美式回望期权定价问题的有限体积法

随着金融市场的不断发展, 期权作为一种能够规避风险的金融衍生产品越来越引起投资者的青睐, 成交量呈逐年上升的趋势, 期权定价问题已经成为金融数学领域中一个重要的研究课题. 本文主要研究Black-Scholes模型下美式回望期权定价问题的数值解法. 美式回望期权定价问题是一个二维非线性抛物问题, 难以直接应用数值方法进行求解. 通过分析该问题的求解难点, 本文给出解决该困难的有效方法. 首先利用计价单位变换将定价问题转换为一维自由边值问题, 并采用Landau's变换将求解区域规范化; 而后针对问题的非线性特点,利用有限体积法和Newton法交替迭代求解期权价格和最佳实施边界, 并对数值解的非负性进行了分析. 最后, 通过与二叉树方法进行比较, 验证了本文方法的正确性和有效性, 为实际应用提供了理论基础.

A finite volume method for pricing the American lookback option

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.