No abstract

Integral transforms of the lognormal distribution are of great importance in statistics and probability, yet closed-form expressions do not exist. A wide variety of methods have been employed to provide approximations, both analytical and numerical. In this paper, we analyze a closed-form approximation L(θ) of the Laplace transform L(θ) which is obtained via a modified version of Laplace's method. This approximation, given in terms of the Lambert W (•) function, is tractable enough for applications. We prove that L(θ) is asymptotically equivalent to L(θ) as θ → ∞. We apply this result to construct a reliable Monte Carlo estimator of L(θ) and prove it to be logarithmically efficient in the rare event sense as θ → ∞.

Let X be lognormal(μ,σ2) with density f(x); let θ > 0 and define L(θ)=double-struckEnormale−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 double-struckE[Xnormale−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 the sum of lognormals Sn=X1+⋯+Xn: a saddlepoint approximation and an exponential tilting 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 and x motivated by portfolio value‐at‐risk calculations.

We consider the problem of efficient estimation of tail probabilities of sums of correlated lognormals via simulation. This problem is motivated by the tail analysis of portfolios of assets driven by correlated Black-Scholes models. We propose two estimators that can be rigorously shown to be efficient as the tail probability of interest decreases to zero. The first estimator, based on importance sampling, involves a scaling of the whole covariance matrix and can be shown to be asymptotically optimal. A further study, based on the Cross-Entropy algorithm, is also performed in order to adaptively optimize the scaling parameter of the covariance. The second estimator decomposes the probability of interest in two contributions and takes advantage of the fact that large deviations for a sum of correlated lognormals are (asymptotically) caused by the largest increment. Importance sampling is then applied to each of these contributions to obtain a combined estimator with asymptotically vanishing relative error.