Randomized Stopping Times and Early Exercise for American Derivatives in Dry Markets

Abstract

This paper studies the impact of dry markets for underlying assets on the optimal stopping time and optimal exercise policy of American derivatives. We consider that the underlying is transacted at all points in time except for a subset of dates, for which there is an exogenous probability that trading may exist. Using superreplicating strategies, we derive expectation representations for the range of arbitrage-free values of the derivatives. For arbitrary probability, an enlarged filtration jointly induced by the price process and the market existence process makes ordinary stopping times sufficient to characterize such representation. For the deterministic case where the probability is zero, randomized stopping times are required. Several comparisons of the ranges obtained with the two market restrictions are performed. Finally, we conclude that market incompleteness caused by dryness may delay the optimal exercise of American derivatives.