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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.
With the continuous development of financial market, options as a kind of financial derivative products that can evade risks are attracting more and more investors. The transaction volume has been increasing year by year. The option pricing problem has become an important research topic in the field of financial mathematics This paper mainly studies the numerical solution of American lookback option pricing under the Black-Scholes model.American lookback option pricing problem is a two-dimensional nonlinear parabolic problem that can not be directly solved by numerical methods.By analyzing the difficulty of solving this problem, In this paper, an effective method to solve this problem is given. Firstly, the pricing problem is transformed into a one-dimensional free-boundary problem by using the unit-of-change-unit transformation. Landau’s transformation is used to normalize the solution region. Then, based on the nonlinear characteristics of the problem, Method iteratively solves the option price and the optimal implementation boundary and analyzes the nonnegativeness of the numerical solution.Finally, by comparing with the binary tree method, the correctness and validity of the method in this paper are verified, which provides a theory for practical application basis.