04-SEP-2023
The Russian option, introduced by Shepp and Shiryaev (1993), is a perpetual American option whose payoff is the running maximum $M_\tau = \sup_{s \leq \tau} X_s$ of the underlying stock price, discounted to the stopping time $\tau$. We analyse the perpetual problem for a geometric Brownian motion with drift $\mu < r$, and exploit the homogeneity of the payoff to reduce the two-dimensional state space $(X_t, M_t)$ to the one-dimensional ratio process $Y_t = X_t/M_t \in (0,1]$. In the continuation region $(y^*, 1)$ the reduced value function $v(y)$ solves an Euler--Cauchy ODE whose general solution is $v(y) = Ay^{\beta_+} + By^{\beta_-}$, and the four constants $(A, B, y^*)$ are determined by a Neumann reflection condition at $y = 1$ and smooth-pasting conditions at $y^*$. We present Peskir's maximality principle, which characterises $y^*$ as the largest candidate threshold for which the ODE solution dominates the obstacle, and we identify the two Doob--Meyer compensators of the discounted value process as the local times of $Y$ at $y^*$ and at the reflecting barrier $y = 1$.