Measuring Market Risk - Reserve Bank of India

MEASURING MARKET RISK 13
distributional forms considered would be quite wide including, say,
hyperbolic distribution, t-distribution, mixture of two or more normal
distributions, Laplace distribution or so forth, (van den Goorbergh
and Vlaar, 1999; Bauer 2000; Linden, 2001).
In our study we consider symmetric hyperbolic distribution as an
alternative fat-tailed distribution for returns 6 . A d-dimensional random
variable ‘r’ is said to follow a symmetric hyperbolic distribution if it
has density function as below;
where, K ν
is the modified Bessel function of the third kind, the
parameters δ and ∆ are for multivariate scales, µ for location and ζ
mainly changes the tails.
For the presence of Bessel functions in above density function,
closed form expression for maximum likelihood estimators are not
possible. Bauer (2000) suggests an approach to have maximum
likelihood estimators 7 . Once estimates of the parameters become
available, one can estimate the required percentile of the distribution
following numerical iteration method.
3.2.4 Methods under Extreme Value Theory – Use of Tail-Index
The fat tails of unconditional return distribution can also be
handled through extreme value theory using, say, tail-index, which
measures the amount of tail fatness. One can therefore, estimate the
tail-index and measure VaR based on the underlying distribution. The
basic premise of this idea stems from the result that the tails of every
fat-tailed distribution converge to the tails of Pareto distribution. In a
simple case, upper tail of such a distribution can be modeled as,