Drawdowns and the Speed of Market Crash
 Hongzhong Zhang,
 Olympia Hadjiliadis
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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 prespecified level and has been used extensively as a pathdependent 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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 Title
 Drawdowns and the Speed of Market Crash
 Journal

Methodology and Computing in Applied Probability
Volume 14, Issue 3 , pp 739752
 Cover Date
 20120901
 DOI
 10.1007/s1100901192627
 Print ISSN
 13875841
 Online ISSN
 15737713
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Drawdown
 Speed of market crash
 Diffusions
 Drawdown insurance
 Primary 60G40; Secondary 91A60
 Industry Sectors
 Authors

 Hongzhong Zhang ^{(1)}
 Olympia Hadjiliadis ^{(2)}
 Author Affiliations

 1. Department of Statistics, Columbia University, 1255 Amsterdam ave, Room SSW 1010, New York, NY, 10027, USA
 2. Department of Mathematics, Brooklyn College and the Graduate Center C.U.N.Y., Brooklyn, NY, USA