A variation of the Azéma martingale and drawdown options

Angelos Dassios, Jia Wei Lim*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

1 Citation (Scopus)


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, Wτ, Mτ), 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.

Original languageEnglish
Pages (from-to)1116-1130
Number of pages15
JournalMathematical Finance
Issue number4
Early online date28 Feb 2019
Publication statusPublished - 1 Oct 2019


  • Azéma martingale
  • Brownian excursions
  • drawdown duration
  • drawdown options
  • local time


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