Optimal Double Stopping Problems for Maxima and Minima of Geometric Brownian Motions

Abstract We present closed-form solutions to some double optimal stopping problems with payoffs representing linear functions of the running maxima and minima a geometric Brownian motion. It is shown that times are th first at which underlying process reaches lower or upper stochastic boundaries depending on current values its maximum minimum. The proof based reduction original sequences single for resulting three-dimensional continuous Markov process. latter solved as equivalent free-boundary by means smooth-fit normal-reflection conditions value edges state space. show determined extremal associated first-order nonlinear ordinary differential equations. obtained results related valuation perpetual real lookback options floating sunk costs in Black-Merton-Scholes model.