Abstract
In this paper we examine the probabilistic behavior of two quantities closely related to market crashes. The first is the drawdown of an asset and the second is the duration of time between the last reset of the maximum before the drawdown and the time of the drawdown. The former is the first time the current drop of an investor’s wealth from its historical maximum reaches a pre-specified level and has been used extensively as a path-dependent measure of a market crash in the financial risk management literature. The latter is the speed at which the drawdown occurs and thus provides a measure of how fast a market crash takes place. We call this the speed of market crash. In this work we derive the joint Laplace transform of the last visit time of the maximum of a process preceding the drawdown, the speed of market crash, and the maximum of the process under general diffusion dynamics. We discuss applications of these results in the pricing of insurance claims related to the drawdown and its speed. Our applications are developed under the drifted Brownian motion model and the constant elasticity of variance (CEV) model.
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References
Borodin AN, Salminen P (2002) Handbook of Brownian motion—facts and formulae. Basel-Boston-Berlin, Birkhäuser Verlag
Carr P, Madan DB (1999) Option valuation using the fast Fourier transform. J Comput Financ 2(4):61–73
Carr P, Zhang H, Hadjiliadis O (2011) Maximum drawdown insurance. Int J Theor Appl Financ (to appear). http://userhome.brooklyn.cuny.edu/ohadjiliadis/maximumdrawdown-1.pdf
Chekhlov A, Uryasev S, Zabarankin M (2005) Drawdown measure in portfolio optimization. Int J Theor Appl Financ 8(1):13–58
Cvitanic J, Karatzas I (1995) On portfolio optimization under drawdown constraints. IMA Lecture Notes in Mathematical Applications, vol 65, pp 77–78
Douady R, Shiryaev A, Yor M (2000) On probability characteristics of downfalls in a standard Brownian motion. Theory Prob Appl 44:29–38
Graversen SE, Shiryaev A (2000) An extension of P. Lévy’s distributional properties to the case of a Brownian motion with drift. Bernoulli 6:615–620
Grossman SJ, Zhou Z (1993) Optimal investment strategies for controlling drawdowns. Math Financ 3(3):241–276
Hadjiliadis O, Vecer J (2006) Drawdowns preceding rallies in a Brownian motion model. J Quant Financ 5(5):403–409
Jeanblanc M, Yor M, Chesney M (2009) Mathematical methods for financial markets. Springer-Verlag, London
Jeulin T (1980) Semi-martingales et grossissement d’une filtration. Lecture notes in mathematics, vol 833, Springer-Verlag, Berlin
Jeulin T, Yor M (1978) Grossissement d’une filtration et semi-martingales: formules explicites. Lecture notes in mathematics, vol 649. Springer-Verlag, Berlin, pp 78–97
Jeulin T, Yor M (1985) Grossissements de filtrations: exemples et applications. Lecture notes in mathematics, vol 1118. Springer-Verlag, Berlin
Karatzas I, Shreve SE (1991) Stochastic calculus and Brownian motion. Springer-Verlag, New York
Kardaras C (2010) On random times. arXiv:1007.1124
Lebedev NN (1965) Special functions and their applications. Prentice Hall, New Jersey
Lehoczky JP (1977) Formulas for stopped diffusion processes with stopping times based on the maximum. Ann Probab 5(4):601–607
Magdon-Ismail M, Atiya A (2004) Maximum drawdown. Risk 17(10):99–102
Magdon-Ismail M, Atiya A, Pratap A, Abu-Mostafa Y (2004) On the maximum drawdown of Brownian motion. J Appl Probab 41(1):147–161
McKean HP (1956) Elementary solutions for certain parabolic partial differential equations. Trans Am Math Soc 82(2):519–548
Meilijson I (2003) The time to a given drawdown in Brownian motion. Seminaire de probabilités XXXVII, pp 94–108
Nikeghbali A (2006) A class of remarkable submartingales. Stoch Process Their Appl 116(6):917–938
Pospisil L, Vecer J (2010) Portfolio sensitivities to the changes in the maximum and the maximum drawdown. Quantitative Finance 10(6):617–627
Pospisil L, Vecer J, Hadjiliadis O (2009) Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups. Stoch Process Their Appl 119(8):2563–2578
Protter PE (2005) Stochastic integration and differential equations. Springer-Verlag, New York
Revuz D, Yor M (1999) Continuous martingales and Brownian motion. Springer-Verlag, New York
Salminen P, Vallois P (2007) On maximum increase and decrease of Brownian motion. Ann Inst Henri Poincaré B Probab Stat 43(6):655–676
Sornette D (2003) Why stock markets crash: critical events in complex financial systems. Princeton University Press, Princeton
Tanré E, Vallois P (2006) Range of Brownian motion with drift. J Theor Probab 19(1):45–69
Taylor HM (1975) A stopped Brownian motion formula. Ann Probab 3(2):234–246
Vecer J (2006) Maximum drawdown and directional trading. Risk 19(12):88–92
Vecer J (2007) Preventing portfolio losses by hedging maximum drawdown. Wilmott 5(4):1–8
Zhang H, Hadjiliadis O (2010) Drawdowns and rallies in a finite time-horizon. Methodol Comput Appl Probab 12(2):293–308
Zhang H, Hadjiliadis O (2011) Formulas for the Laplace transform of stopping times based on drawdowns and drawups. Stoch Process Their Appl. arXiv:0911.1575
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This research was supported by the NSF grant number 0929317.
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Zhang, H., Hadjiliadis, O. Drawdowns and the Speed of Market Crash. Methodol Comput Appl Probab 14, 739–752 (2012). https://doi.org/10.1007/s11009-011-9262-7
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DOI: https://doi.org/10.1007/s11009-011-9262-7