Fast Mean Reversion and Loss Estimation in Large Stochastic Volatility Portfolios

Summary

This paper studies losses in a large portfolio of assets whose prices follow stochastic volatility models and default when they hit a lower barrier. A stochastic partial differential equation represents the portfolio limit, with systemic Brownian motions linking asset prices and volatility. The quantity of interest is loss as a function of the solution’s total mass.

The analysis considers fast mean reversion under two assumptions about volatility. When volatility converges to a limiting distribution, the system converges weakly; when only the mean-reversion rate increases, a stronger form of convergence is obtained. These results support approximating the loss distribution with a simpler constant-volatility model in a fast mean-reversion setting. The description gives theoretical convergence conclusions but no numerical examples, calibration guidance, or evidence about approximation accuracy for particular portfolios.

Key ideas

• The model represents a large defaultable portfolio through a stochastic partial differential equation.
• Systemic Brownian motions create dependence between asset prices and their volatilities.
• Loss is modeled as a function of the total mass remaining in the system.
• Fast mean reversion permits approximation with a constant-volatility model.
• The convergence result depends on how volatility behaves in the limiting regime.

Tags

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


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









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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