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<h1 id="parent-fieldname-title" class="documentFirstHeading">
Stochastic volatility models
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Stochastic volatility models are used to model financial time series with time-varying volatility.
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<p>An important
property of financial time series is that volatility, defined as the
standard deviation of log-returns, is time-varying. There are two
types of models used for analyzing time series with varying
volatility: GARCH models and stochastic volatility (SV) models. In
GARCH models the conditional volatility is a deterministic function
of previous returns and volatilities, while in SV-models volatility
is assumed to follow a latent stochastic process. Although SV-models
have favorable properties compared to GARCH, they are less used in
practice. The reason for this is that they are much harder to fit.
The problem is that due to non-linear dependence between returns and
latent volatilities, it is not possible to find a closed form
expression for the likelihood function for SV models. Much work has
been done to overcome this problem, but most approaches are either
inefficient or hard to implement. Here it is shown how several
SV-models can be easily fit to financial returns using ADMB-RE.</p>
<ol><li>
<a href="stochastic-volatility-collection/the-basic-sv-model-and-simple-extensions-1/modeller.html" class="internal-link" title="The basic SV model and simple extensions"><strong>The
basic SV-model and simple extensions</strong>.</a>
In this section the basic SV-model is introduced, and it is shown how
simple modifications can produce models that also can capture
skewness and higher kurtosis than the basic model. </li><li><a href="stochastic-volatility-collection/sv-models-with-leverage-effect/sv-models-with-leverage-effect.html" class="internal-link" title="SV models with leverage effect"><strong>SV-models
with leverage effect</strong>.
</a>We often see an increase in volatility after a drop in returns. This
phenomenon is called leverage effect and seems to be particularly
important for stocks. Leverage effect can be modeled by correlated noise terms in the retun and the volatility equations. Models with correlated noise terms have a more
complicated structure, and are therefore treated in a separate
section.</li></ol>
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