π 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 non-linear ordinary differential equation (ODE). It turns out that the associated variational inequality has uncountable solutions that satisfy the so-called “principle of smooth fit”, and that identifying the value function relies on the so-called transversality condition in a rather non-trivial way.