In this work we derive analytical formulas for the joint distribution of the drawdown, the last visit time of the maximum of a process preceding the drawdown and the maximum of the process under general diffusion dynamics.
The initial motivation of this work arises in the financial risk management of drawdowns. Drawdowns measure the first time the current drop of an investor’s wealth from its historical maximum reaches a pre-specified level. Thus, drawdowns capture the time of distress and have as such been used as path dependent measures of risk. However, in order to encapsulate the time of distress one needs to measure the duration of time between the drawdown and the last time at which the maximum was achieved. We call this time the speed of market crash and study its distribution under general diffusion dynamics in detail. We further examine the sensitivity of the speed of market crash to the drift parameter of a drifted Brownian motion model. Our results suggest that the speed of market crash can serve as sensitive online estimator of changes in the drift which can be used in algorithmic trading. We finally discuss the connection the drawdown and its speed to the Cumulative sum statistic and its speed of reaction and in particular to the problem of optimal quickest detection and identification of a drift. ( Slides)
Speaker: Olympia Hadjiliadis, Department of Mathematics, Graduate Center of CUNY