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.
Highlights
Derivatives were originally priced by assuming that markets were complete, computing the value of a derivative as the value of a self-financing, and replicating portfolio on the underlying risky asset and risk-free bond
This paper studies the impact of dry markets for underlying assets on the optimal stopping time and optimal exercise policy of American derivatives
The upper bound for the value of an American derivative is the maximum value for which the derivative would be traded without allowing for arbitrage opportunities
Summary
Derivatives were originally priced by assuming that markets were complete, computing the value of a derivative as the value of a self-financing, and replicating portfolio on the underlying risky asset and risk-free bond. A rational exercise policy may even not be well defined if the state-price deflator depends on the exercise policy This is their argument for using optimal randomized stopping times when characterizing the superreplication bounds of American derivatives under proportional transaction costs. Able to show that this type of market incompleteness may delay the optimal exercise of American derivatives as compared to the case where the underlying asset may be traded at every point in time. Consider at any node ( j, t ) the portfolio constituted by ∆ ( j, t ) shares of the underlying asset and an amount B ( j, t ) invested in the bond Such portfolio is denoted by ∆ ( j,t ), B ( j,t ) and its value process is given by. In order to do that, we first define some necessary mathematical objects
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