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

J.Q. Smith and Antônio Santos

4 Stochastic Volatility models and Particle Fil-
ters

To develop efficient particle filter procedures which may be applied to predict the
variance evolution in financial markets, we propose to use the characteristics of the
model to find better approximations of the target distribution described above.

It is straightforward to simulate from a Gaussian distribution and, for a given
value αt-1,k, k = 1, . . . , m, the transition density in (2) assumes the Gaussian form.
To obtain the posterior distribution, this must be combined with the likelihood
function, and in this case, the conjugate property does not apply. It is not pos-
sible to sample directly from the target distribution, but we are able to define an
approximating Gaussian distribution from which it is easy to sample.

The way to implement these procedures is to perform a first or second order
Taylor approximation of the log-likelihood. The log-likelihood function associated
to model (1) as a function of αt is

2

^l ( ^αt ) = const^- 2^- X                           ^<12)

This function is concave in α_t and so first and second order Taylor series approxi-
mations may work.

Based on a first order Taylor approximation, Pitt and Shephard (1999, 2001)
developed a rejection sampler which was used to implement the particle filter. If
this approximation is defined around some arbitrary value μ_t,_k, g yy₁∖οι_tμμ,_k1^, it can
easily be seen that g yyt∖at,μ_t,k) ≥ f (yt∖α_t) due to the assumed log-concavity of
f (y_t∖α_t). This allows the definition of a perfect envelope of the target density and
a rejection sampler can be implemented. These arguments can be summarized by

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

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