Duration of Drawdowns Under Lévy Models

An alternative way to model the time under distress for an asset price is through the length of excursions from its running maximum process, namely, the amount of time it takes for the asset price to revisit the previous peak. Such quantities are of interest to investors who are more concerned about the duration of drawdown than the actual magnitude of drawdowns experienced. In Chapter 6, we study the duration of drawdown for a wide class of L´evy models, by deriving the distribution of a Parisian time of the drawdown process. Given a time threshold b, this Parisian time is defined as the first time when the underlying is continuously below the running maximum for more than b units of time. Our analysis makes use of renewal equations, weak convergence, asymptotics of the magnitude of drawdowns, and some recent work on the asymptotic analysis of the running maximum of L´evy process. We illustrate how our results can be applied to popular models including Brownian motion with drift, spectrally negative α-stable process, spectrally negative Gamma process, and Kou’s jump diffusion.