Abstract
We define the drawdown stopping time of a Brownian motion as the first time its drawdown reaches a duration of length 1. In this paper, we propose an efficient algorithm to efficiently simulate the drawdown stopping time and the associated maximum at this time. The method is straightforward and fast to implement, and avoids simulating sample paths thus eliminating discretisation bias. We show how the simulation algorithm is useful for pricing more complicated derivatives such as multiple drawdown options.
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Anderluh JHM (2008) Pricing Parisians and barriers by hitting time simulation. Eur J Fin 14(2):137–156
Bernard C, Boyle P (2011) Monte Carlo methods for pricing discrete Parisian options. Eur J Fin 17(3):169–196
Carr P, Zhang H, Hadjiliadis O (2011) Maximum drawdown insurance. Int J Theor Appl Fin 14(8):1195–1230
Chesney M, Jeanblanc-Picqué M, Yor M (1997) rownian excursions and Parisian barrier options. Adv Appl Probab 29(1):165–184
Dassios A, Wu S (2010) Perturbed Brownian motion and its application to Parisian option pricing. Fin Stoch 14:473–494
Dassios A, Lim JW (2013) Parisian option pricing: a recursive solution for the density of the parisian stopping time. SIAM J Fin Math 4(1):599–615
Dassios A, Lim JW (2015) An analytical solution for the two-sided Parisian stopping time, its asymptotics and the pricing of Parisian options. Math Fin 00(00):1–17
Dassios A, Lim JW (2016). An extension to the Azéma martingale and drawdown options
Douady R, Shiryaev AN, Yor M (2000) On probability characteristics of ”downfalls” in a standard Brownian motion. Theory Probab Appl 44(1):29–38
Landriault D, Li B, Zhang H (2015) On the frequency of drawdowns for Brownian motion processes. J Appl Probab 52(1):191–208
Landriault D, Li B, Zhang H (2017) On magnitude, asymptotics and duration of drawdowns for Levy models. Bernoulli 23(1):432–458
Taylor H (1975) A stopped Brownian motion formula. Ann Probab 3:234–246
Zhang H, Hadjiliadis O (2012) Drawdowns and speed of market crash. Methodol Comput Appl Probab 14(3):739–752
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Dassios, A., Lim, J.W. An Efficient Algorithm for Simulating the Drawdown Stopping Time and the Running Maximum of a Brownian Motion. Methodol Comput Appl Probab 20, 189–204 (2018). https://doi.org/10.1007/s11009-017-9542-y
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DOI: https://doi.org/10.1007/s11009-017-9542-y