.. 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_01_jarrow_turnbull.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_01_jarrow_turnbull.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_01_jarrow_turnbull.py:


Jarrow and Turnbull: hazard rates and survival curves
=====================================================

Jarrow and Turnbull (1995) modelled the timing of default rather than
its cause: default is the first jump of a Poisson process with
intensity :math:`\lambda(t)`, so the survival probability is
:math:`Q(t) = e^{-\int_0^t \lambda}`. A risky zero-coupon bond that
recovers a fraction :math:`\delta` of a riskless one is worth
:math:`P(0,T)[\delta + (1-\delta)Q(T)]`, and its spread is close to
:math:`\lambda(1 - \delta)`: the credit triangle.

.. GENERATED FROM PYTHON SOURCE LINES 15-27

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.credit import HazardCurve, jarrow_turnbull_zero, spread_from_hazard
    from financialkit.credit.visualizers import plot_survival

    curve = HazardCurve([1.0, 3.0, 5.0, 10.0], [0.010, 0.015, 0.025, 0.030])
    fig, ax = plt.subplots(figsize=(7, 4))
    plot_survival(curve, T=10.0, ax=ax, label="piecewise-constant hazard")
    plot_survival(HazardCurve.flat(0.02), T=10.0, ax=ax, label="flat 2%", hazard=False)
    ax.legend(loc="lower left")




.. image-sg:: /api/gallery/credit/reduced_form/images/sphx_glr_plot_01_jarrow_turnbull_001.png
   :alt: plot 01 jarrow turnbull
   :srcset: /api/gallery/credit/reduced_form/images/sphx_glr_plot_01_jarrow_turnbull_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x11612b380>



.. GENERATED FROM PYTHON SOURCE LINES 28-33

Simulated default times
-----------------------

Inverting the cumulative hazard of a uniform draw gives default times
whose histogram is the default density :math:`\lambda(t) Q(t)`.

.. GENERATED FROM PYTHON SOURCE LINES 33-42

.. code-block:: Python


    tau = curve.sample_default_times(100_000, rng=0)
    t = np.linspace(0, 10, 400)
    fig, ax = plt.subplots(figsize=(7, 4))
    ax.hist(tau[tau < 10], bins=80, density=True, weights=np.full((tau < 10).sum(), (tau < 10).mean()), color="0.7", label="simulated")
    ax.plot(t, curve.default_density(t), "C3", lw=2, label="lambda(t) Q(t)")
    ax.set_xlabel("default time (years)")
    ax.legend()




.. image-sg:: /api/gallery/credit/reduced_form/images/sphx_glr_plot_01_jarrow_turnbull_002.png
   :alt: plot 01 jarrow turnbull
   :srcset: /api/gallery/credit/reduced_form/images/sphx_glr_plot_01_jarrow_turnbull_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x116492ba0>



.. GENERATED FROM PYTHON SOURCE LINES 43-45

Risky bond prices and the credit triangle
-----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 45-59

.. code-block:: Python


    T = np.linspace(0.25, 10, 40)
    r, delta = 0.04, 0.4
    P = jarrow_turnbull_zero(T, r, curve, delta)
    spread = -np.log(P) / T - r
    fig, ax = plt.subplots(figsize=(7, 4))
    ax.plot(T, 1e4 * spread, label="zero-coupon spread")
    avg_hazard = np.asarray(curve.cumulative_hazard(T)) / T
    ax.plot(T, 1e4 * np.asarray(spread_from_hazard(avg_hazard, delta)), "--", label="credit triangle, average hazard x (1 - R)")
    ax.set_xlabel("maturity")
    ax.set_ylabel("bp")
    ax.legend()

    plt.show()



.. image-sg:: /api/gallery/credit/reduced_form/images/sphx_glr_plot_01_jarrow_turnbull_003.png
   :alt: plot 01 jarrow turnbull
   :srcset: /api/gallery/credit/reduced_form/images/sphx_glr_plot_01_jarrow_turnbull_003.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_credit_reduced_form_plot_01_jarrow_turnbull.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_01_jarrow_turnbull.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_jarrow_turnbull.ipynb <plot_01_jarrow_turnbull.ipynb>`

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

      :download:`Download Python source code: plot_01_jarrow_turnbull.py <plot_01_jarrow_turnbull.py>`

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

      :download:`Download zipped: plot_01_jarrow_turnbull.zip <plot_01_jarrow_turnbull.zip>`


.. only:: html

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

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