Pi options

We consider a discretionary stopping problem that arises in the context of pricing a class of perpetual American-type call options, which include the perpetual American, Russian and lookback-American call options as special cases. We solve this genuinely two-dimensional optimal stopping problem by means of an explicit construction of its value function. In particular, we fully characterise the free-boundary that provides the optimal strategy, and which involves the analysis of a highly nonlinear ordinary differential equation (ODE). In accordance with other optimal stopping problems involving a running maximum process that have been studied in the literature, it turns out that the associated variational inequality has an uncountable set of solutions that satisfy the so-called principle of smooth fit.