Perpetual Options for Lévy Processes in the Bachelier Model

Solution to the optimal stopping problem V (x) = sup τ Eeg(x+Xτ ) is given, where X = {Xt}t≥0 is a Lévy process, τ is an arbitrary stopping time, δ ≥ 0 is a discount rate, and the reward function g takes the form gc(x) = (x−K) or gp(x) = (K−x) Results, interpreted as option prices of perpetual options in Bachelier’s model are expressed in terms of the distribution of the overall supremum in case g = gc and overall infimum in case g = gp of the process X killed at rate δ. Closed form solutions are obtained under mixed exponentially distributed positive jumps with arbitrary negative jumps for gc, and under arbitrary positive jumps and mixed exponentially distributed negative jumps for gp. In case g = gc a prophet inequality comparing prices of perpetual look-back call options and perpetual call options is obtained.