.. 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_04_brenner_subrahmanyam.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_04_brenner_subrahmanyam.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_04_brenner_subrahmanyam.py:


The Brenner-Subrahmanyam approximation
======================================

Menachem Brenner and Marti Subrahmanyam (1988) gave traders a
pocket-calculator implied volatility. At the money forward,

.. math:: c = S e^{-qT}\left[2N\left(\tfrac{\sigma\sqrt{T}}{2}\right) - 1\right]
          \approx S e^{-qT} \sigma \sqrt{\frac{T}{2\pi}},

so :math:`\sigma \approx \sqrt{2\pi/T}\, c / S`, roughly :math:`2.5\,c/(S\sqrt{T})`.
The relative error is about :math:`-\sigma^2 T / 24`.

.. GENERATED FROM PYTHON SOURCE LINES 16-21

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.options import black_scholes_price, brenner_subrahmanyam_vol, implied_volatility








.. GENERATED FROM PYTHON SOURCE LINES 22-24

A quick estimate against the exact inversion
--------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 24-30

.. code-block:: Python


    S, r, T = 100.0, 0.03, 0.5
    F = S * np.exp(r * T)
    c = black_scholes_price(S, F, T, r, 0.25)
    print(f"ATM-forward call {c:.4f}: approximation {brenner_subrahmanyam_vol(c, S, T, r):.5f}, exact {implied_volatility(c, S, F, T, r):.5f}")





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

 .. code-block:: none

    ATM-forward call 7.0432: approximation 0.24967, exact 0.25000




.. GENERATED FROM PYTHON SOURCE LINES 31-33

Accuracy across volatility and maturity
---------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 33-46

.. code-block:: Python


    sigmas = np.linspace(0.05, 1.0, 60)
    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    for T in (0.1, 0.5, 2.0):
        F = S * np.exp(r * T)
        approx = brenner_subrahmanyam_vol(black_scholes_price(S, F, T, r, sigmas), S, T, r)
        axes[0].plot(sigmas, 100 * (approx / sigmas - 1), label=f"T = {T}")
        axes[0].plot(sigmas, -100 * sigmas**2 * T / 24, "k:", lw=1)
    axes[0].set_xlabel("true volatility")
    axes[0].set_ylabel("relative error (%)")
    axes[0].set_title(r"error, with $-\sigma^2 T/24$ dotted")
    axes[0].legend()




.. image-sg:: /api/gallery/options/implied_vol/images/sphx_glr_plot_04_brenner_subrahmanyam_001.png
   :alt: error, with $-\sigma^2 T/24$ dotted
   :srcset: /api/gallery/options/implied_vol/images/sphx_glr_plot_04_brenner_subrahmanyam_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x1167d1be0>



.. GENERATED FROM PYTHON SOURCE LINES 47-49

Away from the money the approximation breaks down
-------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 49-61

.. code-block:: Python


    K = np.linspace(80, 120, 41)
    T, sigma = 0.5, 0.25
    prices = black_scholes_price(S, K, T, r, sigma)
    axes[1].plot(K / (S * np.exp(r * T)), brenner_subrahmanyam_vol(prices, S, T, r), label="Brenner-Subrahmanyam")
    axes[1].axhline(sigma, color="k", ls=":", label="true volatility")
    axes[1].set_xlabel("moneyness K / F")
    axes[1].set_ylabel("estimated volatility")
    axes[1].legend()
    fig.tight_layout()

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.064 seconds)


.. _sphx_glr_download_api_gallery_options_implied_vol_plot_04_brenner_subrahmanyam.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_04_brenner_subrahmanyam.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_brenner_subrahmanyam.ipynb <plot_04_brenner_subrahmanyam.ipynb>`

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

      :download:`Download Python source code: plot_04_brenner_subrahmanyam.py <plot_04_brenner_subrahmanyam.py>`

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

      :download:`Download zipped: plot_04_brenner_subrahmanyam.zip <plot_04_brenner_subrahmanyam.zip>`


.. only:: html

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

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