Pricings and hedgings of the perpetual Russian options

In this paper, we study the optimal stopping time and the optimal stopping boundary for the perpetual Russian option under the diffusion process. The general continuation region is characterized by a function b(p,t) depending on both variables t and the maximum value of the stock and initial starting value P0. Previous studies assume that the continuation region is given by a function depending upon the time t only. This is unreal hypothesis for the diffusion to achieve. Our result shows that the perpetual Russian option can be described by a Black–Scholes equation over the continuation region and smooth boundary conditions on the optimal stopping boundary. Furthermore, we develop an evaluation method from the lookback option on a stopping time, and establish the Greek letters for the perpetual Russian option. We obtain the exact upper bound for the prices of the perpetual Russian options and demonstrate that both the payoff and the optimal stopping time are path-dependent by Monte Carlo simulations.