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Brownian Excursions and Parisian Barrier Options

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Marc Chesney ,
Monique Jeanblanc-Picqué  and
Marc Yor
Marc Chesney*
Affiliation:
HEC
Monique Jeanblanc-Picqué*
Affiliation:
Université d'Evry Val d'Essonne
Marc Yor*
Affiliation:
Université Pierre et Marie Curie
*
∗ Postal address: HEC, Département Finance et Economie, 1, rue de la libération, 78351 Jouy en Josas Cedex, France.
∗∗ Postal address: Equipe d'analyse et probabilités, Université d'Evry Val d'Essonne, Boulevard des Coquibus, 91025 Evry Cedex, France.
∗∗∗ Postal address: Laboratoire de Probabilités, Tour 56, 3-ième étage, Université Pierre et Marie Curie, 4, Place Jussieu, 75252 Paris Cedex, France.
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Abstract

In this paper we study a new kind of option, called hereinafter a Parisian barrier option. This option is the following variant of the so-called barrier option: a down-and-out barrier option becomes worthless as soon as a barrier is reached, whereas a down-and-out Parisian barrier option is lost by the owner if the underlying asset reaches a prespecified level and remains constantly below this level for a time interval longer than a fixed number, called the window. Properties of durations of Brownian excursions play an essential role. We also study another kind of option, called here a cumulative Parisian option, which becomes worthless if the total time spent below a certain level is too long.

Keywords

BROWNIAN EXCURSIONBROWNIAN MEANDERBARRIER OPTIONS

MSC classification

Primary: 60G46: Martingales and classical analysis
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 1 , March 1997 , pp. 165 - 184
Copyright
Copyright © Applied Probability Trust 1997 

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References

[1] Akahori, J. (1995) Some formulae for a new type of path-dependent option. Ann. Appl. Prob. 5, 383–388.Google Scholar
[2] Azema, J. and Yor, M. (1989) Etude d'une martingale remarquable. Séminaire de probabilités XXIII (Lecture Notes in Mathematics 1372). Springer, Berlin.Google Scholar
[3] Black, F. and Scholes, M. (1973) The pricing of options and corporate liabilities. J. Political Econ. 81, 637–654.Google Scholar
[4] Chung, K. L. (1976) Excursions in Brownian motion. Ark. Math. 14, 155–177.Google Scholar
[5] Cornwall, M. J., Kentwell, G. W., Chesney, M., Jeanblanc-Picque, M. and Yor, M. (1997) Parisian barrier option: a discussion. Risk Mag. To appear.Google Scholar
[6] Darling, D. A. (1952) The influence of the maximum term in the addition of independent random variables. Trans. Amer. Math. Soc. 73, 95–107.CrossRefGoogle Scholar
[7] Dassios, A. (1995) The distribution of the quantiles of a Brownian motion with drift. Ann. Appl. Prob. 5, 389–398.Google Scholar
[8] Dellacherie, C., Maisonneuve, B. and Meyer, P. A. (1992) Probabilités et Potentiel. Processus de Markov (fin). Compléments de Calcul Stochastique. Hermann, Paris.Google Scholar
[9] Embrechts, P., Rogers, L. C. G. and Yor, M. (1995) A proof of Dassios' representation of the a-quantile of Brownian motion with drift. Ann. Appl. Prob. 5, 757–767.CrossRefGoogle Scholar
[10] Geman, H. and Yor, M. (1993) Bessel processes, Asian options and perpetuities. Math. Finance 3, 349–375.Google Scholar
[11] Geman, H. and Yor, M. (1996) Pricing and hedging double-barrier options: a probabilistic approach. Math. Finance 6, 365–378.Google Scholar
[12] Grabbe, J. (1983) The pricing of call and put options on foreign exchanges. J. Int. Money Finance 2, 39–254.Google Scholar
[13] Horowitz, J. (1972) Semi-linear Markov processes, subordinator and renewal theory. Z. Wahrscheinlichkeitsth. 24, 167–193.Google Scholar
[14] Karatzas, I. and Shreve, S. (1984) Trivariate density of Brownian motion, its local and occupation times, with application to stochastic control. Ann. Prob. 12, 819–828.Google Scholar
[15] Kingman, J. E. (1975) Random distributions. J. R. Statist. Soc. B 37, pp. 1–22.Google Scholar
[16] Knight, F. B. (1986) On the duration of the longest excursion. In Seminar on Stochastic Processes 1985. Birkhäuser, Boston.Google Scholar
[17] Lamperti, J. (1961) On contribution to renewal theory. Proc. Amer. Math. Soc. 12, 724–731.Google Scholar
[18] Pitman, J. and Yor, M. (1997) The two parameter Poisson-Dirichlet distribution derived from stable subordinators. Ann. Prob. To appear.CrossRefGoogle Scholar
[19] Port, S. (1963) An elementary probability approach to fluctuation theory. Ann. Math. Statist. 31, 1034–1044.Google Scholar
[20] Reiner, E. and Rubinstein, M. (1992) Exotic options. Working paper. Google Scholar
[21] Resnick, S. I. (1986) Point process, regular variation and weak convergence. Adv. Appl. Prob. 18, 66–138.CrossRefGoogle Scholar
[22] Revuz, D. and Yor, M. (1994) Continuous Martingales and Borwmian Motion. 2nd edn. Springer, Berlin.Google Scholar
[23] Wendel, J. G. (1963) Order statistics of partial sums. J. Math. Anal. Appl. 6, 109–151.Google Scholar
[24] Wendel, J. G. (1964) Zero-free intervals of semi-stable Markov-processes. Math. Scand. 14, 21–34.CrossRefGoogle Scholar
[25] Yor, M. (1993) Some remarks on Akahori's generalized arc sine formula for Brownian motion with drift. Preprint. Laboratoire de Probabilités, Paris 6.Google Scholar
[26] Yor, M. (1995) The distribution of Brownian quantiles. J. Appl. Prob. 32, 405–416.CrossRefGoogle Scholar