Stochastic PDEs for large portfolios with general mean-reverting volatility processes

Abstract

We consider a structural stochastic volatility model for the loss from a large portfolio of credit risky assets. Both the asset value and the volatility processes are correlated through systemic Brownian motions, with default determined by the asset value reaching a lower boundary. We prove that if our volatility models are picked from a class of mean-reverting diffusions, the system converges as the portfolio becomes large and, when the vol-of-vol function satisfies certain regularity and boundedness conditions, the limit of the empirical measure process has a density given in terms of a solution to a stochastic initial-boundary value problem on a half-space. The problem is defined in a special weighted Sobolev space. Regularity results are established for solutions to this problem, and then we show that there exists a unique solution. In contrast to the CIR volatility setting covered by the existing literature, our results hold even when the systemic Brownian motions are taken to be correlated.

Keywords

Stochastic PDEs / Large portfolios / General mean-reverting volatility processes / Stochastic volatility model / Credit risk

Acknowledgements

The second author’s work was supported financially by the United Kingdom Engineering and Physical Sciences Research Council (Grant No. EP/L015811/1), and by the Foundation for Education and European Culture (founded by Nicos & Lydia Tricha).

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