Title: Accurate Computation of the Distribution of Sums of Dependent Log-Normals with Applications to the Black-Scholes Model,
Abstract: 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. In all of these three cases our methodology
delivers fast and highly accurate estimators 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 pre-limit. 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.