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Exponential family techniques in the lognormal left tail, with applications to portfolio VaR

CSGB seminar
Thursday, 9 October, 2014, at 14:15-15:00, in Koll. G3 (1532-218)
Søren Asmussen & Jens Ledet Jensen (Department of Mathematics, Aarhus University)
Sums S_n=X_1+...+X_n of lognormals arises in a wide variety of disciplines such as engineering, economics, insurance or finance, and are often employed in modeling across the sciences. The right lognormal tail P(S_n>y) is heavy-tailed and typically analyzed by subexponential techniques. 

The left tail P(S_n<z) is of interest for example in portfolio VaR valculations . 

The typical tool would be applying saddlepoint or large deviations techniques. This faces, however, the problem that the Laplace transform L(\theta)=Ee^{-\theta X} is not explicit.

We present an approximation for L(\theta) in terms of the Lambert W function.

This is used to describe the shape of the exponentially tilted distribution  \tilde P(X\in dx)=e^{-\theta x}P(X\in dx)/L(\theta) and to derive a saddlepoint type approximation for P(S_n<z). 

Also related importance sampling algorithms are presented. 

Numerical examples are presented in a range of parameters that we consider realistic for portfolio VaR calculations.
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Revised 03.06.2016