Discounted optimal stopping zero-sum games in diffusion-type models with maxima and minima

We present closed-form solutions to a discounted optimal stopping zero-sum game in a model with a generalised geometric Brownian motion with coecients depending on its running maximum and minimum processes. The optimal stopping times forming a Nash equilibrium are shown to be the rst times at which the original process hits certain boundaries depending on the running values of the associated maximum and minimum processes. The proof is based on the reduction of the original game to the equivalent free-boundary problem and the solution of the latter problem by means of the smooth- t and normal-re ection conditions. We show that the optimal stopping boundaries are partially determined as either unique solutions of the appropriate system of arithmetic equations or unique solutions of the appropriate rst-order nonlinear ordinary dierential equations. The obtained results are related to the valuation of the perpetual lookback game options with oating strikes in the appropriate diusion-type extension of the Black- Merton-Scholes model