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Derivatives pricing via p-optimal martingale measures: some extreme cases

Part of: Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

Marina Santacroce
Marina Santacroce*
Affiliation:
Politecnico di Torino
*
∗Postal address: Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. Email address: [email protected]
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Abstract

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In an incomplete financial market in which the dynamics of the asset prices is driven by a d-dimensional continuous semimartingale X, we consider the problem of pricing European contingent claims embedded in a power utility framework. This problem reduces to identifying the p-optimal martingale measure, which can be given in terms of the solution to a semimartingale backward equation. We use this characterization to examine two extreme cases. In particular, we find a necessary and sufficient condition, written in terms of the mean-variance trade-off, for the p-optimal martingale measure to coincide with the minimal martingale measure. Moreover, if and only if an exponential function of the mean-variance trade-off is a martingale strongly orthogonal to the asset price process, the p-optimal martingale measure can be simply expressed in terms of a Doléans-Dade exponential involving X.

Keywords

Semimartingale backward equationp-optimal martingale measureincomplete marketdiffusion

MSC classification

Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.) 90C39: Dynamic programming
Type
Research Article
Information
Journal of Applied Probability , Volume 43 , Issue 3 , September 2006 , pp. 634 - 651
Copyright
© Applied Probability Trust 2006 

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