.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/credit/structural/plot_02_black_cox.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_credit_structural_plot_02_black_cox.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_credit_structural_plot_02_black_cox.py:


Black and Cox: default at the first passage
===========================================

Black and Cox (1976) let safety covenants force default the first time
the asset value touches a barrier, not only at maturity. The reflection
principle gives the first-passage probability in closed form. Here it is
checked against simulated asset paths (with a Brownian-bridge
correction for crossings between grid points) and compared with
Merton's maturity-only default probability.

.. GENERATED FROM PYTHON SOURCE LINES 14-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy import stats

    from financialkit.credit import black_cox_default_probability, first_passage_crossing_probability
    from financialkit.integrators import gbm_paths

    V0, K, mu, sigma, T = 100.0, 70.0, 0.05, 0.25, 5.0
    t, paths = gbm_paths(V0, mu, sigma, T=T, n_steps=100, n_paths=4000, rng=0)








.. GENERATED FROM PYTHON SOURCE LINES 25-27

Paths that touch the barrier
----------------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-38

.. code-block:: Python


    hit_grid = (paths <= K).any(axis=1)
    fig, ax = plt.subplots(figsize=(7, 4))
    for i in range(60):
        ax.plot(t, paths[i], color="C3" if hit_grid[i] else "C0", lw=0.6, alpha=0.7)
    ax.axhline(K, color="k", lw=2, label="barrier K")
    ax.set_xlabel("years")
    ax.set_ylabel("asset value")
    ax.set_title("red: defaulted before maturity")
    ax.legend()




.. image-sg:: /api/gallery/credit/structural/images/sphx_glr_plot_02_black_cox_001.png
   :alt: red: defaulted before maturity
   :srcset: /api/gallery/credit/structural/images/sphx_glr_plot_02_black_cox_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x1163b3e00>



.. GENERATED FROM PYTHON SOURCE LINES 39-41

First passage versus Merton
---------------------------

.. GENERATED FROM PYTHON SOURCE LINES 41-57

.. code-block:: Python


    horizons = np.linspace(0.1, T, 50)
    bc = black_cox_default_probability(V0, K, horizons, mu, sigma)
    merton = stats.norm.cdf((np.log(K / V0) - (mu - 0.5 * sigma**2) * horizons) / (sigma * np.sqrt(horizons)))
    p_bridge = first_passage_crossing_probability(paths, K, sigma, T / 100)

    fig, ax = plt.subplots(figsize=(7, 4))
    ax.plot(horizons, bc, lw=2, label="Black-Cox first passage")
    ax.plot(horizons, merton, lw=2, ls="--", label="Merton: below K at the horizon")
    ax.errorbar(T, p_bridge.mean(), yerr=4 * p_bridge.std() / np.sqrt(p_bridge.size), fmt="o", color="k", label="Monte Carlo, bridge-corrected")
    ax.plot(T, hit_grid.mean(), "x", color="C3", ms=8, label="Monte Carlo, grid only (biased low)")
    ax.set_xlabel("horizon (years)")
    ax.set_ylabel("default probability")
    ax.legend()

    plt.show()



.. image-sg:: /api/gallery/credit/structural/images/sphx_glr_plot_02_black_cox_002.png
   :alt: plot 02 black cox
   :srcset: /api/gallery/credit/structural/images/sphx_glr_plot_02_black_cox_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.080 seconds)


.. _sphx_glr_download_api_gallery_credit_structural_plot_02_black_cox.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/credit/structural/plot_02_black_cox.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_black_cox.ipynb <plot_02_black_cox.ipynb>`

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

      :download:`Download Python source code: plot_02_black_cox.py <plot_02_black_cox.py>`

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

      :download:`Download zipped: plot_02_black_cox.zip <plot_02_black_cox.zip>`


.. only:: html

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

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