Efficient Algorithms for Tail Probabilities of Exchangeable Lognormal Sums

For the estimation of left and right-tail probabilities and the pdf of the sum of exchangeable lognormal random vectors a new conditional Monte Carlo (CMC) algorithm is developed. It removes the randomness of the sum of all input variables and is simple and fast. For estimating the left-tail probabilities the CMC algorithm is logarithmically efficient. A further improvement of the algorithm by removing also the randomness of the radius of the normal input using close to optimal one dimensional importance sampling, results in the CMC.RCMC algorithm. For the sum of independent and identically distributed (i.i.d.) and exchangeable lognormal vectors it is the first algorithm that has bounded relative error for the left-tail probabilities. The CMC.RCMC algorithm is logarithmically efficient for the right-tail probabilities. Numerical experiments verify that it has a very good performance for all left-tail estimation problems and a good performance for the right tail for probabilities not smaller than 10(-10). When estimating the pdf the relative errors observed are all very close to those of the corresponding probability estimates.