STOCHASTIC VOLATILITY MODELS, CORRELATION, AND THE q ‐OPTIMAL MEASURE

The aim of this paper is to study the minimal entropy and variance‐optimal martingale measures for stochastic volatility models. In particular, for a diffusion model where the asset price and volatility are correlated, we show that the problem of determining the q ‐optimal measure can be reduced to finding a solution to a representation equation. The minimal entropy measure and variance‐optimal measure are seen as the special cases q = 1 and q = 2 respectively. In the case where the volatility is an autonomous diffusion we give a stochastic representation for the solution of this equation. If the correlation ρ between the traded asset and the autonomous volatility satisfies ρ 2 < 1/ q , and if certain smoothness and boundedness conditions on the parameters are satisfied, then the q ‐optimal measure exists. If ρ 2 ≥ 1/ q , then the q ‐optimal measure may cease to exist beyond a certain time horizon. As an example we calculate the q ‐optimal measure explicitly for the Heston model.