On the drawdown of completely asymmetric Lévy processes

Abstract. The drawdown process Y = X − X of a completely asymmetric Lévy process X is given by X reflected at its running supremum X. In this paper we explicitly express the law of the sextuple (τa, Gτa, Xτa, Xτa, Yτa−, Yτa − a) in terms of the scale function and the Lévy measure of X, where τa denotes the first-passage time of Y over the level a> 0, Gτa is the time of the last supremum of X prior to τa and X is the running infimum of X. We also explicitly identify the distribution of the drawup Ŷτa at the moment τa, where Y ̂ = X −X, and derive the probability of a large drawdown preceding a small rally. These results are applied to the Carr & Wu [10] model for S&P 500. 1