.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/time_series/volatility/plot_03_bollerslev_garch.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_time_series_volatility_plot_03_bollerslev_garch.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_time_series_volatility_plot_03_bollerslev_garch.py:


Bollerslev's GARCH
==================

Pure ARCH models needed many lags to capture how slowly volatility decays.
Tim Bollerslev (1986), Engle's student, added the lagged variance itself,

.. math:: \sigma_t^2 = \omega + \alpha\,\varepsilon_{t-1}^2 + \beta\,\sigma_{t-1}^2,

an ARCH(:math:`\infty`) with geometrically declining weights in three
parameters. Fitted to daily returns almost anywhere, GARCH(1,1) gives
:math:`\alpha \approx 0.05\text{-}0.10` and :math:`\alpha + \beta` just
below one: shocks to volatility are small but long-lived. Variance
forecasts revert to :math:`\bar\sigma^2 = \omega/(1-\alpha-\beta)` at rate
:math:`(\alpha+\beta)^h`.

.. GENERATED FROM PYTHON SOURCE LINES 19-26

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.constants import TRADING_DAYS_PER_YEAR
    from financialkit.time_series import GARCH
    from financialkit.time_series.visualizers import plot_conditional_volatility








.. GENERATED FROM PYTHON SOURCE LINES 27-31

Fitting GARCH(1,1) by maximum likelihood
----------------------------------------
Eight years of daily returns from a GARCH(1,1) with typical equity-index
parameters (16% long-run volatility, persistence 0.98).

.. GENERATED FROM PYTHON SOURCE LINES 31-44

.. code-block:: Python


    ann = TRADING_DAYS_PER_YEAR
    true = GARCH(omega=0.16**2 / ann * 0.02, alpha=0.08, beta=0.90, mu=3e-4)
    r = true.simulate(2000, rng=1986).returns
    fit = GARCH.fit(r)
    for k, v in fit.params.items():
        print(f"{k:5s} true {true.params[k]:.3g}  fitted {v:.3g} +/- {fit.std_errors[k]:.2g}")
    print(f"persistence {fit.model.persistence:.3f}, half-life {fit.model.half_life:.0f} days")

    fig, ax = plt.subplots(figsize=(8, 3.5))
    plot_conditional_volatility(r, fit.variance, ax=ax)
    ax.set_title("Fitted conditional volatility")




.. image-sg:: /api/gallery/time_series/volatility/images/sphx_glr_plot_03_bollerslev_garch_001.png
   :alt: Fitted conditional volatility
   :srcset: /api/gallery/time_series/volatility/images/sphx_glr_plot_03_bollerslev_garch_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    mu    true 0.0003  fitted 0.000604 +/- 0.00021
    omega true 2.03e-06  fitted 2.44e-06 +/- 8.6e-07
    alpha true 0.08  fitted 0.0691 +/- 0.012
    beta  true 0.9  fitted 0.908 +/- 0.017
    persistence 0.978, half-life 31 days

    Text(0.5, 1.0, 'Fitted conditional volatility')



.. GENERATED FROM PYTHON SOURCE LINES 45-49

The term structure of volatility forecasts
------------------------------------------
From a calm day and from a turbulent day, forecasts converge to the
long-run volatility at the rate set by the persistence.

.. GENERATED FROM PYTHON SOURCE LINES 49-62

.. code-block:: Python


    h = 250
    calm = int(np.argmin(fit.variance[500:])) + 500
    wild = int(np.argmax(fit.variance[500:])) + 500
    fig, ax = plt.subplots()
    for name, t in (("after a calm day", calm), ("after a turbulent day", wild)):
        f = fit.model.forecast(r[: t + 1], h)
        ax.plot(np.arange(1, h + 1), 100 * np.sqrt(ann * f), label=name)
    ax.axhline(100 * np.sqrt(ann * fit.model.unconditional_variance), color="k", ls="--", lw=1, label="long-run")
    ax.set_xlabel("horizon (days)")
    ax.set_ylabel("forecast volatility (% a year)")
    ax.legend()
    plt.show()



.. image-sg:: /api/gallery/time_series/volatility/images/sphx_glr_plot_03_bollerslev_garch_002.png
   :alt: plot 03 bollerslev garch
   :srcset: /api/gallery/time_series/volatility/images/sphx_glr_plot_03_bollerslev_garch_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.071 seconds)


.. _sphx_glr_download_api_gallery_time_series_volatility_plot_03_bollerslev_garch.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/time_series/volatility/plot_03_bollerslev_garch.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_bollerslev_garch.ipynb <plot_03_bollerslev_garch.ipynb>`

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

      :download:`Download Python source code: plot_03_bollerslev_garch.py <plot_03_bollerslev_garch.py>`

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

      :download:`Download zipped: plot_03_bollerslev_garch.zip <plot_03_bollerslev_garch.zip>`


.. only:: html

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

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