  In this short paper, we study the simulation of a large system of stochastic
processes subject to a common driving noise and fast mean-reverting stochastic
volatilities. This model may be used to describe the firm values of a large
pool of financial entities. We then seek an efficient estimator for the
probability of a default, indicated by a firm value below a certain threshold,
conditional on common factors. We first analyse the convergence of the
Euler--Maruyama scheme applied to the fast Ornstein--Uhlenbeck SDE for the
volatility, and show that the first order strong error is robust with respect
to the mean reversion speed (only) if the step size is scaled appropriately.
Next, we consider approximations where coefficients containing the fast
volatility are replaced by certain ergodic averages (a type of law of large
numbers), and study a correction term (of central limit theorem-type). The
accuracy of these approximations is assessed by numerical simulation of
pathwise losses and the estimation of payoff functions as they appear in basket
credit derivatives.
