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 \mbox{%DIFAUXCMD 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 .
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 .
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an SPDE description of
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where the
model where the underlying
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to
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When the
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Motions, that limit exist and it is described by a SPDE
motions, and the resulting SPDE is defined
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which has been studied in \mbox{%DIFAUXCMD HK17
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loss from the system, a function of the
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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
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
R
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
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