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Theory of Probability and Mathematical Statistics

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The multivariate Black & Scholes market: conditions for completeness and no-arbitrage


Authors: J. Dhaene, A. Kukush and D. Linders
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 88 (2013).
Journal: Theor. Probability and Math. Statist. 88 (2014), 85-98
MSC (2010): Primary 91B24; Secondary 60H10, 60J60, 60J65
DOI: https://doi.org/10.1090/S0094-9000-2014-00920-5
Published electronically: July 24, 2014
MathSciNet review: 3112636
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Abstract | References | Similar Articles | Additional Information

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 multivariate Black & Scholes model is arbitrage-free and complete.


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Additional Information

J. Dhaene
Affiliation: KU Leuven, Leuven, Belgium
Email: jan.dhaene@kuleuven.be

A. Kukush
Affiliation: Taras Shevchenko National University, Kyiv, Ukraine
Email: alexander{\textunderscore}kukush@univ.kiev.ua

D. Linders
Affiliation: KU Leuven, Leuven, Belgium
Email: daniel.linders@kuleuven.be

DOI: https://doi.org/10.1090/S0094-9000-2014-00920-5
Keywords: Black \& Scholes, multivariate asset price models, arbitrage-free, completeness, Brownian motion, risk-neutral probability measure
Received by editor(s): October 27, 2012
Published electronically: July 24, 2014
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society