In this paper we study the Russian option, which is a non-standard path-dependent option of American type. There is no fixed exercise date since it is a perpetual option. The owner of a Russian option receives the maximum value that the price of the underlying asset (a stock paying a continuous dividend yield) has ever achieved until whatever stopping time (which can be indefinitely long) he may select. The Russian option is also said to be a no regret option: the holder may look at the fluctuations only occasionally and has no regret if he did not exercise the option at an earlier time. We note that the Russian option is not currently traded in any existing market despite its appeal for the holder. Shepp and Shiryaev (1993) give a closed form solution for the fair price (the optimal expected present value of the payoff) of the Russian option and the unique optimal exercise strategy, under the assumption that the asset fluctuations follow the Black-Scholes exponential Brownian motion model. Building on the Shepp and Shiryaev analysis, we present some applications of the pricing formula and the sensitivity analysis for the option price. A purpose of this paper is to show that the price of a Russian option can also be estimated using simple Monte Carlo simulation techniques; in this way we are also able to give an estimation of the mean stopping time.