Processes of class Sigma, last passage times and drawdowns

Patrick Cheridito, Ashkan Nikeghbali, Eckhard Platen

Published 2009-10-28Version 1

We propose a general framework to study last passage times, suprema and drawdowns of a large class of stochastic processes. A central role in our approach is played by processes of class Sigma. After investigating convergence properties and a family of transformations that leave processes of class Sigma invariant, we provide three general representation results. The first one allows to recover a process of class Sigma from its final value and the last time it visited the origin. In many situations this gives access to the distribution of the last time a stochastic process hit a certain level or was equal to its running maximum. It also leads to a formula recently discovered by Madan, Roynette and Yor expressing put option prices in terms of last passage times. Our second representation result is a stochastic integral representation of certain functionals of processes of class Sigma, and the third one gives a formula for their conditional expectations. From the latter one can deduce the laws of a variety of interesting random variables such as running maxima, drawdowns and maximum drawdowns of suitably stopped processes. As an application we discuss the pricing and hedging of options that depend on the running maximum of an underlying price process and are triggered when the underlying price drops to a given level or alternatively, when the drawdown or relative drawdown of the underlying price attains a given height.