Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T08:17:45.873Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

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

References

1. Harrison, J. M. and Pliska, S. R. (1981) Martingales and stochastic integrals in the theory of continuous trading. Stoch. Proc. Appl. 11, 215–260.Google Scholar
2. Harrison, J. M. and Pliska, S. R. (1983) A stochastic calculus model of continuous trading: complete markets. Stoch. Proc. Appl. 15, 313–316.Google Scholar
3. Harrison, J. M. and Kreps, D. M. (1979) Martingales and arbitrage in multi-period securities markets. J. Econom. Theory 20, 381–408.Google Scholar
4. Ingersoll, J. (1986) Theory of Financial Decision Making. Rowman and Allanheld, Totowa, NJ.Google Scholar
5. Karr, A. F. (1983) Extreme points of certain sets of probability measures with applications. Math. Operat. Res. 8, 79–85.CrossRefGoogle Scholar
6. Kreps, D. M. (1979) Multiperiod securities and the efficient allocation of risk: a comment on the Black-Scholes option pricing model. U-NBER Conf. Economics of Information and Uncertainty, Boston, Mass.Google Scholar
7. Murty, K. G. (1976) Linear and Combinatorial Programming. Wiley, New York.Google Scholar
8. Rockafellar, R. T. (1970) Convex Analysis. Princeton University Press, Princeton, NJ.Google Scholar
9. Ross, S. A. (1978) A simple approach to the valuation of risky streams. J. Business 51, 453–475.Google Scholar