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Pricing path-dependent options in a Black-Scholes market from the distribution of homogeneous Brownian functionals

Part of: Markov processes Mathematical economics

Published online by Cambridge University Press:  14 July 2016

T. Fujita ,
F. Petit  and
M. Yor
T. Fujita*
Affiliation:
Hitotsubashi University
F. Petit*
Affiliation:
Université Paris VI
M. Yor*
Affiliation:
Université Paris VI
*
∗ Postal address: Graduate School of Commerce and Management, Hitotsubashi University, Naka 2-1, Kunitachi, Tokyo, 186-8601, Japan. Email address: fujita@math.hit-u.ac.jp
∗∗ Postal address: Laboratoire de Probabilités, Université Paris VI, casier 188, 4, Place Jussieu, 75252 Paris Cedex 05, France.
∗∗ Postal address: Laboratoire de Probabilités, Université Paris VI, casier 188, 4, Place Jussieu, 75252 Paris Cedex 05, France.
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Abstract

We give some explicit formulae for the prices of two path-dependent options which combine Brownian averages and penalizations. Because these options are based on both the maximum and local time of Brownian motion, obtaining their prices necessitates some involved study of homogeneous Brownian functionals, which may be of interest in their own right.

Keywords

Brownian motionBrownian supremumBrownian local timeoptions with barriers and penalties

MSC classification

Primary: 60J55: Local time and additive functionals 60J65: Brownian motion 91B70: Stochastic models

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 1 , March 2004 , pp. 1 - 18
Copyright
Copyright © Applied Probability Trust 2004 

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References

Bertoin, J., and Pitman, J. W. (1994). Path transformation connecting Brownian bridge, excursion and meander. Bull. Sci. Math. 118, 147–166.Google Scholar
Bertoin, J., Chaumont, L., and Pitman, J. W. (2003). Path transformations of first passage bridges. Electron. Commun. Prob. 8, 155–166.Google Scholar
Biane, P., and Yor, M. (1987). Valeurs principales associées aux temps locaux browniens. Bull. Sci. Math. 111, 23–101.Google Scholar
Biane, P., and Yor, M. (1988). Sur la loi des temps locaux browniens pris en un temps exponentiel. In Séminaire de Probabilités XXII (Lecture Notes Math. 1321), Springer, Berlin, pp. 454–466.Google Scholar
Carmona, P., Petit, F., Pitman, J., and Yor, M. (1999). On the laws of homogeneous functionals of the Brownian bridge. Studia Sci. Math. Hungar. 35, 445–455.Google Scholar
Cartan, H. (1961). Théorie Élémentaire des Fonctions Analytiques. Hermann, Paris.Google Scholar
Chesney, M., Geman, H., Jeanblanc-Piqué, M., and Yor, M. (1997). Some combinations of Asian, Parisian and barrier options. In Mathematics of Derivative Securities, eds Dempster, M. H. A. and Pliska, S., Cambridge University Press, pp. 61–87.Google Scholar
Chung, K. L. (1982). A cluster of great formulas. Acta Math. Acad. Sci. Hungar. 39, 65–67.Google Scholar
Durrett, R., and Iglehart, D. L. (1977). Functionals of Brownian meander and Brownian excursion. Ann. Prob. 5, 130–135.Google Scholar
Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. (1954). Tables of Integral Transforms. McGraw-Hill, New York.Google Scholar
Gittenberger, B., and Louchard, G. (1999). The Brownian excursion multidimensional local time density. Ann. Prob. 36, 350–373.Google Scholar
Jeanblanc, M., Pitman, J. W., and Yor, M. (1997). The Feynman–Kac formula and decomposition of Brownian paths. Mat. Apl. Comput. 16, 27–52.Google Scholar
Lebedev, N. N. (1972). Special Functions and Their Applications. Dover, New York.Google Scholar
Leuridan, C. (1996). Une démonstration élémentaire d'une identité de Biane et Yor. In Séminaire de Probabilités XXX (Lecture Notes Math. 1626), Springer, Berlin, pp. 255–260.Google Scholar
Petiau, G. (1955). La théorie des fonctions de Bessel exposée en vue de ses applications à la physique mathématique. Centre National de la Recherche Scientifique, Paris.Google Scholar
Pitman, J. W., and Yor, M. (1996). Quelques identités en loi pour les processus de Bessel. Astérisque 236, 249–276.Google Scholar
Pitman, J. W., and Yor, M. (1996). Decomposition at the maximum for excursions and bridges of one-dimensional diffusions. In Ito's Stochastic Calculus and Probability Theory, eds Ikeda, N., Watanabe, S., Fukushima, M. and Kunita, H., Springer, Berlin, pp. 293–310.Google Scholar
Pitman, J. W., and Yor, M. (1998). Random Brownian scaling identities and splicing of Bessel processes. Ann. Prob. 26, 1683–1702.Google Scholar
Pitman, J. W., and Yor, M. (1999). The law of the maximum of a Bessel bridge. Electron. J. Prob. 4, No. 15.Google Scholar
Revuz, D., and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.Google Scholar
Yor, M. (1994–1995). Local times and excursions for Brownian motion: a concise introduction (Lecciones en Matemáticas 1). Universidad Central de Venezuela, Caracas.Google Scholar