Title: Fast mean-reversion asymptotics for large portfolios of stochastic volatility models,
Abstract: We consider a large portfolio limit where the asset prices evolve according
certain stochastic volatility models with default upon hitting a lower barrier.
When the asset prices and the volatilities are correlated via systemic Brownian
Motions, that limit exist and it is described by a SPDE on the positive
half-space with Dirichlet boundary conditions which has been studied in
\cite{HK17}. We study the convergence of the total mass of a solution to this
stochastic initial-boundary value problem when the mean-reversion coefficients
of the volatilities are multiples of a parameter that tends to infinity. When
the volatilities of the volatilities are multiples of the square root of the
same parameter, the convergence is extremely weak. On the other hand, when the
volatilities of the volatilities are independent of this exploding parameter,
the volatilities converge to their means and we can have much better
approximations. Our aim is to use such approximations to improve the accuracy
of certain risk-management methods in markets where fast volatility
mean-reversion is observed.