Simulation of conditional expectations under fast mean-reverting stochastic volatility models
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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.
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