Perpetual American Compound Fixed-Strike Lookback Options on Maxima Drawdowns

Abstract We present closed-form solutions to the problem of pricing of the perpetual American compound lookback options on the maximum drawdown with fixed strikes in the Black-Merton-Scholes model. It is shown that the optimal exercise times are the first times at which the underlying risky asset price process reaches either lower or upper stochastic boundaries depending on the current values of its running maximum and maximum drawdown processes. The proof is based on the reduction of the original double optimal stopping problem to a sequence of two single optimal stopping problems for the resulting three-dimensional continuous Markov process. The latter problems are solved as the equivalent free-boundary problems by means of the smooth-fit and normal-reflection conditions for the value functions at the optimal stopping boundaries and at the edges of the three-dimensional state space. We show that the optimal exercise boundaries are determined as the maximal and minimal solutions to the appropriate first-order nonlinear ordinary differential equations.