Optimal stopping problems and pricing of american exotic options

Abstract

We find closed form formulas for the value functions of the several types of American lookback spread option through solving optimal stopping problems of running maximum and minimum of an underlying process. First of all, we provide formulas for fixed strike, floating strike lookback options and lookback spread option whose payoff is the difference of the running maximum and minimum prices of a single asset. Therefore, an optimal stopping problem related to both maximum and minimum is posed. Using the monotonicity of the maximum and minimum processes we show that the spread option is equivalent to some fixed strike options on some domains, find the exact form of the optimal stopping region, and obtain the solution of the resulting partial differential equations. We also solve an optimal stopping problem whose gain is the ratio between the maximum and minimum. Finally, we address the optimal selling strategy for an investor who has regret on the maximum value.