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Perturbed Brownian motion and its application to Parisian option pricing. (English) Zbl 1226.91073
Summary: We study the excursion times of a Brownian motion with drift below and above a given level by using a simple two-state semi-Markov model. In mathematical finance, these results have an important application in the valuation of path-dependent options such as Parisian options. Based on our results, we introduce a new type of Parisian options, single-barrier two-sided Parisian options, and give an explicit expression for the Laplace transform of its price formula.

MSC:
91G20 Derivative securities (option pricing, hedging, etc.)
60J65 Brownian motion
60K15 Markov renewal processes, semi-Markov processes
60J27 Continuous-time Markov processes on discrete state spaces
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[1] Akahori, J.: Some formulae for a new type of path-dependent option. Ann. Appl. Probab. 5, 383–388 (1995) · Zbl 0834.90026 · doi:10.1214/aoap/1177004769
[2] Anderluh, J.H.M., van der Weide, J.A.M.: Double-sided Parisian option pricing. Finance Stoch. 13, 205–238 (2009) · Zbl 1199.91199 · doi:10.1007/s00780-009-0090-3
[3] Azéma, J., Yor, M.: Etude d’une martingale remarquable, Séminaire de Probabilités XXIII. Lecture Notes in Mathematics, vol. 1372, pp. 88–130. Springer, Berlin (1989) · Zbl 0743.60045
[4] Chesney, M., Jeanblanc-Picqué, M., Yor, M.: Brownian excursions and Parisian barrier options. Adv. Appl. Probab. 29, 165–184 (1997) · Zbl 0882.60042 · doi:10.2307/1427865
[5] Chung, K.L.: Excursions in Brownian motion. Ark. Math. 14, 155–177 (1976) · Zbl 0356.60033 · doi:10.1007/BF02385832
[6] Dassios, A.: The distribution of the quantiles of a Brownian motion with drift. Ann. Appl. Probab. 5, 389–398 (1995) · Zbl 0837.60076 · doi:10.1214/aoap/1177004770
[7] Labart, C., Lelong, J.: Pricing double Parisian options using Laplace transforms. Int. J. Theor. Appl. Finance 12, 19–44 (2009) · Zbl 1190.91143 · doi:10.1142/S0219024909005154
[8] Pechtl, A.: Some applications of occupation times of Brownian motion with drift in mathematical finance. J. Appl. Math. Decis. Sci. 3, 63–73 (1999) · Zbl 0933.91017 · doi:10.1155/S1173912699000048
[9] Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999) · Zbl 0917.60006
[10] Schröder, M.: Brownian excursions and Parisian barrier options: a note. J. Appl. Probab. 40, 855–864 (2003) · Zbl 1056.60040 · doi:10.1239/jap/1067436086
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