Valuing finite-lived Russian options

This paper deals with the valuation of the Russian option with finite time horizon in the framework of the Black–Scholes–Merton model. On the basis of the PDE approach to a parabolic free boundary problem, we derive Laplace transforms of the option value, the early exercise boundary and some hedging parameters. Using Abelian theorems of Laplace transforms, we characterize the early exercise boundary at a time to close to expiration as well as the well-known perpetual case in a unified way. Furthermore, we obtain a symmetric relation in the perpetual early exercise boundary. Combining the Gaver–Stehfest inversion method and the Newton method, we develop a fast algorithm for computing both the option value and the early exercise boundary in the finite time horizon.