.. 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_07_g2pp.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_07_g2pp.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_07_g2pp.py:


Brigo-Mercurio G2++: two Gaussian factors
=========================================

In a one-factor model a single shock moves every yield, so the 1-year
and 30-year rates are perfectly correlated, which the data flatly
reject. Damiano Brigo and Fabio Mercurio's G2++ (2001) adds a second
Gaussian factor,

.. math:: r = x + y + \varphi(t), \quad dx = -a x\,dt + \sigma\,dW_1, \quad
          dy = -b y\,dt + \eta\,dW_2, \quad dW_1 dW_2 = \rho\,dt,

with :math:`\varphi` fitted to today's curve. Bonds and bond options
stay in closed form, and a negative :math:`\rho` between a fast and a
slow factor decorrelates the ends of the curve.

.. GENERATED FROM PYTHON SOURCE LINES 19-28

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.fixed_income import NelsonSiegelCurve
    from financialkit.interest_rate_models import G2PlusPlusModel, monte_carlo_bond_prices

    curve = NelsonSiegelCurve(0.045, -0.02, 0.01, 2.0)
    model = G2PlusPlusModel(curve, a=0.8, sigma=0.012, b=0.05, eta=0.008, rho=-0.75)








.. GENERATED FROM PYTHON SOURCE LINES 29-31

Correlation between yields
--------------------------

.. GENERATED FROM PYTHON SOURCE LINES 31-41

.. code-block:: Python


    taus = np.array([0.5, 1, 2, 5, 10, 20, 30])
    fig, axes = plt.subplots(1, 3, figsize=(14, 4))
    im = axes[0].imshow(model.yield_correlation(taus[:, None], taus[None, :]), vmin=0, vmax=1, cmap="viridis")
    axes[0].set_xticks(range(taus.size), [f"{t:g}" for t in taus])
    axes[0].set_yticks(range(taus.size), [f"{t:g}" for t in taus])
    axes[0].set_title("instantaneous yield correlation")
    fig.colorbar(im, ax=axes[0])
    print(f"corr(1y, 30y): G2++ {model.yield_correlation(1.0, 30.0):.2f}; any one-factor model 1.00")




.. image-sg:: /api/gallery/interest_rate_models/short_rate/images/sphx_glr_plot_07_g2pp_001.png
   :alt: instantaneous yield correlation
   :srcset: /api/gallery/interest_rate_models/short_rate/images/sphx_glr_plot_07_g2pp_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    corr(1y, 30y): G2++ 0.37; any one-factor model 1.00




.. GENERATED FROM PYTHON SOURCE LINES 42-44

The same correlation from simulated yield changes
-------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 44-60

.. code-block:: Python


    _, xy = model.simulate_factors(1.0, 50, 4000, rng=0)
    dx = xy[:, 1, :] - xy[:, 0, :]  # factor moves over the first step


    def dyield(tau):
        """Change in the tau-year yield, -d ln P / tau = (B_a dx + B_b dy) / tau."""
        Ba, Bb = -np.expm1(-model.a * tau) / model.a, -np.expm1(-model.b * tau) / model.b
        return (Ba * dx[:, 0] + Bb * dx[:, 1]) / tau


    axes[1].scatter(1e4 * dyield(1.0), 1e4 * dyield(30.0), s=2, alpha=0.3)
    axes[1].set_xlabel("1y yield change (bp)")
    axes[1].set_ylabel("30y yield change (bp)")
    axes[1].set_title(f"sample corr {np.corrcoef(dyield(1.0), dyield(30.0))[0, 1]:.2f}")





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

 .. code-block:: none


    Text(0.5, 1.0, 'sample corr 0.37')



.. GENERATED FROM PYTHON SOURCE LINES 61-63

The fitted model reprices the curve
-----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 63-74

.. code-block:: Python


    res = monte_carlo_bond_prices(model, np.arange(1.0, 11.0), n_steps=100, n_paths=8000, rng=1)
    axes[2].plot(res.maturities, 100 * -np.log(res.exact) / res.maturities, "k-", label="initial curve")
    yield_se = 100 * res.std_errors / res.prices / res.maturities
    axes[2].errorbar(res.maturities, 100 * -np.log(res.prices) / res.maturities, yerr=1.96 * yield_se, fmt="o", ms=3, label="G2++ Monte Carlo")
    axes[2].set_xlabel("maturity (years)")
    axes[2].set_ylabel("zero yield (%)")
    axes[2].legend()
    fig.tight_layout()

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.601 seconds)


.. _sphx_glr_download_api_gallery_interest_rate_models_short_rate_plot_07_g2pp.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_07_g2pp.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_07_g2pp.ipynb <plot_07_g2pp.ipynb>`

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

      :download:`Download Python source code: plot_07_g2pp.py <plot_07_g2pp.py>`

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

      :download:`Download zipped: plot_07_g2pp.zip <plot_07_g2pp.zip>`


.. only:: html

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

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