.. 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_05_cir.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_05_cir.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_05_cir.py:


Cox-Ingersoll-Ross: the square-root short rate
==============================================

John Cox, Jonathan Ingersoll and Stephen Ross (1985) derived the short
rate from a general-equilibrium economy and found

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

Volatility shrinks as the rate falls, so rates stay positive (under
Feller's condition), and bonds are still exponential-affine, with
:math:`B(\tau) = 2(e^{\gamma\tau} - 1)/[(\gamma + \kappa)(e^{\gamma\tau} - 1) + 2\gamma]`,
:math:`\gamma = \sqrt{\kappa^2 + 2\sigma^2}`. The transition density is a
scaled noncentral chi-square, which allows exact simulation.

.. GENERATED FROM PYTHON SOURCE LINES 18-27

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.interest_rate_models import CIRModel, VasicekModel, monte_carlo_bond_prices, path_discount_factors

    cir = CIRModel(r0=0.03, kappa=0.5, theta=0.05, sigma=0.1)
    # A Vasicek model with the same mean reversion and the same local volatility at theta
    vas = VasicekModel(0.03, 0.5, 0.05, 0.1 * np.sqrt(0.05))








.. GENERATED FROM PYTHON SOURCE LINES 28-30

Level-dependent volatility keeps rates positive
-----------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 30-42

.. code-block:: Python


    fig, axes = plt.subplots(1, 3, figsize=(14, 4))
    _, r_cir = cir.simulate(10.0, 100, 20_000, rng=0)
    _, r_vas = vas.simulate(10.0, 100, 20_000, rng=0)
    bins = np.linspace(-4, 16, 81)
    axes[0].hist(100 * r_vas[:, -1], bins=bins, density=True, alpha=0.5, label="Vasicek (Gaussian)")
    axes[0].hist(100 * r_cir[:, -1], bins=bins, density=True, alpha=0.5, label="CIR (noncentral chi-square)")
    axes[0].axvline(0, color="k", lw=0.5)
    axes[0].set_xlabel("r(10) (%)")
    axes[0].legend()
    print(f"P(r(10) < 0): Vasicek {np.mean(r_vas[:, -1] < 0):.2%}, CIR {np.mean(r_cir[:, -1] < 0):.2%}")




.. image-sg:: /api/gallery/interest_rate_models/short_rate/images/sphx_glr_plot_05_cir_001.png
   :alt: plot 05 cir
   :srcset: /api/gallery/interest_rate_models/short_rate/images/sphx_glr_plot_05_cir_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    P(r(10) < 0): Vasicek 1.23%, CIR 0.00%




.. GENERATED FROM PYTHON SOURCE LINES 43-45

Yield curves
------------

.. GENERATED FROM PYTHON SOURCE LINES 45-54

.. code-block:: Python


    T = np.linspace(0.1, 30, 200)
    for sigma in (0.05, 0.1, 0.2):
        axes[1].plot(T, 100 * CIRModel(0.03, 0.5, 0.05, sigma).yields(T), label=f"sigma = {sigma}")
    axes[1].set_xlabel("maturity (years)")
    axes[1].set_ylabel("zero yield (%)")
    axes[1].legend()
    axes[1].set_title("CIR yield curves")





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

 .. code-block:: none


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



.. GENERATED FROM PYTHON SOURCE LINES 55-57

Exact and Euler simulation against the closed form
--------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 57-70

.. code-block:: Python


    res = monte_carlo_bond_prices(cir, np.arange(1.0, 11.0), n_steps=100, n_paths=10_000, rng=1)
    axes[2].plot(res.maturities, res.z_scores, "o-", label="exact sampling")
    times, r = cir.simulate(10.0, 100, 10_000, rng=1, method="euler")
    D = path_discount_factors(times, r)[:, 10::10]
    axes[2].plot(np.arange(1.0, 11.0), (D.mean(0) - cir.zero_coupon_bond(np.arange(1.0, 11.0))) / (D.std(0) / 100), "s-", label="full-truncation Euler")
    axes[2].axhspan(-2, 2, color="0.9")
    axes[2].set_xlabel("maturity (years)")
    axes[2].set_ylabel("(MC - exact) / SE")
    axes[2].legend()
    fig.tight_layout()

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.512 seconds)


.. _sphx_glr_download_api_gallery_interest_rate_models_short_rate_plot_05_cir.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_05_cir.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_05_cir.ipynb <plot_05_cir.ipynb>`

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

      :download:`Download Python source code: plot_05_cir.py <plot_05_cir.py>`

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

      :download:`Download zipped: plot_05_cir.zip <plot_05_cir.zip>`


.. only:: html

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

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