Complications with Stochastic Volatility Models

We show a class of stochastic volatility price models for which the most natural candidates for martingale measures are only strictly local martingale measures, contrary to what is usually assumed in the finance literature. We also show the existence of martingale measures, however, and give explicit examples. 1 Introduction In the mathematical finance literature there is a growing interest in the study of incomplete markets, and among the efforts in understanding their properties the stochastic volatility models have become increasingly popular: see for example Hull and White (87), Scott (87), Wiggins (87), Johnson and Shanno (87), Stein and Stein (91), Heston (93), Dupire (92), Hofmann, Platen and Schweizer (92) among others. They all start by imposing certain dynamics for a "volatility process" and construct asset prices as stochastic-exponentials of integrals of this volatility with respect to a Brownian motion. If we want to apply results from arbitrage-pricing theory, which char..