.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/credit/cds/plot_01_cds_par_spread.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_cds_plot_01_cds_par_spread.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_cds_plot_01_cds_par_spread.py:


O'Kane and Turnbull: the CDS par spread
=======================================

A credit default swap exchanges a running premium for a payment of
:math:`1 - R` at default. O'Kane and Turnbull (2003) wrote both legs as
integrals over the default density and evaluated them on a grid with
defaults at mid-step, including the premium accrued between the last
payment date and default. The par spread equates the legs; for a flat
hazard and a continuously paid premium it is exactly
:math:`\lambda(1 - R)`.

.. GENERATED FROM PYTHON SOURCE LINES 15-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.credit import HazardCurve, cds_legs, cds_par_spread

    res = cds_legs(HazardCurve.flat(0.02), 5.0, 0.05, 0.4, frequency=1, steps_per_year=1)
    print("Hull's annual setting (hazard 2%, R = 40%, r = 5%):")
    print(f"  premium annuity {res.rpv01 - res.accrued_on_default:.4f}, accrual {res.accrued_on_default:.4f}, protection {res.protection_leg:.4f}")
    print(f"  par spread {1e4 * res.par_spread:.1f} bp  vs  credit triangle {1e4 * 0.02 * 0.6:.0f} bp")





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

 .. code-block:: none

    Hull's annual setting (hazard 2%, R = 40%, r = 5%):
      premium annuity 4.0728, accrual 0.0422, protection 0.0506
      par spread 123.0 bp  vs  credit triangle 120 bp




.. GENERATED FROM PYTHON SOURCE LINES 26-30

Payment frequency and the credit triangle
-----------------------------------------

More frequent premiums and accrual on default converge to the triangle.

.. GENERATED FROM PYTHON SOURCE LINES 30-41

.. code-block:: Python


    freqs = [1, 2, 4, 12, 52]
    fig, ax = plt.subplots(figsize=(7, 4))
    for h in (0.01, 0.03, 0.06):
        s = [cds_par_spread(HazardCurve.flat(h), 5.0, 0.05, 0.4, frequency=f, steps_per_year=max(f, 12)) for f in freqs]
        ax.semilogx(freqs, 1e4 * np.array(s) / (1e4 * h * 0.6), "o-", label=f"hazard {h:.0%}")
    ax.axhline(1.0, color="k", ls=":")
    ax.set_xlabel("premium payments per year")
    ax.set_ylabel("par spread / lambda (1 - R)")
    ax.legend()




.. image-sg:: /api/gallery/credit/cds/images/sphx_glr_plot_01_cds_par_spread_001.png
   :alt: plot 01 cds par spread
   :srcset: /api/gallery/credit/cds/images/sphx_glr_plot_01_cds_par_spread_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x11612ba10>



.. GENERATED FROM PYTHON SOURCE LINES 42-44

The term structure of par spreads
---------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 44-55

.. code-block:: Python


    curve = HazardCurve([1, 3, 5, 10], [0.005, 0.012, 0.02, 0.03])
    mats = np.linspace(0.5, 10, 20)
    fig, ax = plt.subplots(figsize=(7, 4))
    ax.plot(mats, [1e4 * cds_par_spread(curve, T, 0.03) for T in mats], "o-", label="par spread")
    ax.step([0, 1, 3, 5, 10], 1e4 * 0.6 * np.r_[0.005, curve.hazards], where="pre", label="hazard x (1 - R)")
    ax.set_xlabel("CDS maturity")
    ax.set_ylabel("bp")
    ax.legend()

    plt.show()



.. image-sg:: /api/gallery/credit/cds/images/sphx_glr_plot_01_cds_par_spread_002.png
   :alt: plot 01 cds par spread
   :srcset: /api/gallery/credit/cds/images/sphx_glr_plot_01_cds_par_spread_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.648 seconds)


.. _sphx_glr_download_api_gallery_credit_cds_plot_01_cds_par_spread.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/cds/plot_01_cds_par_spread.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_cds_par_spread.ipynb <plot_01_cds_par_spread.ipynb>`

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

      :download:`Download Python source code: plot_01_cds_par_spread.py <plot_01_cds_par_spread.py>`

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

      :download:`Download zipped: plot_01_cds_par_spread.zip <plot_01_cds_par_spread.zip>`


.. only:: html

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

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