The multivariate Black & Scholes market: conditions for completeness and no-arbitrage
Authors:
J. Dhaene, A. Kukush and D. Linders
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
Full-text PDF Free Access
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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.
References
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References
- L. Bachelier, Theorie de la spéculation, Ann. Sci. École Norm. Sup. (3) 17 (1900), 21–86. MR 1508978
- T. Björk, Arbitrage Theory in Continuous Time, Oxford University Press, 1998.
- F. Black and M. Scholes, The pricing of options and corporate liabilities, The Journal of Political Economy 81 (1973), no. 3, 637–654.
- R. Carmona and V. Durrleman, Generalizing the Black–Scholes formula to multivariate contingent claims, Journal of Computational Finance 9 (2006).
- G. Deelstra, J. Liinev, and M. Vanmaele, Pricing of arithmetic basket options by conditioning, Insurance Mathematics and Economics 34 (2004), no. 1, 55–77. MR 2035338 (2004k:91103)
- F. Delbaen and W. Schachermayer, The Mathematics of Arbitrage, Springer, 2006. MR 2200584
- J. Dhaene, M. Denuit, M. J. Goovaerts, R. Kaas, and D. Vyncke, The concept of comonotonicity in actuarial science and finance: applications, Insurance Mathematics & Economics 31 (2002), no. 2, 133–161. MR 1932751 (2004d:91139)
- R. Frey, Financial Mathematics in Continuous Time, University Lectures, Universität Leipzig, 2009, http://statmath.wu.ac.at/~frey/Skript-FimaII.pdf.
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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_kukush@univ.kiev.ua
D. Linders
Affiliation:
KU Leuven, Leuven, Belgium
Email:
daniel.linders@kuleuven.be
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