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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
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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  • 1. L. Bachelier, Theorie de la spéculation, Ann. Sci. École Norm. Sup. (3) 17 (1900), 21-86. MR 1508978
  • 2. T. Björk, Arbitrage Theory in Continuous Time, Oxford University Press, 1998.
  • 3. F. Black and M. Scholes, The pricing of options and corporate liabilities, The Journal of Political Economy 81 (1973), no. 3, 637-654.
  • 4. R. Carmona and V. Durrleman, Generalizing the Black-Scholes formula to multivariate contingent claims, Journal of Computational Finance 9 (2006).
  • 5. 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)
  • 6. F. Delbaen and W. Schachermayer, The Mathematics of Arbitrage, Springer, 2006. MR 2200584
  • 7. 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)
  • 8. R. Frey, Financial Mathematics in Continuous Time, University Lectures, Universität Leipzig, 2009,
  • 9. I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, Springer-Verlag, New York, 1998. MR 1640352 (2000e:91076)
  • 10. R. C. Merton, Theory of Rational Option Pricing, Bell Journal of Economics and Management 4 (1973), no. 1, 141-183. MR 0496534 (58:15058)
  • 11. M. Musiela and M. Rutkowski, Martingale Methods in Financial Modeling, 2nd edition, Springer-Verlag, Berlin, 2005. MR 2107822 (2005m:91004)
  • 12. P. A. Samuelson, Proof that properly anticipated prices fluctuate randomly, Industrial Management Review 6 (1965), 41-49.
  • 13. A. N. Shiryaev, Y. M. Kobanov, O. D. Kramkov, and A. V. Melnikov, Toward a theory of pricing of options of European and American types. II. Continuous time, Theory of Probability and its Applications 39 (1994), 61-102. MR 1348191
  • 14. S. Vanduffel, T. Hoedemakers, and J. Dhaene, Comparing approximations for risk measures of sums of non-independent lognormal random variables, North American Actuarial Journal 9 (2005), no. 4, 71-82. MR 2211905

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Additional Information

J. Dhaene
Affiliation: KU Leuven, Leuven, Belgium

A. Kukush
Affiliation: Taras Shevchenko National University, Kyiv, Ukraine
Email: alexander{\textunderscore}

D. Linders
Affiliation: KU Leuven, Leuven, Belgium

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

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