.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/credit/structural/plot_01_merton_structural.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_01_merton_structural.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_01_merton_structural.py:


Merton's structural model
=========================

Merton (1974) treated a firm's equity as a European call on its assets
:math:`V`, struck at the face value :math:`D` of its zero-coupon debt.
Debt is then a riskless bond minus a put, the risk-neutral default
probability is :math:`N(-d_2)`, and the credit spread follows from
leverage and asset volatility alone. Hull's Example 24.3 backs out the
unobserved asset value and volatility from equity (3) and its
volatility (80%).

.. GENERATED FROM PYTHON SOURCE LINES 15-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.credit import merton_calibrate, merton_model

    m = merton_calibrate(E=3.0, sigma_E=0.80, D=10.0, T=1.0, r=0.05)
    print(f"assets V = {m.asset_value:.2f}, sigma_V = {m.asset_volatility:.4f}")
    print(f"debt = {m.debt:.2f}, PD = {m.default_probability:.1%}, spread = {1e4 * m.credit_spread:.0f} bp, recovery = {m.recovery_rate:.1%}")





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

 .. code-block:: none

    assets V = 12.40, sigma_V = 0.2123
    debt = 9.40, PD = 12.7%, spread = 124 bp, recovery = 90.3%




.. GENERATED FROM PYTHON SOURCE LINES 25-27

Payoffs at maturity
-------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-36

.. code-block:: Python


    VT = np.linspace(0, 25, 300)
    fig, ax = plt.subplots(figsize=(6, 4))
    ax.plot(VT, np.maximum(VT - 10, 0), label="equity: max(V - D, 0)")
    ax.plot(VT, np.minimum(VT, 10), label="debt: min(V, D)")
    ax.axvline(10, color="0.5", ls=":")
    ax.set_xlabel("asset value at maturity")
    ax.legend()




.. image-sg:: /api/gallery/credit/structural/images/sphx_glr_plot_01_merton_structural_001.png
   :alt: plot 01 merton structural
   :srcset: /api/gallery/credit/structural/images/sphx_glr_plot_01_merton_structural_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x116491a90>



.. GENERATED FROM PYTHON SOURCE LINES 37-43

The term structure of credit spreads
------------------------------------

Merton's spreads start at zero for very short maturities (assets cannot
jump below the debt), rise, and for highly levered firms fall again
because surviving firms drift away from default.

.. GENERATED FROM PYTHON SOURCE LINES 43-54

.. code-block:: Python


    T = np.linspace(0.05, 10, 120)
    fig, ax = plt.subplots(figsize=(7, 4))
    for lev in (0.5, 0.7, 0.9, 1.1):
        s = [merton_model(100.0, 100 * lev * np.exp(0.05 * t), t, 0.05, 0.25).credit_spread for t in T]
        ax.plot(T, 1e4 * np.array(s), label=f"quasi-leverage {lev:.1f}")
    ax.set_xlabel("maturity (years)")
    ax.set_ylabel("credit spread (bp)")
    ax.legend()

    plt.show()



.. image-sg:: /api/gallery/credit/structural/images/sphx_glr_plot_01_merton_structural_002.png
   :alt: plot 01 merton structural
   :srcset: /api/gallery/credit/structural/images/sphx_glr_plot_01_merton_structural_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.102 seconds)


.. _sphx_glr_download_api_gallery_credit_structural_plot_01_merton_structural.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_01_merton_structural.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_merton_structural.ipynb <plot_01_merton_structural.ipynb>`

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

      :download:`Download Python source code: plot_01_merton_structural.py <plot_01_merton_structural.py>`

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

      :download:`Download zipped: plot_01_merton_structural.zip <plot_01_merton_structural.zip>`


.. only:: html

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

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