.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/exotic_options/path_dependent/plot_03_variance_swap.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_exotic_options_path_dependent_plot_03_variance_swap.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_exotic_options_path_dependent_plot_03_variance_swap.py:


Demeterfi-Derman-Kamal-Zou variance swaps
=========================================

A variance swap pays the realized variance of the returns minus a fixed
strike. Kresimir Demeterfi, Emanuel Derman, Michael Kamal and Joseph Zou
(1999) showed how to replicate it, whatever the volatility process, with
a static strip of out-of-the-money options weighted by :math:`1/K^2`
plus a delta hedge, which gives the fair strike from today's option
prices alone:

.. math:: K_{\mathrm{var}} = \frac{2e^{rT}}{T}\left[\int_0^{F}\frac{P(K)}{K^2}dK + \int_F^\infty \frac{C(K)}{K^2}dK\right].

The same formula, discretised, is the VIX. With a downward skew the
cheap-looking puts carry high implied volatility, and the fair variance
lies above the at-the-money volatility squared.

.. GENERATED FROM PYTHON SOURCE LINES 20-45

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.exotic_options import plot_variance_strip, realized_variance, variance_swap_strike
    from financialkit.integrators import gbm_paths
    from financialkit.options import black_scholes_price

    S, T, r = 100.0, 1.0, 0.03
    F = S * np.exp(r * T)
    K = np.arange(30.0, 300.0, 2.5)


    def skew(k):
        return np.clip(0.2 - 0.15 * np.log(k / F), 0.08, None)


    def strip(vol):
        c, p = (black_scholes_price(S, K, T, r, vol, option_type=t) for t in ("call", "put"))
        return variance_swap_strike(K, c, p, S, T, r)


    flat, skewed = strip(0.2), strip(skew(K))
    print(f"flat 20% smile: fair volatility {flat.vol_strike:.4f}")
    print(f"skewed smile (ATM 20%): fair volatility {skewed.vol_strike:.4f}")





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

 .. code-block:: none

    flat 20% smile: fair volatility 0.2003
    skewed smile (ATM 20%): fair volatility 0.2115




.. GENERATED FROM PYTHON SOURCE LINES 46-48

The replicating strip under the skew
------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 48-57

.. code-block:: Python


    fig, axes = plt.subplots(1, 3, figsize=(15, 4))
    axes[0].plot(K, skew(K))
    axes[0].axvline(F, color="k", ls="--", lw=1)
    axes[0].set_xlabel("strike K")
    axes[0].set_ylabel("implied volatility")
    plot_variance_strip(skewed, ax=axes[1])
    axes[1].set_xlim(40, 200)




.. image-sg:: /api/gallery/exotic_options/path_dependent/images/sphx_glr_plot_03_variance_swap_001.png
   :alt: fair volatility 21.15%
   :srcset: /api/gallery/exotic_options/path_dependent/images/sphx_glr_plot_03_variance_swap_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    (40.0, 200.0)



.. GENERATED FROM PYTHON SOURCE LINES 58-60

Under Black-Scholes, realized variance averages to the fair strike
------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 60-70

.. code-block:: Python


    _, paths = gbm_paths(S, r, 0.2, T, 252, 5000, rng=0)
    rv = realized_variance(paths, T)
    axes[2].hist(rv, bins=50, density=True, alpha=0.7)
    axes[2].axvline(flat.fair_variance, color="k", lw=2, label=f"fair strike {flat.fair_variance:.4f}")
    axes[2].axvline(rv.mean(), color="C3", ls="--", label=f"mean realized {rv.mean():.4f}")
    axes[2].set_xlabel("realized variance (daily returns)")
    axes[2].legend()
    fig.tight_layout()
    plt.show()








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

   **Total running time of the script:** (0 minutes 0.152 seconds)


.. _sphx_glr_download_api_gallery_exotic_options_path_dependent_plot_03_variance_swap.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/exotic_options/path_dependent/plot_03_variance_swap.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_variance_swap.ipynb <plot_03_variance_swap.ipynb>`

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

      :download:`Download Python source code: plot_03_variance_swap.py <plot_03_variance_swap.py>`

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

      :download:`Download zipped: plot_03_variance_swap.zip <plot_03_variance_swap.zip>`


.. only:: html

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

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