Occupation Times Related to Drawdowns

The amount of time when the underlying price is in distress can characterize investment risks from a different angle. In Chapter 5, we propose to study a number of occupation times that are related to the drawdowns and drawups of a linear diffusion process. To that end, we derive analytical formulas for the Laplace transform of the occupation time of the underlying below a fixed threshold y upon leaving a given finite interval. We then use Lehoczky’s approximation technique and progressive enlargement of filtration to extend these results to occupation times of the drawdown or drawup process. In special cases of drifted Brownian motion and threedimensional Bessel process, our results imply that the occupation time of the drawdown process has the same distribution as the first passage time of the drawdown process. We also demonstrate how our analytical results can be used for computing probabilities of drawdown events under Omega default models, and for evaluation of Parisian-like or α-quantile options written on the drawdowns.