A variation of the Azéma martingale and drawdown options. (28th February 2019)

• Record Type: Journal Article
• Authors: Dassios, Angelos; Lim, Jia Wei
• Is Part Of: Mathematical finance. Volume 29:Number 4(2019)
• Page Start: 1116
• Page End: 1130
• Publication Date: 2019-02-28
• Subjects: Azéma martingale -- Brownian excursions -- drawdown duration -- drawdown options -- local time; Business mathematics -- Periodicals; 332
• DOI: 10.1111/mafi.12202
• Languages: English
• ISSNs: 0960-1627

Abstract

Abstract: In this paper, we derive a variation of the Azéma martingale using two approaches—a direct probabilistic method and another by projecting the Kennedy martingale onto the filtration generated by the drawdown duration. This martingale links the time elapsed since the last maximum of the Brownian motion with the maximum process itself. We derive explicit formulas for the joint densities of ( τ, W τ, M τ ), which are the first time the drawdown period hits a prespecified duration, the value of the Brownian motion, and the maximum up to this time. We use the results to price a new type of drawdown option, which takes into account both dimensions of drawdown risk—the magnitude and the duration.