Multilevel simulation of functionals of Bernoulli random variables with application to basket credit derivatives

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 $N\rightarrow \infty$. 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 $\epsilon^2$ and computational complexity of order $\epsilon^{-2}$, 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.