Fast mean-reversion asymptotics for large portfolios of stochastic volatility models

Abstract

We consider an SPDE description of a large portfolio limit model where the underlying asset prices evolve according to certain stochastic volatility models with default upon hitting a lower barrier. The asset prices and their volatilities are correlated via systemic Brownian motions, and the resulting SPDE is defined on the positive half-space with Dirichlet boundary conditions. We study the convergence of the loss from the system, a function of the total mass of a solution to this stochastic initial-boundary value problem under fast mean reversion of the volatility. We consider two cases. In the first case the volatility converges to a limiting distribution and the convergence of the system is in the sense of weak convergence. On the other hand, when only the mean reversion of the volatility goes to infinity we see a stronger form of convergence of the system to its limit. Our results show that in a fast mean-reverting volatility environment we can accurately estimate the distribution of the loss from a large portfolio by using an approximate constant volatility model which is easier to handle.