Abstract
In this work we derive the probability that a rally of a units precedes a drawdown of equal units in a random walk model and its continuous equivalent, a Brownian motion model in the presence of a finite time-horizon. A rally is defined as the difference of the present value of the holdings of an investor and its historical minimum, while the drawdown is defined as the difference of the historical maximum and its present value. We discuss applications of these results in finance and in particular risk management.
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References
Anderson TW (1960) A modification of the sequential probability ratio test to reduce the sample size. Ann Math Stat 31(1):165–197
Chekhlov A, Uryasev S, Zabarankin M (2005) Drawdown measure in portfolio optimization. Int J Theor Appl Financ 8(1):13–58
Feller W (1951) The asymptotic distribution of the range of sums of independent random variables. Ann Math Stat 22(3):427–432
Hadjiliadis O (2005) Change-point detection of two-sided alternative in a Brownian motion model and its connection to the gambler’s ruin problem with relative wealth perception. Ph.D. Thesis, Columbia University, New York
Hadjiliadis O, Vecer J (2006) Drawdowns preceding rallies in a Brownian motion model. J Quant Finance 5(5):403–409
Luenberger DG (1998) Investment science. Oxford University Press, Oxford
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
Pospisil L, Vecer J (2008) Portfolio sensitivities to the changes in the maximum and the maximum drawdown. http://www.stat.columbia.edu/~vecer/portfsens.pdf
Pospisil L, Vecer J, Hadjiliadis O (2009) Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups. Stoch Process their Appl. http://userhome.brooklyn.cuny.edu/ohadjiliadis
Ross S (2008) A first course in probability. Prentice Hall, Englewood Cliffs
Salkuyeh KD (2006) Positive Toeplitz matrices. Int Math Forum 1(22):1061–1065
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
Vallois P (1996) The range of a simple random walk on Z. Adv Appl Probab 28(4):1014–1033
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
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Zhang, H., Hadjiliadis, O. Drawdowns and Rallies in a Finite Time-horizon. Methodol Comput Appl Probab 12, 293–308 (2010). https://doi.org/10.1007/s11009-009-9139-1
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DOI: https://doi.org/10.1007/s11009-009-9139-1