.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/credit/reduced_form/plot_02_duffie_singleton.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_reduced_form_plot_02_duffie_singleton.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_reduced_form_plot_02_duffie_singleton.py:


Duffie and Singleton: recovery of market value
==============================================

Duffie and Singleton (1999) assumed that at default a bond loses a
fraction :math:`L` of its own market value. Then defaultable claims are
priced like riskless ones discounted at :math:`r + \lambda L`, so any
short-rate machinery prices credit too. With a Cox-Ingersoll-Ross
intensity the risky bond has the CIR closed form; here it is compared
with a simulation, and the three recovery conventions are compared for
a flat hazard.

.. GENERATED FROM PYTHON SOURCE LINES 15-33

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.credit import duffie_singleton_cir_zero, risky_zero_coupon
    from financialkit.interest_rate_models import CIRModel

    T = np.linspace(0.1, 15, 60)
    r, h, R = 0.03, 0.03, 0.4
    fig, ax = plt.subplots(figsize=(7, 4))
    for conv, name in (("rmv", "recovery of market value"), ("rt", "recovery of treasury"), ("rfv", "recovery of face value")):
        P = risky_zero_coupon(T, r, h, R, conv)
        ax.plot(T, 1e4 * (-np.log(P) / T - r), label=name)
    ax.axhline(1e4 * h * (1 - R), color="0.5", ls=":", label="lambda (1 - R)")
    ax.set_xlabel("maturity")
    ax.set_ylabel("zero-coupon spread (bp)")
    ax.set_title("same hazard and recovery rate, three conventions")
    ax.legend()




.. image-sg:: /api/gallery/credit/reduced_form/images/sphx_glr_plot_02_duffie_singleton_001.png
   :alt: same hazard and recovery rate, three conventions
   :srcset: /api/gallery/credit/reduced_form/images/sphx_glr_plot_02_duffie_singleton_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x116492e40>



.. GENERATED FROM PYTHON SOURCE LINES 34-39

A stochastic intensity
----------------------

The intensity follows :math:`d\lambda = \kappa(\theta - \lambda)dt + \sigma\sqrt\lambda dW`.
Monte Carlo averages :math:`e^{-\int (r + L\lambda)}` over CIR paths.

.. GENERATED FROM PYTHON SOURCE LINES 39-57

.. code-block:: Python


    lam0, kappa, theta, sig = 0.01, 0.4, 0.04, 0.12
    mats = np.array([1.0, 3.0, 5.0, 7.0, 10.0])
    closed = duffie_singleton_cir_zero(mats, r, lam0, kappa, theta, sig, R)
    times, lam = CIRModel(lam0, kappa, theta, sig).simulate(10.0, 400, 4000, rng=1)
    dt = times[1] - times[0]
    cum = np.concatenate([np.zeros((lam.shape[0], 1)), np.cumsum(0.5 * (lam[:, 1:] + lam[:, :-1]) * dt, axis=1)], axis=1)
    idx = np.searchsorted(times, mats - 1e-12)
    mc = np.exp(-r * mats - (1 - R) * cum[:, idx])

    fig, ax = plt.subplots(figsize=(7, 4))
    ax.plot(mats, closed, "o-", label="CIR closed form")
    ax.errorbar(mats, mc.mean(axis=0), yerr=4 * mc.std(axis=0) / np.sqrt(mc.shape[0]), fmt="x", color="k", label="Monte Carlo +- 4 se")
    ax.set_xlabel("maturity")
    ax.set_ylabel("risky zero-coupon price")
    ax.legend()

    plt.show()



.. image-sg:: /api/gallery/credit/reduced_form/images/sphx_glr_plot_02_duffie_singleton_002.png
   :alt: plot 02 duffie singleton
   :srcset: /api/gallery/credit/reduced_form/images/sphx_glr_plot_02_duffie_singleton_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.114 seconds)


.. _sphx_glr_download_api_gallery_credit_reduced_form_plot_02_duffie_singleton.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/reduced_form/plot_02_duffie_singleton.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_duffie_singleton.ipynb <plot_02_duffie_singleton.ipynb>`

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

      :download:`Download Python source code: plot_02_duffie_singleton.py <plot_02_duffie_singleton.py>`

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

      :download:`Download zipped: plot_02_duffie_singleton.zip <plot_02_duffie_singleton.zip>`


.. only:: html

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

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