.. 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_01_implied_volatility.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_01_implied_volatility.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_01_implied_volatility.py:


Latané-Rendleman implied volatility
===================================

Every input to Black-Scholes is observable except the volatility. Henry
Latané and Richard Rendleman (1976) turned the formula around: given a
market price, solve for the volatility that reproduces it. The price is
strictly increasing in :math:`\sigma` between Merton's bounds, so the
root is unique; it exists only for prices strictly inside them.

.. GENERATED FROM PYTHON SOURCE LINES 13-18

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.options import black_scholes_price, implied_volatility, no_arbitrage_bounds








.. GENERATED FROM PYTHON SOURCE LINES 19-21

Hull's example (Section 15.11)
------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 21-24

.. code-block:: Python


    print(f"implied vol of a 1.875 call (S = 21, K = 20, r = 10%, 3 months): {implied_volatility(1.875, 21.0, 20.0, 0.25, 0.10):.4f}")





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

 .. code-block:: none

    implied vol of a 1.875 call (S = 21, K = 20, r = 10%, 3 months): 0.2345




.. GENERATED FROM PYTHON SOURCE LINES 25-27

The price is monotone in volatility, so the inverse is unique
-------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-48

.. code-block:: Python


    S, K, T, r = 100.0, 110.0, 0.5, 0.03
    sigmas = np.linspace(0.001, 2.0, 400)
    prices = black_scholes_price(S, K, T, r, sigmas)
    lo, hi = no_arbitrage_bounds(S, K, T, r)
    target = 4.0
    iv = implied_volatility(target, S, K, T, r)
    print(f"a {target} call has implied volatility {iv:.4f}")

    fig, ax = plt.subplots(figsize=(7, 4))
    ax.plot(sigmas, prices, label="Black-Scholes price")
    ax.axhline(lo, color="k", ls=":", label="Merton bounds")
    ax.axhline(hi, color="k", ls=":")
    ax.plot([0, iv, iv], [target, target, 0], "r--", lw=1)
    ax.plot(iv, target, "ro", label=f"implied vol = {iv:.3f}")
    ax.set_xlabel("volatility")
    ax.set_ylabel("call price")
    ax.set_ylim(0, 60)
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/options/implied_vol/images/sphx_glr_plot_01_implied_volatility_001.png
   :alt: plot 01 implied volatility
   :srcset: /api/gallery/options/implied_vol/images/sphx_glr_plot_01_implied_volatility_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    a 4.0 call has implied volatility 0.2538




.. GENERATED FROM PYTHON SOURCE LINES 49-51

Prices outside the bounds have no implied volatility
----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 51-58

.. code-block:: Python


    try:
        implied_volatility(0.5 * lo, S, 80.0, T, r)
    except ValueError as err:
        print("ValueError:", err)

    plt.show()




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

 .. code-block:: none

    ValueError: price 0 violates the no-arbitrage bounds (21.191, 100); no implied volatility exists





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

   **Total running time of the script:** (0 minutes 0.048 seconds)


.. _sphx_glr_download_api_gallery_options_implied_vol_plot_01_implied_volatility.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_01_implied_volatility.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_implied_volatility.ipynb <plot_01_implied_volatility.ipynb>`

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

      :download:`Download Python source code: plot_01_implied_volatility.py <plot_01_implied_volatility.py>`

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

      :download:`Download zipped: plot_01_implied_volatility.zip <plot_01_implied_volatility.zip>`


.. only:: html

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

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