Stochastic volatility models are a category of stochastic processes that have stochastic (random) second moments. Stated another way, they have random volatility or are conditionally heteroskedastic. "Stochastic volatility model" is a technical term. While all stochastic volatility models have stochastic second moments, not all models that have stochastic second moments are called stochastic volatility models. In finance, two categories of stochastic processes are widely used to model stochastic second moments. One is stochastic volatility models. The other is ARCH/GARCH models. Both ARCH/GARCH and stochastic volatility models derive their randomness from white noise processes. The difference is that an ARCH/GARCH process depends on just one white noise W. That white noise directly determines innovations in the ARCH/GARCH process while also indirectly determining innovations in its second moments. Stochastic volatility models generally depend on two white noises, V and W. One directly determines innovations in the stochastic process. The other directly determines innovations in its second moments. Stochastic volatility models come in forms far more diverse than those of ARCH or GARCH models. An example of a simple univariate stochastic volatility model X is (see the notation conventions documentation)
where V and W are independent
standard normal
Gaussian white noises.
Stochastic volatility models often employ logarithms as in [2] to ensure
conditional variances
While our example is of a discrete-time model, continuous-time stochastic volatility models are widely used in financial engineering.
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are nonnegative.
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