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Arbitrage in continuous complete markets

Part of: Applications Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Eckhard Platen
Eckhard Platen*
Affiliation:
University of Technology, Sydney
*
∗ Postal address: School of Finance and Economics and School of Mathematical Sciences, University of Technology Sydney, PO Box 123, Broadway, NSW 2007, Australia. Email address: eckhard.platen@uts.edu.au
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Abstract

This paper introduces a benchmark approach for the modelling of continuous, complete financial markets, when an equivalent risk-neutral measure does not exist. This approach is based on the unique characterization of a benchmark portfolio, the growth optimal portfolio, which is obtained via a generalization of the mutual fund theorem. The discounted growth optimal portfolio with minimum variance drift is shown to follow a Bessel process of dimension four. Some form of arbitrage can be explicitly modelled by arbitrage amounts. Fair contingent claim prices are derived as conditional expectations under the real world probability measure. The Heath-Jarrow-Morton forward rate equation remains valid despite the absence of an equivalent risk neutral measure.

Keywords

Continuous financial marketarbitrage amountmutual fund theoremgrowth optimal portfolionuméraire portfoliocontingent claim pricingforward rate equation

MSC classification

Secondary: 60G30: Continuity and singularity of induced measures 62P20: Applications to economics

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 3 , September 2002 , pp. 540 - 558
Copyright
Copyright © Applied Probability Trust 2002 

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