Efficient tail estimation for sums of correlated lognormals

Our focus is on efficient estimation of tail probabilities of sums of correlated lognormals. This problem is motivated by the tail analysis of portfolios of assets driven by correlated Black-Scholes models. We propose three different procedures that can be rigorously shown to be asymptotically optimal as the tail probability of interest decreases to zero. The first algorithm is based on importance sampling and is as easy to implement as crude Monte Carlo. The second algorithm is based on an elegant conditional Monte Carlo strategy which involves polar coordinates and the third one is an importance sampling algorithm that can be shown to be strongly efficient.