produce.

.. GENERATED FROM PYTHON SOURCE LINES 48-59

.. code-block:: Python


    w = 0.5
    F = S * np.exp(r * T)
    mix = w * black_scholes_price(0.85 * S, K, T, r, 0.1) + (1 - w) * black_scholes_price(1.15 * S, K, T, r, 0.1)
    res_mix = breeden_litzenberger_density(K, mix, T, r)
    plot_density(res_mix, ax=axes[1], label="two-regime prices")
    axes[1].axvline(F, color="k", lw=0.5)
    print(f"total probability: {trapezoid(res_mix.density, res_mix.strikes):.4f}")
    fig.tight_layout()

    plt.show()




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

 .. code-block:: none

    total probability: 1.0000





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

   **Total running time of the script:** (0 minutes 0.054 seconds)


.. _sphx_glr_download_api_gallery_options_implied_vol_plot_02_breeden_litzenberger.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/implied_vol/plot_02_breeden_litzenberger.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_breeden_litzenberger.ipynb <plot_02_breeden_litzenberger.ipynb>`

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

      :download:`Download Python source code: plot_02_breeden_litzenberger.py <plot_02_breeden_litzenberger.py>`

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

      :download:`Download zipped: plot_02_breeden_litzenberger.zip <plot_02_breeden_litzenberger.zip>`


.. only:: html

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

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