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A variation of the Azéma martingale and drawdown options
Author(s) -
Dassios Angelos,
Lim Jia Wei
Publication year - 2019
Publication title -
mathematical finance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.98
H-Index - 81
eISSN - 1467-9965
pISSN - 0960-1627
DOI - 10.1111/mafi.12202
Subject(s) - drawdown (hydrology) , martingale (probability theory) , mathematics , brownian motion , econometrics , local martingale , probabilistic logic , martingale representation theorem , geometric brownian motion , statistics , economics , diffusion process , geology , geotechnical engineering , economy , service (business) , aquifer , groundwater
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.