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On Modeling Log-Volatility |
| The daily returns of
financial assets has become ubiquitous in empirical finance literature.
Similarly, the modeling of its variance (or volatility), conditional or not, a
center of a myriad of research papers. Because these returns are unpredictable
given the past information of the series and its uncertainty depends upon
unknowable future developments, a fair estimate of this volatility requires
forecasts of the distribution of the returns based on our best information up to
today (Engle, 2004). Although the expected value of the returns, given the
current set of information, is zero, their variance seems to provide observable
`footprints' that change through time in very complicated ways (Shephard, 2005).
Several researchers agree that this volatility follows a nonlinear stochastic
process of indefinite form. Such a complicate process provides the researcher
with the colossal task of its modeling. The caveat is that if the model
presupposes an inadequate stochastic process that does not render the entire
complexity of the real underlying generating mechanism the pricing of any
financial products that depend on this measure will be incorrect and people and
institutions may have imperfect hedges which expose them to unwanted risks (Ammermann,
1999). In this document we will propose and alternative way of modeling volatility directly as the logarithm of the squared daily returns using a beta-like regression model. This process would render a conditional variance equation similar to that obtained by the Stochastic Volatility Models and easily transformable to the conditional variance obtained through the GARCH methodology. We will then compare our estimates by contrasting the implied standard deviations obtained by our model, the SV framework, and the GARCH framework.
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