Stochastic evolution equations for large portfolios of stochastic volatility models

Ben Hambly, Nikolaos Kolliopoulos

Published 2017-01-19Version 1

We consider a large market model of defaultable assets in which the asset price processes are modelled as Heston stochastic volatility models with default upon hitting a lower boundary. We assume that both the asset prices and their volatilities are correlated through systemic Brownian motions. We are interested in the loss process that arises in this setting and consider the large portfolio limit of the empirical measure for this system. This limit evolves as a measure valued process and we show that it will have a density that satisfies a stochastic partial differential equation of filtering type with Dirichlet boundary conditions. We are able to show uniqueness and regularity of that solution. We employ Malliavin calculus to establish the existence of a regular density for the volatility component and an approximation by models of piecewise constant volatilities combined with a kernel smoothing technique to obtain the desired results for the full two-dimensional problem.