.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/interest_rate_models/short_rate/plot_04_vasicek.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_interest_rate_models_short_rate_plot_04_vasicek.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_interest_rate_models_short_rate_plot_04_vasicek.py:


Vasicek's mean-reverting short rate
===================================

Oldrich Vasicek (1977) derived the term structure from a short rate
pulled back towards a long-run mean,

.. math:: dr = \kappa(\theta - r)\,dt + \sigma\,dW,

an Ornstein-Uhlenbeck process, together with the market price of risk
that makes bond prices consistent. Bonds are exponential-affine in the
rate, :math:`P = A(\tau) e^{-B(\tau) r}`, and the curve is upward
sloping, humped or inverted depending on where :math:`r_0` stands
against the long yield :math:`\theta - \sigma^2/(2\kappa^2)`.

.. GENERATED FROM PYTHON SOURCE LINES 18-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.interest_rate_models import VasicekModel, monte_carlo_bond_prices
    from financialkit.interest_rate_models.visualizers import plot_model_yields, plot_short_rate_paths








.. GENERATED FROM PYTHON SOURCE LINES 25-27

Three curve shapes from one model
---------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-35

.. code-block:: Python


    fig, axes = plt.subplots(1, 3, figsize=(14, 4))
    for r0, shape in ((0.01, "normal"), (0.0555, "humped"), (0.09, "inverted")):
        m = VasicekModel(r0, 0.25, 0.06, 0.02)
        plot_model_yields(m, t_max=30, ax=axes[0], label=f"r0 = {100 * r0:.2f}% ({shape})")
    axes[0].axhline(100 * m.long_yield, color="k", ls=":", lw=1)
    axes[0].set_title("Vasicek yield curves")




.. image-sg:: /api/gallery/interest_rate_models/short_rate/images/sphx_glr_plot_04_vasicek_001.png
   :alt: Vasicek yield curves
   :srcset: /api/gallery/interest_rate_models/short_rate/images/sphx_glr_plot_04_vasicek_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'Vasicek yield curves')



.. GENERATED FROM PYTHON SOURCE LINES 36-38

Exact simulation and mean reversion
-----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 38-45

.. code-block:: Python


    model = VasicekModel(0.01, 0.25, 0.06, 0.02)
    times, r = model.simulate(20.0, 200, 5000, rng=0)
    plot_short_rate_paths(times, r, n_show=12, ax=axes[1], label="simulated")
    axes[1].plot(times, 100 * (0.06 + (0.01 - 0.06) * np.exp(-0.25 * times)), "w--", lw=1, label="E[r(t)]")
    axes[1].legend()





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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x11b508440>



.. GENERATED FROM PYTHON SOURCE LINES 46-48

Closed form against Monte Carlo
-------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 48-58

.. code-block:: Python


    res = monte_carlo_bond_prices(model, np.arange(1.0, 21.0), n_steps=200, n_paths=8000, rng=1)
    axes[2].errorbar(res.maturities, res.z_scores, yerr=1, fmt="o", ms=3)
    axes[2].axhspan(-2, 2, color="0.9")
    axes[2].set_xlabel("maturity (years)")
    axes[2].set_ylabel("(MC - exact) / SE")
    axes[2].set_title(f"max |z| = {np.abs(res.z_scores).max():.2f}")
    fig.tight_layout()

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.116 seconds)


.. _sphx_glr_download_api_gallery_interest_rate_models_short_rate_plot_04_vasicek.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/interest_rate_models/short_rate/plot_04_vasicek.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_vasicek.ipynb <plot_04_vasicek.ipynb>`

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

      :download:`Download Python source code: plot_04_vasicek.py <plot_04_vasicek.py>`

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

      :download:`Download zipped: plot_04_vasicek.zip <plot_04_vasicek.zip>`


.. only:: html

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

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