Formulas for the Laplace Transform of Stopping Times based on Drawdowns and Drawups

TL;DR

This paper derives formulas for the Laplace transform of the distribution of drawdowns and drawups in diffusion processes, with applications in financial risk management and signal detection.

Contribution

It provides a closed-form Laplace transform formula for drawdowns and drawups, including a specific case for drifted Brownian motion, advancing analysis of diffusion process extremes.

Findings

Closed-form Laplace transform for drawdowns and drawups

Explicit density formula for drifted Brownian motion

Applications in financial risk and signal detection

Abstract

In this work we study drawdowns and drawups of general diffusion processes. The drawdown process is defined as the current drop of the process from its running maximum, while the drawup process is defined as the current increase over its running minimum. The drawdown and the drawup are the first hitting times of the drawdown and the drawup processes respectively. In particular, we derive a closed-form formula for the Laplace transform of the probability density of the drawdown of a units when it precedes the drawup of b units. We then separately consider the special case of drifted Brownian motion, for which we derive a closed form formula for the above-mentioned density by inverting the Laplace transform. Finally, we apply the results to a problem of interest in financial risk-management and to the problem of transient signal detection and identification of two-sided changes in the drift of general diffusion processes.