Due to the characteristic of risk aversion, option has become one of the most fashionable derivatives in the financial field. More and more investigators are attracted to devote themselves to exploring the option pricing problem. In this paper, we are concerned with the valuation of American lookback options in terms of the Black-Scholes model. It is well known that the American lookback option satisfies a two-dimensional nonlinear partial differential equation in an unbounded domain, which couldn't be numerically solved directly. Based on the analysis of the issues for solving this problem, this paper introduces an approach to settle it. First, we transform the problem into a one-dimensional form by the numeraire transformation. And then, the Landau's transformation is applied to normalize the defined domain. For the nonlinear feature of the resulting problem, we propose a finite volume method coupled with Newton iterative method to obtain the optional value and the optimal exercise boundary simultaneously. We also give a proof on the nonnegativity of the numerical solutions under some appropriate assumptions. Finally, some numerical simulations are presented using the proposed method in this paper. Comparing with the binomial method, we can conclude that the proposed method is an effective one, which provides a theoretical basis for practical applications.