An Asymptotic Expansion Approach to Numerical Problems in Finance

金融数值问题的渐近展开法

基本信息

批准号：
13680509
负责人：
TAKAHASHI Akihiko
金额：
$ 1.98万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

项目摘要

1.We developed mathematical validity of the asymptotic expansion approach to valuation of contingent claims in Markovian and non-Markovian setting. ("On Validity of the Asymptotic Expansion Approach in Contingent Claim Analysis," Annals of Applied Probability(2003).)2.We derived numerically tractable formulas of optimal portfolio in the dynamic investment problems when state variables follow general diffusion processes. Moreover, we provided analytic approximation formulas based on the asymptotic expansion approach and showed their effectiveness through numerical examples. ("An Asymptotic Expansion Scheme for the Optimal Investment Problems," forthcoming in Statistical Inference for Stochastic Processes.)3.We showed a decomposition of the value of an American option into the corresponding European value and the early exercise premium when the underlying price follow a general diffusion process. Then, we applied the asymptotic expansion approach to develop a semi-analytic computational scheme. ("An Asymptotic Expansion Approach to American Options," Monetary and Economic Studies(2003).)4.We proposed a new computational scheme with the asymptotic method to achieve variance reduction of Monte Carlo simulation for numerical analysis in finance. We provided general scheme and mathematical validity of our method. The examples of the application include pricing options under jump-diffusion processes, pricing interest rate derivatives in HJM framework and computing optimal portfolios in the dynamic investment problems. ("Applications of the Asymptotic Expansion Approach based on Malliavin-Watanabe Calculus in Financial Problems," forthcoming in Stochastic Processes and Applications to Mathematical Finance.)5.We published a book on the asymptotic expansion approach to finance.("Foundation of Mathematical Finance-Application of Malliavin Calculus and Asymptotic Expansion Method-," (2003).)

1.我们开发了马尔可夫和非马尔可夫环境中或有债权估值的渐近展开方法的数学有效性。（“论或有债权分析中渐近展开法的有效性”，《应用概率年鉴》（2003）。） 2.当状态变量遵循一般扩散过程时，我们推导了动态投资问题中最优投资组合的数值可处理公式。此外，我们提供了基于渐近展开方法的解析近似公式，并通过数值例子证明了其有效性。（“最优投资问题的渐近扩展方案”，即将发表于《随机过程统计推断》中。）3.我们展示了当标的价格遵循以下条件时，将美式期权的价值分解为相应的欧式期权价值和早期履约溢价一般的扩散过程。然后，我们应用渐近展开方法来开发半解析计算方案。（“美国期权的渐近扩展方法”，《货币与经济研究》（2003）。）4.我们提出了一种新的渐近方法计算方案，以实现金融数值分析中蒙特卡罗模拟的方差减少。我们提供了我们方法的一般方案和数学有效性。应用示例包括跳跃扩散过程下的期权定价、HJM框架下的利率衍生品定价以及动态投资问题中的最优投资组合计算。（“基于 Malliavin-Watanabe 微积分的渐近展开方法在金融问题中的应用”，即将发表于《随机过程及其在数学金融中的应用》。）5.我们出版了一本关于金融渐近展开方法的书。（《数学金融基础》 -Malliavin 微积分和渐近展开法的应用-，”（2003）。）