.. 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_03_volatility_smile.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_03_volatility_smile.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_03_volatility_smile.py:


The volatility smile
====================

Under Black-Scholes every strike has the same implied volatility. Before
October 1987 index options roughly agreed; after the crash, low-strike
puts became permanently more expensive, and implied volatility fell
steeply with the strike (Rubinstein 1994). Inverting a strip of option
prices strike by strike shows the market's departure from the
lognormal. Here the prices come from two simple arbitrage-free models:
a mixture of volatilities (a symmetric smile) and a mixture of a normal
regime with a crash regime (a skew).

.. GENERATED FROM PYTHON SOURCE LINES 16-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.options import black_scholes_price, implied_vol_smile
    from financialkit.options.visualizers import plot_smile

    S, T, r = 100.0, 0.5, 0.02
    K = np.linspace(65, 135, 29)








.. GENERATED FROM PYTHON SOURCE LINES 26-28

Flat: Black-Scholes prices
--------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-32

.. code-block:: Python


    flat = implied_vol_smile(black_scholes_price(S, K, T, r, 0.2), K, S, T, r)
    print("Black-Scholes prices, implied vol range:", np.round([flat.vols.min(), flat.vols.max()], 6))





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

 .. code-block:: none

    Black-Scholes prices, implied vol range: [0.2 0.2]




.. GENERATED FROM PYTHON SOURCE LINES 33-35

Smile: uncertain volatility
---------------------------

.. GENERATED FROM PYTHON SOURCE LINES 35-39

.. code-block:: Python


    smile_prices = 0.5 * black_scholes_price(S, K, T, r, 0.12) + 0.5 * black_scholes_price(S, K, T, r, 0.32)
    smile = implied_vol_smile(smile_prices, K, S, T, r)








.. GENERATED FROM PYTHON SOURCE LINES 40-45

Skew: a small chance of a crash
-------------------------------

With probability 10% the forward is 25% lower; the other regime's
forward is raised so the overall forward is unchanged.

.. GENERATED FROM PYTHON SOURCE LINES 45-58

.. code-block:: Python


    p_crash, drop = 0.10, 0.25
    up = (1 - p_crash * (1 - drop)) / (1 - p_crash)
    skew_prices = p_crash * black_scholes_price(S * (1 - drop), K, T, r, 0.3) + (1 - p_crash) * black_scholes_price(S * up, K, T, r, 0.15)
    skew = implied_vol_smile(skew_prices, K, S, T, r)

    fig, ax = plt.subplots(figsize=(7, 4))
    plot_smile(flat, ax=ax, label="Black-Scholes: flat", moneyness=True)
    plot_smile(smile, ax=ax, label="vol mixture: smile", moneyness=True)
    plot_smile(skew, ax=ax, label="crash mixture: skew", moneyness=True)
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/options/implied_vol/images/sphx_glr_plot_03_volatility_smile_001.png
   :alt: plot 03 volatility smile
   :srcset: /api/gallery/options/implied_vol/images/sphx_glr_plot_03_volatility_smile_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.089 seconds)


.. _sphx_glr_download_api_gallery_options_implied_vol_plot_03_volatility_smile.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_03_volatility_smile.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_volatility_smile.ipynb <plot_03_volatility_smile.ipynb>`

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

      :download:`Download Python source code: plot_03_volatility_smile.py <plot_03_volatility_smile.py>`

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

      :download:`Download zipped: plot_03_volatility_smile.zip <plot_03_volatility_smile.zip>`


.. only:: html

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

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