.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/pde_methods/schemes/plot_04_heat_equation_transform.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_pde_methods_schemes_plot_04_heat_equation_transform.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_pde_methods_schemes_plot_04_heat_equation_transform.py:


Black-Scholes as the heat equation
==================================

Black and Scholes (1973) found their formula by changing variables until
the pricing equation became the heat equation. With
:math:`x = \ln(S/K)` and :math:`\tau = \tfrac12\sigma^2(T - t)`,
:math:`V = K e^{\alpha x + \beta\tau} u` turns Black-Scholes into
:math:`u_\tau = u_{xx}`, whose initial condition is the transformed
payoff. :func:`~financialkit.pde_methods.heat_equation_price` solves the
heat equation by Crank-Nicolson on a uniform grid, watches the payoff
diffuse, and transforms back.

.. GENERATED FROM PYTHON SOURCE LINES 16-27

.. code-block:: Python

    import matplotlib.pyplot as plt

    from financialkit.options import black_scholes_price
    from financialkit.pde_methods import heat_equation_price
    from financialkit.pde_methods.visualizers import plot_heat_solution

    S, K, T, r, sigma, q = 100.0, 100.0, 1.0, 0.05, 0.25, 0.02
    res = heat_equation_price(S, K, T, r, sigma, q, n_space=400, n_time=100)
    print(f"alpha = {res.alpha:.3f}, beta = {res.beta:.3f}, tau_end = {res.tau[-1]:.4f}")
    print(f"heat equation {res.price:.5f}, Black-Scholes {black_scholes_price(S, K, T, r, sigma, q):.5f}")





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

 .. code-block:: none

    alpha = 0.020, beta = -1.600, tau_end = 0.0312
    heat equation 11.12264, Black-Scholes 11.12376




.. GENERATED FROM PYTHON SOURCE LINES 28-30

The payoff diffusing in heat-equation variables, and back in prices
-------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 30-44

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    plot_heat_solution(res, n_curves=5, ax=axes[0])
    axes[0].set_xlim(-1.0, 1.0)
    axes[0].set_title(r"$u_\tau = u_{xx}$")

    near = (res.S > 50) & (res.S < 180)
    axes[1].plot(res.S[near], black_scholes_price(res.S[near], K, T, r, sigma, q), "k", lw=1, label="Black-Scholes")
    axes[1].plot(res.S[near][::6], res.V[near][::6], "o", ms=3, label="heat equation, transformed back")
    axes[1].set_xlabel("spot S")
    axes[1].set_ylabel("call value")
    axes[1].legend()
    fig.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/pde_methods/schemes/images/sphx_glr_plot_04_heat_equation_transform_001.png
   :alt: $u_\tau = u_{xx}$
   :srcset: /api/gallery/pde_methods/schemes/images/sphx_glr_plot_04_heat_equation_transform_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.092 seconds)


.. _sphx_glr_download_api_gallery_pde_methods_schemes_plot_04_heat_equation_transform.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/pde_methods/schemes/plot_04_heat_equation_transform.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_heat_equation_transform.ipynb <plot_04_heat_equation_transform.ipynb>`

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

      :download:`Download Python source code: plot_04_heat_equation_transform.py <plot_04_heat_equation_transform.py>`

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

      :download:`Download zipped: plot_04_heat_equation_transform.zip <plot_04_heat_equation_transform.zip>`


.. only:: html

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

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