Magnitude and Speed of Consecutive Market Crashes in a Diffusion Model

In this paper, in a time-homogeneous diffusion setting, we study a sequence of last passage times of generalized drawdown processes before the first passage time of another monitoring generalized drawdown process. These quantities are closely related to consecutive small market downward movements before a final big market crash modeled by the first passage time of the monitoring generalized drawdown process. Our theoretical framework simultaneously incorporates the first passage time, the absolute and relative drawdown times, and generalizes the scope of (Zhang and Hadjiliadis Methodol Comput Appl Probab 14(8):739–752 2012). We explicitly determine the path decompositions bridging these random times, the stochastic differential equations (SDE) governing each of the path fragments, and derive the Laplace transform of durations between any two of these random times. Applications include designing various risk measures summarizing several market information (e.g. historical maximum/minimum, absolute drawdown, relative drawdown) altogether. They allow us to study, in a microscopic way, a sequence of smaller drawdowns (signals) before a big drawdown (market crash) actually realizes. They can have applications to the detection of a potential market crash from signals of market drawdowns. We illustrate our results using a drifted Brownian motion.