Moment Explosions in Stochastic Volatility Models

Abstract

In this paper, we demonstrate that many stochastic volatility models have the undesirable property that moments of order higher than one can become infinite in finite time. As arbitrage-free price computation for a number of important fixed income products involves forming expectations of functions with super-linear growth, such lack of moment stability is of significant practical importance. For instance, we demonstrate that reasonably parameterized models can produce infinite prices for Eurodollar futures and for swaps with floating legs paying either Libor-in-arrears or a constant maturity swap (CMS) rate. We systematically examine the moment explosion property across a spectrum of stochastic volatility models. Related properties such as the failure of the martingale property, and asymptotics of the volatility smile are also considered.