10-FEB-2024
The Russian option, introduced by Shepp and Shiryaev (1993), is a perpetual American lookback contract whose payoff is the running maximum of the underlying asset. We price it via two independent frameworks: a Hamilton-Jacobi-Bellman (HJB) variational inequality and a singly reflected backward stochastic differential equation (BSDE) with barrier $M_t$. The HJB approach yields a closed-form solution through a scalar Euler ODE, while the BSDE Bellman iteration — solved by Gauss-Hermite quadrature — converges to the same value function with mean relative error $0.10\%$. A third result ties the two together: the expected stopping time $u(x) = \mathbb{E}[\tau^* \mid X_0 = x]$ satisfies a companion Poisson equation $\mathcal{L}u = -1$ under the same operator and boundary conditions, and its closed form reveals that the option premium equals, to leading order, the discount rate times the expected wait times the current value.