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Exponential Family Techniques for the Lognormal Left Tail

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Let X be lognormal(μ,σ2) with density f(x), let θ>0 and define L(θ)=Ee−θX. We study properties of the exponentially tilted density (Esscher transform) fθ(x)=e−θxf(x)/L(θ), in particular its moments, its asymptotic form as θ→∞ and asymptotics for the saddlepoint θ(x) determined by E[Xe−θX]/L(θ)=x. The asymptotic formulas involve the Lambert W function. The established relations are used to provide two different numerical methods for evaluating the left tail probability of lognormal sum Sn=X1+⋯+Xn: a saddlepoint approximation and an exponential twisting importance sampling estimator. For the latter we demonstrate logarithmic efficiency. Numerical examples for the cdf Fn(x) and the pdf fn(x) of Sn are given in a range of values of σ2,n,x motivated from portfolio Value-at-Risk calculations.
OriginalsprogEngelsk
UdgiverT.N. Thiele Centre, Department of Mathematics, Aarhus University
Antal sider19
StatusUdgivet - 2014
SerietitelThiele Research Reports
Nummer01

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