The Multivariate Black & Scholes Market: Conditions for Completeness and No-Arbitrage

Theory of Probability and Mathematical Statistics, vol 88, pages 1-14, 2013

13 Pages Posted: 9 Dec 2012 Last revised: 6 Nov 2013

See all articles by Jan Dhaene

Jan Dhaene

Katholieke Universiteit Leuven

Alexander Kukush

Catholic University of Leuven (KUL)

Daniël Linders

University of Illinois

Date Written: December 8, 2012

Abstract

In order to price multivariate derivatives, there is need for a multivariate stock price model. To keep the simplicity and attractiveness of the one-dimensional Black & Scholes model, one often considers a multivariate model where each individual stock follows a Black & Scholes model, but the underlying Brownian motions might be correlated. Although the classical one-dimensional Black & Scholes model is always arbitrage-free and complete, this statement does not hold true in a multivariate setting.

In this paper, we derive conditions under which the the multivariate Black & Scholes model is arbitrage-free and complete.

Keywords: Black & Scholes, multivariate asset price models, arbitrage-free, completeness, Brownian motion, risk-neutral probability measure

Suggested Citation

Dhaene, Jan and Kukush, Alexander and Linders, Daniël, The Multivariate Black & Scholes Market: Conditions for Completeness and No-Arbitrage (December 8, 2012). Theory of Probability and Mathematical Statistics, vol 88, pages 1-14, 2013. Available at SSRN: https://ssrn.com/abstract=2186830 or http://dx.doi.org/10.2139/ssrn.2186830

Jan Dhaene

Katholieke Universiteit Leuven ( email )

Naamsestraat 69
Leuven, 3000
Belgium

Alexander Kukush

Catholic University of Leuven (KUL) ( email )

Leuven, B-3000
Belgium

Daniël Linders (Contact Author)

University of Illinois ( email )

306 Altgeld Hall,
1409 West Green Street
Champaign, IL 61822
United States

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