Second Order Filter Distribution Approximations for Financial Time Series with Extreme Outlier

J.Q. Smith and Antonio Santos

tions appear in the tails than for Gaussian processes, giving rise to high kurtosis.
The second is volatility clustering, indicating the need to model the variance evolu-
tion of the series. It is a well established fact from empirical as well as theoretical
financial literature that with short term series, variances as measures of volatility in
financial markets are time varying and present some degree of predictability (Boller-
slev et al. 1994; Taylor 1994; Diebold and Lopez 1995; Engle 1995; Campbell et al.
1997; Christoffersen and Diebold 1997; Diebold et al. 1997; Ait-Sahalia 1998; An-
dersen et al. 1999). Variances are used as a measure of risk in a variety of senses:
Value-at-Risk (VaR) calculations, portfolio allocation and pricing options.

To model variance dynamics it is usually necessary to use non-linear models
(Gallant et al. 1993; Hsieh 1993; Bollerslev et al. 1994; Asbrink 1997; Campbell
et al. 1997), which, in turn, usually require numerical algorithms to make estima-
tions and predictions. The two most common classes of models used in financial
time series are the Auto-Regressive Conditional Heteroscedastic (ARCH) models
and Stochastic Volatility (SV) models. The focus of this paper is on the prediction
of the variance evolution in SV models. The method used here is the Particle Fil-
ter method as described in Kong et al. (1994), Carpenter et al. (1998), Fearnhead
(1998), Liu and Chen (1998), Carpenter et al. (1999), Freitas (1999), Doucet (2000),
Doucet et al. (2000), Godsill et al. (2000), Doucet et al. (2001) and Liu (2001).

The SV model (Taylor 1986) is a nonlinear state space model. Financial returns
yt are related to unobserved states which are serially correlated. Thus we may write
yt  =  ^β exp (^αt´ ^εt    ^εt ~ N (0^, 1)                    (1)

at  =  φαt-1 + σηη₁.     η₁. ~ N (0, 1)                       (2)

where αt are the states of the process for t = 1, . . . , n. Note that the model is
characterized by the vector of parameters θ = (β, φ, ση).

Assuming that the parameters are known or have been previously estimated,

G.E.M.F - F.E.U.C.

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