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A PDE representation of the density of the minimal entropy martingale measure in stochastic volatility markets |
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| Authors: | Fred Espen Benth Kenneth Hvistendahl Karlsen ¶ |
| Institution: | 1. Department of Mathematics , Centre of Mathematics for Applications, University of Oslo , P.O. Box 1053, Blindern, N-0316, Oslo, Norway;2. Department of Economics and Business Administration , Agder University College , Serviceboks 422, N-4604, Kristiansand, Norway fredb@math.uio.no;4. Department of Mathematics , Centre of Mathematics for Applications, University of Oslo , P.O. Box 1053, Blindern, N-0316, Oslo, Norway |
| Abstract: | Under general conditions stated in Rheinländer An entropy approach to the stein/stein model with correlation. Preprint, 2003, ETH Zürich.], we prove that in a stochastic volatility market the Radon–Nikodym density of the minimal entropy martingale measure (MEMM) can be expressed in terms of the solution of a semilinear PDE. The semilinear PDE is suggested by the dynamic programming approach to the utility indifference pricing problem of contingent claims. One of our main results is the existence and uniqueness of a classical solution of the semilinear PDE in the case of a general stochastic volatility model with additive noise correlated with the asset price. Our results are applied to the Stein–Stein and Heston stochastic volatility models. |
| Keywords: | Stochastic volatility Incomplete market Pricing of contingent claims Minimal entropy martingale measure Utility optimization Semilinear PDE 93E20 35K55 91B28 |