.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/options/greeks/plot_02_delta_hedging.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_greeks_plot_02_delta_hedging.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_greeks_plot_02_delta_hedging.py:


Delta hedging in discrete time
==============================

Black and Scholes priced an option by showing that a dealer who sells
it and holds :math:`\Delta` shares, rebalancing continuously, carries
no risk. Real hedges are rebalanced a finite number of times. The
dealer's terminal P&L then has mean zero but a spread that shrinks like
:math:`1/\sqrt{n}` in the number of rebalances (Boyle and Emanuel 1980),
and the hedge works whatever the stock's real-world drift.

.. GENERATED FROM PYTHON SOURCE LINES 14-20

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.options import delta_hedge
    from financialkit.options.visualizers import plot_hedge_pnl








.. GENERATED FROM PYTHON SOURCE LINES 21-26

Hedging error against rebalancing frequency
-------------------------------------------

The stock drifts at 12% a year, far from the 5% risk-free rate, yet
the hedged P&L is centred near zero.

.. GENERATED FROM PYTHON SOURCE LINES 26-46

.. code-block:: Python


    S0, K, T, r, sigma = 100.0, 100.0, 0.5, 0.05, 0.2
    counts = [4, 16, 64, 256]
    stds = []
    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    for n in counts:
        res = delta_hedge(S0, K, T, r, sigma, n_rebalance=n, n_paths=4000, mu=0.12, rng=1)
        stds.append(res.pnl.std())
        print(f"n = {n:3d}: mean P&L {res.pnl.mean():+.3f}, std {res.pnl.std():.3f}  (premium {res.option_price:.3f})")
        if n in (16, 256):
            plot_hedge_pnl(res, ax=axes[0])
    axes[0].set_title("terminal P&L of the hedged short call")

    axes[1].loglog(counts, stds, "o-", label="std of hedged P&L")
    axes[1].loglog(counts, stds[0] * np.sqrt(counts[0] / np.array(counts)), "k:", label=r"$\propto 1/\sqrt{n}$")
    axes[1].set_xlabel("rebalances n")
    axes[1].set_ylabel("std of P&L")
    axes[1].legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/options/greeks/images/sphx_glr_plot_02_delta_hedging_001.png
   :alt: terminal P&L of the hedged short call
   :srcset: /api/gallery/options/greeks/images/sphx_glr_plot_02_delta_hedging_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    n =   4: mean P&L -0.009, std 2.297  (premium 6.889)
    n =  16: mean P&L +0.011, std 1.171  (premium 6.889)
    n =  64: mean P&L +0.001, std 0.588  (premium 6.889)
    n = 256: mean P&L +0.008, std 0.304  (premium 6.889)




.. GENERATED FROM PYTHON SOURCE LINES 47-49

One path: the hedge tracks the option
-------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 49-60

.. code-block:: Python


    res = delta_hedge(S0, K, T, r, sigma, n_rebalance=64, n_paths=1, rng=5)
    fig, ax = plt.subplots(figsize=(7, 3.5))
    ax.plot(res.times, res.spot[0])
    ax.axhline(K, color="k", lw=0.5)
    ax.set_xlabel("time (years)")
    ax.set_ylabel("stock price")
    ax.set_title(f"a single path: hedged P&L {res.pnl[0]:+.3f}")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/options/greeks/images/sphx_glr_plot_02_delta_hedging_002.png
   :alt: a single path: hedged P&L +0.282
   :srcset: /api/gallery/options/greeks/images/sphx_glr_plot_02_delta_hedging_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.271 seconds)


.. _sphx_glr_download_api_gallery_options_greeks_plot_02_delta_hedging.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/greeks/plot_02_delta_hedging.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_delta_hedging.ipynb <plot_02_delta_hedging.ipynb>`

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

      :download:`Download Python source code: plot_02_delta_hedging.py <plot_02_delta_hedging.py>`

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

      :download:`Download zipped: plot_02_delta_hedging.zip <plot_02_delta_hedging.zip>`


.. only:: html

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

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