.. 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_07_barrier_pde.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_07_barrier_pde.py>`
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.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_pde_methods_schemes_plot_07_barrier_pde.py:


Zvan-Forsyth-Vetzal barrier options
===================================

A knock-out option is a boundary condition: the value is zero on the
barrier. Zvan, Forsyth and Vetzal (2000) showed that finite differences
price barriers accurately when a grid node sits exactly on the barrier
and the solver handles the discontinuous data smoothly, rather than
letting the barrier fall between nodes.
:meth:`~financialkit.pde_methods.ThetaScheme.price_barrier` starts the
grid at the barrier, and the result matches the method of images,
:math:`C_{do}(S) = C(S) - (H/S)^{2\lambda - 2} C(H^2/S)`.

.. GENERATED FROM PYTHON SOURCE LINES 16-34

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.options import black_scholes_price
    from financialkit.pde_methods import CrankNicolson

    S, K, H, T, r, sigma, q = 100.0, 100.0, 90.0, 1.0, 0.05, 0.25, 0.0
    lam = (r - q + 0.5 * sigma**2) / sigma**2


    def images(s):
        """Down-and-out call by the method of images (Hull, 10th ed., Section 26.9)."""
        return black_scholes_price(s, K, T, r, sigma, q) - (H / s) ** (2 * lam - 2) * black_scholes_price(H**2 / s, K, T, r, sigma, q)


    res = CrankNicolson(400, 200, rannacher_steps=2).price_barrier(S, K, H, T, r, sigma, q, "call", "down-and-out")
    print(f"PDE {res.price:.4f}, method of images {images(S):.4f}, vanilla {black_scholes_price(S, K, T, r, sigma, q):.4f}")





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

 .. code-block:: none

    PDE 9.1106, method of images 9.1112, vanilla 12.3360




.. GENERATED FROM PYTHON SOURCE LINES 35-41

A node on the barrier, against a barrier between nodes
------------------------------------------------------
On a grid that ignores the barrier, the knock-out condition can only be
imposed at the nearest node, typically half a spacing away. Moving the
barrier by h/2 mimics that, and costs an error of order h; the aligned
grid keeps the scheme's second order.

.. GENERATED FROM PYTHON SOURCE LINES 41-64

.. code-block:: Python


    ms = np.array([50, 100, 200, 400])
    aligned, between = [], []
    for m in ms:
        cn = CrankNicolson(m, 200, rannacher_steps=2)
        h = (400.0 - H) / m
        aligned.append(abs(cn.price_barrier(S, K, H, T, r, sigma).price - images(S)))
        between.append(abs(cn.price_barrier(S, K, H + 0.5 * h, T, r, sigma).price - images(S)))

    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    near = res.S < 160
    axes[0].plot(res.S[near], res.V[near], label="down-and-out call (PDE)")
    axes[0].plot(res.S[near], black_scholes_price(res.S[near], K, T, r, sigma, q), "--", label="vanilla call")
    axes[0].axvline(H, color="k", lw=0.8)
    axes[0].set_xlabel("spot S")
    axes[0].legend()
    axes[1].loglog(ms, aligned, "o-", label="node on the barrier")
    axes[1].loglog(ms, between, "s-", label="barrier half a node away")
    axes[1].set_xlabel("spot intervals M")
    axes[1].set_ylabel("absolute error")
    axes[1].legend()
    fig.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/pde_methods/schemes/images/sphx_glr_plot_07_barrier_pde_001.png
   :alt: plot 07 barrier pde
   :srcset: /api/gallery/pde_methods/schemes/images/sphx_glr_plot_07_barrier_pde_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.130 seconds)


.. _sphx_glr_download_api_gallery_pde_methods_schemes_plot_07_barrier_pde.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
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        :alt: Launch JupyterLite
        :width: 150 px

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

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

      :download:`Download Python source code: plot_07_barrier_pde.py <plot_07_barrier_pde.py>`

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