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Stochastic volatility models, correlation, and the \(q\)-optimal measure. (English) Zbl 1169.60317

Summary: 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 between the traded asset and the autonomous volatility satisfies \(\rho^2 < 1/q\), and if certain smoothness and boundedness conditions on the parameters are satisfied, then the \(q\)-optimal measure exists. If \(\rho^2\geq 1/q\), 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.

MSC:

60H30 Applications of stochastic analysis (to PDEs, etc.)
60G44 Martingales with continuous parameter
91B28 Finance etc. (MSC2000)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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