Perpetual Options for Lévy Processes in the Bachelier Model

A solution to the optimal stopping problem $V(x)=\sup_\tau\mathsf Ee^{-\delta\tau}g(x+X_\tau)$ is given, where $X=\{X_t\}_{t\ge 0}$ is a Lévy process, $\tau$ is an arbitrary stopping time, $\delta\ge 0$ is a discount rate, and the reward function $g$ takes the form $g_c(x)=(x-K)^+$ or $g_p(x)=(K-x)^+$. The results interpreted as option prices of perpetual options in Bachelier's model are expressed in terms of the distribution of the overall supremum in the case $g=g_c$ and overall infimum in the case $g=g_p$ of the process $X$ killed at rate $\delta$. Closed-form solutions are obtained under mixed exponentially distributed positive jumps with arbitrary negative jumps for $g_c$ and under arbitrary positive jumps and mixed exponentially distributed negative jumps for $g_p$. In the case $g=g_c$, a prophet inequality comparing the prices of perpetual look-back call options and perpetual call options is obtained.