Abstract: The problem of an American option pricing is considered in the financial market model of discrete time. Under the conditions of stock price submitting to the exponential hypothesis we obtain an upper bound of the value of the American option on this stock price by using the martingale method of option pricing. The upper bound is irrelative to the time selected by the holder, and hence it can give the option writer an estimation to assets his average loss.

Key words: option pricing, Black-Scholes mode, martingale approach, American option