On the drawdown of completely asymmetric Levy processes

TL;DR

This paper derives explicit formulas for the distribution of key drawdown-related functionals of completely asymmetric Lévy processes, enhancing understanding of their extremal behavior in financial modeling.

Contribution

It provides explicit expressions for the joint distribution of first-passage times and related functionals of drawdowns in asymmetric Lévy processes, using scale functions and Lévy measures.

Findings

Explicit formulas for the law of first-passage times of drawdowns.

Distribution of the last supremum before drawdown crossing.

Law of drawdowns and rallies in exponential Lévy models.

Abstract

The {\em drawdown} process $Y$ of a completely asymmetric L\'{e}vy process $X$ is equal to $X$ reflected at its running supremum $\bar{X}$: $Y = \bar{X} - X$. In this paper we explicitly express in terms of the scale function and the L\'{e}vy measure of $X$ the law of the sextuple of the first-passage time of $Y$ over the level $a>0$, the time $\bar{G}_{\tau_a}$ of the last supremum of $X$ prior to $\tau_a$, the infimum $\unl X_{\tau_a}$ and supremum $\ovl X_{\tau_a}$ of $X$ at $\tau_a$ and the undershoot $a - Y_{\tau_a-}$ and overshoot $Y_{\tau_a}-a$ of $Y$ at $\tau_a$. As application we obtain explicit expressions for the laws of a number of functionals of drawdowns and rallies in a completely asymmetric exponential L\'{e}vy model.