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Processes of Class Sigma, Last Passage Times, and Drawdowns

Author(s): Cheridito, Patrick; Nikeghbali, Ashkan; Platen, Eckhard

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dc.contributor.authorCheridito, Patrick-
dc.contributor.authorNikeghbali, Ashkan-
dc.contributor.authorPlaten, Eckhard-
dc.identifier.citationCheridito, Patrick, Ashkan Nikeghbali, and Eckhard Platen. "Processes of class sigma, last passage times, and drawdowns." SIAM Journal on Financial Mathematics 3, no. 1 (2012): 280-303. doi:10.1137/09077878Xen_US
dc.description.abstractWe propose a general framework for studying last passage times, suprema, and drawdowns of a large class of continuous-time stochastic processes. Our approach is based on processes of class Sigma and the more general concept of two processes, one of which moves only when the other is at the origin. After investigating certain transformations of such processes and their convergence properties, we provide three general representation results. The first allows the recovery of 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 attains a certain level or is equal to its running maximum. It also leads to recently discovered formulas expressing option prices in terms of last passage times. Our second representation result is a stochastic integral representation that will allow us to price and hedge options on the running maximum of an underlying that are triggered when the underlying drops to a given level or, alternatively, when the drawdown or relative drawdown of the underlying attains a given height. The third representation gives conditional expectations of certain functionals of processes of class Sigma. It can be used to deduce the distributions of a variety of interesting random variables such as running maxima, drawdowns, and maximum drawdowns of suitably stopped processes.en_US
dc.format.extent280 - 303en_US
dc.relation.ispartofSIAM Journal on Financial Mathematicsen_US
dc.rightsAuthor's manuscripten_US
dc.titleProcesses of Class Sigma, Last Passage Times, and Drawdownsen_US
dc.typeJournal Articleen_US

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