Multilevel Simulation of Functionals of Bernoulli Random Variables with Application to Basket Credit Derivatives

Abstract

We consider N Bernoulli random variables, which are independent conditional on a common random factor determining their probability distribution. We show that certain expected functionals of the proportion L N of variables in a given state converge at rate 1/N as N → ∞. Based on these results, we propose a multi-level simulation algorithm using a family of sequences with increasing length, to obtain estimators for these expected functionals with a mean-square error of ϵ 2 and computational complexity of order ϵ −2, independent of N. In particular, this optimal complexity order also holds for the infinite-dimensional limit. Numerical examples are presented for tranche spreads of basket credit derivatives.