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The density process of the minimal entropy martingale measure in a stochastic volatility model with jumps

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Abstract.

We derive the density process of the minimal entropy martingale measure in the stochastic volatility model proposed by Barndorff-Nielsen and Shephard [2]. The density is represented by the logarithm of the value function for an investor with exponential utility and no claim issued, and a Feynman-Kac representation of this function is provided. The dynamics of the processes determining the price and volatility are explicitly given under the minimal entropy martingale measure, and we derive a Black & Scholes equation with integral term for the price dynamics of derivatives. It turns out that the minimal entropy price of a derivative is given by the solution of a coupled system of two integro-partial differential equations.

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Author notes

  1. Fred Espen Benth

    Present address: Department of Economics and Business Administration, Agder University College, Serviceboks 422, 4604, Kristiansand, Norway

Authors and Affiliations

  1. Department of Mathematics, University of Oslo, Centre of Mathematics for Applications, Blindern, P.O. Box 1053, 0316, Oslo, Norway

    Fred Espen Benth

  2. Department of Mathematics, University of Oslo, Centre of Mathematics for Applications, Blindern, P.O. Box 1053, 0316, Oslo, Norway

    Thilo Meyer-Brandis

Authors
  1. Fred Espen Benth
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  2. Thilo Meyer-Brandis
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Corresponding author

Correspondence to Fred Espen Benth.

Additional information

Mathematics Subject Classification:

91B28, 60G51, 60G57

JEL Classification:

D46, D52, G13

We are grateful to Kenneth Hvistendahl Karlsen and Thorsten Rheinländer for interesting and fruitful discussions. Two anonymous referees and an associate editor are thanked for their careful reading of an earlier version of the paper leading to a significant improvement of the presentation

Manuscript received: March 2004; final version received: March 2005

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Benth, F.E., Meyer-Brandis, T. The density process of the minimal entropy martingale measure in a stochastic volatility model with jumps. Finance Stochast. 9, 563–575 (2005). https://doi.org/10.1007/s00780-005-0161-z

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  • DOI: https://doi.org/10.1007/s00780-005-0161-z

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