RESEARCH on PRICING DERIVATIVES BASED ON RISK MEASURES

基于风险度量的衍生品定价研究

• 批准号: 13440029
• 负责人: KUSUOKA Shigeo
• 金额: $ 7.1万
• 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 (B)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-13440029/
• 关键词: Mathematical Finance; Derivatives; Risk measures; Numerical Analysis; Malliavin Calculus; Free Lie Algebra; Stochastic Differential Equations; Runge-KUtta Method; アクチュアリー; 生命保険; 数値解析; ファイナンス; リスク; 法則不変; 多期間モデル; 連続極限; フィルトレーション; ルンゲ・クッタ

This research focused on the pricing of American or European derivatives in the case where the market is not complete or where there exist frictions such as transaction cost, short sale constraint, tax etc., from view point of Asset Liability Management by using risk measures.First, we did research on coherent risk measures proposed by Artzner, Delbaen, Eber and Heath, which is quite practical for banks. We introduced a new notion, law invariance, and characterized law invariant coherent risk measures.Then we did research on a new effective numerical computation method for pricing derivatives.We gave a rigorous proof of the effectiveness of the method proposed by Longstaff-Schwartz such that the value function is approximated with polynomials by the least square method. We also discussed the bound of this method.We also introduced a new numerical computation method for pricing of European derivatives based on Malliavin calculus applied to stochastic differential equations and on free Lie algebra, and we developed this method. The details are the following. We introduced new notions, m-similar Markov operators and m-L similar random variables, and then we gave flexibility of the approximation methods and studied the algebraic structure of iterated stochastic integrals. Also, we used the Runge-Kutta method which is effective in numerical computation in ordinary differential equations, to construct definite m-similar Markov operators and m-L-similar random variables, and we did research on practical algorithm.

本文主要研究在市场不完全或存在交易成本、卖空销售约束、税收等摩擦的情况下，美国或欧洲衍生品的定价问题。本文首先研究了Artzner、Delbaen、Eber和Heath提出的一致性风险度量方法，该方法对银行具有较强的实用性。引入了律不变性的概念，刻画了律不变性的相干风险测度，研究了衍生品定价的一种新的有效的数值计算方法，并对Longstaff-Schwartz提出的用最小二乘法多项式逼近价值函数的方法的有效性给出了严格的证明.我们还讨论了该方法的界，并基于Malliavin演算应用于随机微分方程和自由李代数，提出了一种新的欧式衍生品定价的数值计算方法，并将该方法加以发展。详情如下。引入了m-相似Markov算子和m-L相似随机变量的概念，给出了逼近方法的灵活性，并研究了迭代随机积分的代数结构。利用常微分方程数值计算中行之有效的Runge-Kutta方法，构造了确定的m-相似Markov算子和m-L-相似随机变量，并对实用算法进行了研究。

项目成果

Yoshida, Nakahiro: "Conditional expansions and their applications"Stochastic Processes and their Applications. 107. 53-81 (2003)

Yoshida、Nakahiro：“条件展开及其应用”随机过程及其应用。

Yoshida, Nakahiro: "Malliavin calculus and martingale expansions"Bull. Sci. math.. 125. 431-456 (2001)

吉田中弘：“Malliavin 演算和鞅展开式”Bull。

高橋明彦: "モンテカルロフィルタを用いた金利モデルの推定"統計数理. 50・2. (2003)

高桥明彦：“使用蒙特卡罗滤波器的利率模型估计”统计数学50・2。

Kusuoka, Shigeo: "On a certain Metric on the Space of Pairs of a Random Variable and a Probability Measure"Journal of Mathenatical Sciences University Tokyo. 8. 343-356 (2001)

Kusuoka, Shigeo：“关于随机变量对空间的某个度量和概率测度”东京数学科学大学学报。

Kusuoka, Shigeo: "Laplace Approximations for Diffusion Processes on Torus : Nondegenerate Case"Journal of Mathenatical Sciences University Tokyo. 8. 43-70 (2001)

Kusuoka, Shigeo：“环面扩散过程的拉普拉斯近似：非简并情况”东京数学科学大学学报。