Multilevel simulation of functionals of Bernoulli random variables with application to basket credit derivatives
This work provides an optimal-complexity simulation method for credit derivative pricing, a domain-specific problem in computational finance.
The authors prove that expected functionals of the proportion of Bernoulli variables converge at rate 1/N, and propose a multilevel simulation algorithm achieving ε^2 mean-square error with O(ε^{-2}) complexity, independent of N. Numerical results for basket credit derivative tranche spreads demonstrate the method.
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 $ε^2$ 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.