Fast and accurate computation of the distribution of sums of dependent log-normals
We present a new Monte Carlo methodology for the accurate estimation of the distribution of the sum of dependent log-normal random variables. The methodology delivers statistically unbiased estimators for three distributional quantities of significant interest in finance and risk management: the left tail, or cumulative distribution function; the probability density function; and the right tail, or complementary distribution function of the sum of dependent log-normal factors. For the right tail our methodology delivers a fast and accurate estimator in settings for which existing methodology delivers estimators with large variance that tend to underestimate the true quantity of interest. We provide insight into the computational challenges using theory and numerical experiments, and explain their much wider implications for Monte Carlo statistical estimators of rare-event probabilities. In particular, we find that theoretically strongly efficient estimators should be used with great caution in practice, because they may yield inaccurate results in the prelimit. Further, this inaccuracy may not be detectable from the output of the Monte Carlo simulation, because the simulation output may severely underestimate the true variance of the estimator.
KeywordsLog-normal distribution Lognormal Rare-event simulation Logarithmic efficiency Large deviations Conditional Monte Carlo Quasi Monte Carlo Second-order efficiency
We would like to thank Patrick Laub for his valuable feedback and comments on earlier drafts of this work and for sharing his computer code for pdf estimation. Zdravko Botev has been supported by the Australian Research Council Grant DE140100993. Robert Salomone has been supported by the Australian Research Council Centre of Excellence for Mathematical & Statistical Frontiers (ACEMS), under Grant No. CE140100049.
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