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
Date Written: December 8, 2012
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
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