# Efficient simulation of tail probabilities of sums of correlated lognormals

By Søren Asmussen, José Blanchet, Sandeep Juneja and Leonardo Rojas-Nandayapa
Thiele Research Reports
No. 11, September 2008
Abstract:

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.

Keywords: Black-Scholes model, correlated lognormals, Importance sampling, Cross-Entropy method, efficiency, rare-event simulation, vanishing relative error.

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