.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/options/greeks/plot_01_greeks.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_01_greeks.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_01_greeks.py:


The Black-Scholes Greeks
========================

The Black-Scholes argument hedges an option with :math:`\Delta =
\partial V/\partial S` shares; the other partial derivatives measure
how that hedge and the option's value respond to everything else. The
Greeks, named by traders after the 1973 paper, satisfy the pricing PDE
itself,

.. math:: \Theta + \tfrac12 \sigma^2 S^2 \Gamma + (r - q) S \Delta = r V,

so time decay is the price of positive gamma. Each analytic Greek is
checked here against a finite difference of the price.

.. GENERATED FROM PYTHON SOURCE LINES 18-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.options import black_scholes_greeks, black_scholes_price, finite_difference_greeks
    from financialkit.options.visualizers import plot_greeks








.. GENERATED FROM PYTHON SOURCE LINES 25-27

Hull's Example 19.1, analytic and by finite differences
-------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-34

.. code-block:: Python


    args = (49.0, 50.0, 0.3846, 0.05, 0.20)
    g = black_scholes_greeks(*args)
    fd = finite_difference_greeks(black_scholes_price, *args)
    for name in ("delta", "gamma", "vega", "theta", "rho", "epsilon", "vanna", "volga"):
        print(f"{name:>7}: analytic {getattr(g, name):+10.5f}   finite difference {getattr(fd, name):+10.5f}")





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

 .. code-block:: none

      delta: analytic   +0.52160   finite difference   +0.52160
      gamma: analytic   +0.06555   finite difference   +0.06555
       vega: analytic  +12.10524   finite difference  +12.10524
      theta: analytic   -4.30539   finite difference   -4.30539
        rho: analytic   +8.90657   finite difference   +8.90657
    epsilon: analytic   -9.82979   finite difference   -9.82979
      vanna: analytic   +0.13914   finite difference   +0.13914
      volga: analytic   -0.22906   finite difference   -0.22906




.. GENERATED FROM PYTHON SOURCE LINES 35-37

The PDE holds across spot
-------------------------

.. GENERATED FROM PYTHON SOURCE LINES 37-44

.. code-block:: Python


    S = np.linspace(50, 150, 201)
    K, T, r, q, sigma = 100.0, 0.5, 0.05, 0.0, 0.25
    g = black_scholes_greeks(S, K, T, r, sigma, q)
    residual = g.theta + 0.5 * sigma**2 * S**2 * g.gamma + r * S * g.delta - r * black_scholes_price(S, K, T, r, sigma)
    print(f"max PDE residual: {np.max(np.abs(residual)):.1e}")





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

 .. code-block:: none

    max PDE residual: 2.7e-15




.. GENERATED FROM PYTHON SOURCE LINES 45-47

The Greeks of a call against spot
---------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 47-54

.. code-block:: Python


    axes = plot_greeks(S, g)
    for ax in axes:
        ax.axvline(K, color="k", lw=0.5)
    plt.gcf().tight_layout()

    plt.show()



.. image-sg:: /api/gallery/options/greeks/images/sphx_glr_plot_01_greeks_001.png
   :alt: delta, gamma, vega, theta, rho, vanna
   :srcset: /api/gallery/options/greeks/images/sphx_glr_plot_01_greeks_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.133 seconds)


.. _sphx_glr_download_api_gallery_options_greeks_plot_01_greeks.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_01_greeks.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_greeks.ipynb <plot_01_greeks.ipynb>`

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

      :download:`Download Python source code: plot_01_greeks.py <plot_01_greeks.py>`

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

      :download:`Download zipped: plot_01_greeks.zip <plot_01_greeks.zip>`


.. only:: html

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

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