.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/interest_rate_models/hjm/plot_01_hjm_drift_condition.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_hjm_plot_01_hjm_drift_condition.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_hjm_plot_01_hjm_drift_condition.py:


Heath-Jarrow-Morton: the drift condition
========================================

David Heath, Robert Jarrow and Andrew Morton (1992) modelled the whole
forward curve at once,

.. math:: df(t, T) = \alpha(t, T)\,dt + \sigma(t, T)\,dW(t),

starting from today's market curve. Their central result is that the
drift is not free: discounted bond prices are martingales only if

.. math:: \alpha(t, T) = \sigma(t, T)\int_t^T \sigma(t, u)\,du.

The volatility is the whole model; a constant :math:`\sigma` is Ho-Lee
and :math:`\sigma e^{-a(T-t)}` is Hull-White. Leaving the drift out
creates an arbitrage, visible as a mispricing of the very bonds the
simulation started from.

.. GENERATED FROM PYTHON SOURCE LINES 22-32

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.fixed_income import NelsonSiegelCurve
    from financialkit.interest_rate_models import hjm_drift, simulate_hjm
    from financialkit.interest_rate_models.visualizers import plot_forward_curves

    curve = NelsonSiegelCurve(0.05, -0.02, 0.01, 2.0)
    sigma, a = 0.015, 0.1








.. GENERATED FROM PYTHON SOURCE LINES 33-35

The drift each volatility requires
----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 35-49

.. code-block:: Python


    T = np.linspace(0, 20, 81)
    fig, axes = plt.subplots(1, 3, figsize=(15, 4))
    vols = {
        "constant (Ho-Lee)": lambda t, T: sigma,
        "exponential (Hull-White)": lambda t, T: sigma * np.exp(-a * (T - t)),
        "humped": lambda t, T: sigma * (1 + 2 * (T - t)) * np.exp(-0.5 * (T - t)),
    }
    for name, vol in vols.items():
        axes[0].plot(T, 1e4 * hjm_drift(vol, 0.0, T), label=name)
    axes[0].set_xlabel("maturity T (years)")
    axes[0].set_ylabel(r"$\alpha(0, T)$ (bp per year)")
    axes[0].legend()




.. image-sg:: /api/gallery/interest_rate_models/hjm/images/sphx_glr_plot_01_hjm_drift_condition_001.png
   :alt: plot 01 hjm drift condition
   :srcset: /api/gallery/interest_rate_models/hjm/images/sphx_glr_plot_01_hjm_drift_condition_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x11a4fe120>



.. GENERATED FROM PYTHON SOURCE LINES 50-52

Simulated forward curves
------------------------

.. GENERATED FROM PYTHON SOURCE LINES 52-57

.. code-block:: Python


    res = simulate_hjm(curve, sigma, a, horizon=10.0, n_steps=50, n_paths=3000, rng=0)
    plot_forward_curves(res, path=0, every=5, ax=axes[1])
    axes[1].set_title("one path of f(t, T), t = 0, 1, ..., 9")





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

 .. code-block:: none


    Text(0.5, 1.0, 'one path of f(t, T), t = 0, 1, ..., 9')



.. GENERATED FROM PYTHON SOURCE LINES 58-62

With and without the drift condition
------------------------------------

Re-price today's bonds by averaging the simulated discount factors.

.. GENERATED FROM PYTHON SOURCE LINES 62-77

.. code-block:: Python


    wrong = simulate_hjm(curve, sigma, a, horizon=10.0, n_steps=50, n_paths=3000, drift_condition=False, rng=0)
    exact = curve.discount(res.times)
    for r, name in ((res, "HJM drift"), (wrong, "zero drift")):
        p, se = r.bond_prices()
        axes[2].fill_between(res.times, 1e4 * (p - exact - 2 * se), 1e4 * (p - exact + 2 * se), alpha=0.3)
        axes[2].plot(res.times, 1e4 * (p - exact), label=name)
        print(f"{name}: 10y bond off by {1e4 * (p[-1] - exact[-1]):+.1f} bp of face ({(p[-1] - exact[-1]) / se[-1]:+.1f} standard errors)")
    axes[2].axhline(0, color="k", lw=0.5)
    axes[2].set_xlabel("maturity T (years)")
    axes[2].set_ylabel("Monte Carlo - market price (bp of face)")
    axes[2].legend()
    fig.tight_layout()

    plt.show()




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

 .. code-block:: none

    HJM drift: 10y bond off by +17.1 bp of face (+0.8 standard errors)
    zero drift: 10y bond off by +138.5 bp of face (+6.2 standard errors)





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

   **Total running time of the script:** (0 minutes 1.632 seconds)


.. _sphx_glr_download_api_gallery_interest_rate_models_hjm_plot_01_hjm_drift_condition.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/hjm/plot_01_hjm_drift_condition.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_hjm_drift_condition.ipynb <plot_01_hjm_drift_condition.ipynb>`

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

      :download:`Download Python source code: plot_01_hjm_drift_condition.py <plot_01_hjm_drift_condition.py>`

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

      :download:`Download zipped: plot_01_hjm_drift_condition.zip <plot_01_hjm_drift_condition.zip>`


.. only:: html

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

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