.. _sphx_glr_api_gallery_time_series_volatility:

Volatility
----------

Range-based and realized volatility, and the ARCH family of conditional-variance models.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="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 \sigma,">

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  .. image:: /api/gallery/time_series/volatility/images/thumb/sphx_glr_plot_01_parkinson_range_thumb.png
    :alt:

  :doc:`/api/gallery/time_series/volatility/plot_01_parkinson_range`

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      <div class="sphx-glr-thumbnail-title">Parkinson's range volatility</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Forecast errors for UK inflation were sometimes small and sometimes large, in clusters. Robert Engle (1982) let the conditional variance depend on the last squared shock,">

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  .. image:: /api/gallery/time_series/volatility/images/thumb/sphx_glr_plot_02_engle_arch_thumb.png
    :alt:

  :doc:`/api/gallery/time_series/volatility/plot_02_engle_arch`

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      <div class="sphx-glr-thumbnail-title">Engle's ARCH</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Pure ARCH models needed many lags to capture how slowly volatility decays. Tim Bollerslev (1986), Engle&#x27;s student, added the lagged variance itself,">

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  .. image:: /api/gallery/time_series/volatility/images/thumb/sphx_glr_plot_03_bollerslev_garch_thumb.png
    :alt:

  :doc:`/api/gallery/time_series/volatility/plot_03_bollerslev_garch`

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      <div class="sphx-glr-thumbnail-title">Bollerslev's GARCH</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="GARCH reacts to the size of yesterday&#x27;s shock but not its sign, and its coefficients must be kept positive. Daniel Nelson (1991) modelled the logarithm of the variance instead,">

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  .. image:: /api/gallery/time_series/volatility/images/thumb/sphx_glr_plot_04_nelson_egarch_thumb.png
    :alt:

  :doc:`/api/gallery/time_series/volatility/plot_04_nelson_egarch`

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      <div class="sphx-glr-thumbnail-title">Nelson's EGARCH</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Lawrence Glosten, Ravi Jagannathan and David Runkle (1993) asked whether expected stock returns rise with volatility, and found that the answer depended on modelling volatility&#x27;s asymmetric response to news. Their GARCH variant adds one term for negative shocks,">

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  .. image:: /api/gallery/time_series/volatility/images/thumb/sphx_glr_plot_05_gjr_garch_thumb.png
    :alt:

  :doc:`/api/gallery/time_series/volatility/plot_05_gjr_garch`

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      <div class="sphx-glr-thumbnail-title">Glosten-Jagannathan-Runkle</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="In 1994 J.P. Morgan published RiskMetrics, a free methodology and data set for Value-at-Risk. Its volatility model was deliberately simple: an exponentially weighted moving average">

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  .. image:: /api/gallery/time_series/volatility/images/thumb/sphx_glr_plot_06_riskmetrics_ewma_thumb.png
    :alt:

  :doc:`/api/gallery/time_series/volatility/plot_06_riskmetrics_ewma`

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      <div class="sphx-glr-thumbnail-title">RiskMetrics EWMA</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="GARCH models fit beautifully in-sample, yet regressions of squared daily returns on their variance forecasts gave R^2 of a few percent, and sceptics concluded the models were useless for forecasting. Torben Andersen and Tim Bollerslev (1998) showed that the fault lay with the yardstick. A squared return \sigma_t^2 z_t^2 is an unbiased but very noisy measure of the day&#x27;s variance, and even a perfect forecast explains only a small part of it. Summing squared five-minute returns (realized variance) measures the variance almost exactly, and against it the same forecasts explain many times more.">

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  .. image:: /api/gallery/time_series/volatility/images/thumb/sphx_glr_plot_07_realized_volatility_thumb.png
    :alt:

  :doc:`/api/gallery/time_series/volatility/plot_07_realized_volatility`

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      <div class="sphx-glr-thumbnail-title">Andersen-Bollerslev realized volatility</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/time_series/volatility/plot_01_parkinson_range
   /api/gallery/time_series/volatility/plot_02_engle_arch
   /api/gallery/time_series/volatility/plot_03_bollerslev_garch
   /api/gallery/time_series/volatility/plot_04_nelson_egarch
   /api/gallery/time_series/volatility/plot_05_gjr_garch
   /api/gallery/time_series/volatility/plot_06_riskmetrics_ewma
   /api/gallery/time_series/volatility/plot_07_realized_volatility