.. 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_01_parkinson_range.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_01_parkinson_range.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_01_parkinson_range.py:


Parkinson's range volatility
============================

A daily close-to-close return uses two prices out of the thousands traded
that day. Michael Parkinson (1980) used two more, the high and the low:
for Brownian motion with volatility :math:`\sigma`,

.. math:: E\bigl[(\ln H_t / L_t)^2\bigr] = 4 \ln 2 \,\sigma^2\,\Delta t,

so :math:`(\ln H/L)^2 / (4\ln 2)` is an unbiased variance estimate whose
variance is about 5.2 times smaller than the squared return's. One month
of ranges is as informative as about five months of closes.

.. GENERATED FROM PYTHON SOURCE LINES 17-24

.. 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 close_to_close_variance, parkinson_variance
    from financialkit.time_series.utils import simulate_ohlc








.. GENERATED FROM PYTHON SOURCE LINES 25-29

One year of daily bars
----------------------
Prices follow a geometric Brownian motion with 25% volatility, sampled
every minute of a 390-minute session.

.. GENERATED FROM PYTHON SOURCE LINES 29-44

.. code-block:: Python


    sigma = 0.25
    ohlc = simulate_ohlc(1000, sigma, steps_per_day=390, rng=1980)
    pk = parkinson_variance(ohlc.high, ohlc.low)
    cc = close_to_close_variance(np.r_[ohlc.open[0], ohlc.close])

    fig, ax = plt.subplots(figsize=(8, 3))
    d = np.arange(60)
    ax.vlines(d, ohlc.low[:60], ohlc.high[:60], color="0.4")
    ax.plot(d, ohlc.close[:60], "_", color="C0", ms=6, label="close")
    ax.set_xlabel("day")
    ax.set_ylabel("price")
    ax.set_title("High-low bars carry more information than the closes")
    ax.legend()




.. image-sg:: /api/gallery/time_series/volatility/images/sphx_glr_plot_01_parkinson_range_001.png
   :alt: High-low bars carry more information than the closes
   :srcset: /api/gallery/time_series/volatility/images/sphx_glr_plot_01_parkinson_range_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x1167fb620>



.. GENERATED FROM PYTHON SOURCE LINES 45-49

Rolling 21-day volatility estimates
-----------------------------------
Both estimators are centred on the true 25%, but the range-based one is
far less noisy.

.. GENERATED FROM PYTHON SOURCE LINES 49-65

.. code-block:: Python


    window = 21
    kernel = np.ones(window) / window
    ann = TRADING_DAYS_PER_YEAR
    fig, ax = plt.subplots(figsize=(8, 3.5))
    ax.plot(100 * np.sqrt(ann * np.convolve(cc, kernel, "valid")), lw=1, label="close-to-close")
    ax.plot(100 * np.sqrt(ann * np.convolve(pk, kernel, "valid")), lw=1, label="Parkinson")
    ax.axhline(100 * sigma, color="k", ls="--", lw=1)
    ax.set_xlabel("day")
    ax.set_ylabel("annualized volatility (%)")
    ax.legend()

    print(f"mean annualized vol: close-to-close {100 * np.sqrt(ann * cc.mean()):.1f}%, Parkinson {100 * np.sqrt(ann * pk.mean()):.1f}%")
    print(f"relative efficiency Var(cc) / Var(Parkinson) = {cc.var() / pk.var():.1f} (theory 5.2)")
    plt.show()




.. image-sg:: /api/gallery/time_series/volatility/images/sphx_glr_plot_01_parkinson_range_002.png
   :alt: plot 01 parkinson range
   :srcset: /api/gallery/time_series/volatility/images/sphx_glr_plot_01_parkinson_range_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    mean annualized vol: close-to-close 24.8%, Parkinson 24.1%
    relative efficiency Var(cc) / Var(Parkinson) = 5.0 (theory 5.2)




.. GENERATED FROM PYTHON SOURCE LINES 66-69

With one-minute sampling the observed high and low slightly understate the
true extremes, which biases the Parkinson estimate down by a few percent;
finer sampling removes the bias.


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

   **Total running time of the script:** (0 minutes 0.059 seconds)


.. _sphx_glr_download_api_gallery_time_series_volatility_plot_01_parkinson_range.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_01_parkinson_range.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_parkinson_range.ipynb <plot_01_parkinson_range.ipynb>`

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

      :download:`Download Python source code: plot_01_parkinson_range.py <plot_01_parkinson_range.py>`

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

      :download:`Download zipped: plot_01_parkinson_range.zip <plot_01_parkinson_range.zip>`


.. only:: html

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

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