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The analysis of finite security markets using martingales

Published online by Cambridge University Press:  01 July 2016

Murad S. Taqqu  and
Walter Willinger
Murad S. Taqqu*
Affiliation:
Cornell University
Walter Willinger*
Affiliation:
Cornell University
*
∗Present address: Department of Mathematics, 111 Cummington St, Boston University, Boston, MA 02215, USA.
∗∗ Present address: Bell Communications Research, Inc., Morristown, NJ 07960, USA.
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Abstract

The theory of finite security markets developed by Harrison and Pliska [1] used the separating hyperplane theorem to establish the relationship between the lack of arbitrage opportunities and the existence of a certain martingale measure. In this paper we treat this theory by examining certain geometric properties of the sample paths of the price process, that is, we focus on the price increments of the stocks between one time period to the next and convert them to martingale differences through an equivalent change of measure. Thus, in contrast to Harrison and Pliska&s functional analytic derivation, our approach is based on probabilistic methods and allows a geometric interpretation which not only provides a connection to linear programming but also yields an algorithm for analyzing finite security markets. Moreover, we can make precise the connection between diverse expressions of economic equilibrium such as ‘absence of arbitrage’, ‘martingale property, and ‘complementary slackness property’.

Keywords

ARBITRAGECHANGE OF MEASURECONVEX HULLCOMPLETE MARKETSEXTREMAL MEASURESMARTINGALE REPRESENTATION PROPERTYLINEAR PROGRAMMINGDUALITYEQUILIBRIUM
Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 1 , March 1987 , pp. 1 - 25
Copyright
Copyright © Applied Probability Trust 1987 

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Footnotes

Research supported by the National Science Foundation grant ECS-84-08524 at Cornell University.

References

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