Constructions of Low Discrepancy Sequences for Computation of Derivative Pricing

用于衍生品定价计算的低差异序列的构造

• 批准号: 13558046
• 负责人: FUJI-HARA Ryoh
• 金额: $ 5.95万
• 依托单位: University of Tsukuba
• 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-13558046/
• 关键词: Low Descrepancy sequence; (T,M,S)-net; (T,S)-sequence; Teta-configuration; derandomization; i-binomial property; digital net; Difference set; i-binominal property; nested design; オプション価格; 準モンテカルロ法; (T,M,S)-net; 有限射影幾何; Theta-Configureation /

When we compute a price of option derivative in financial engineering, an integration over very high dimension is required. We usually use Monte Carlo method for such computation problems with ordinal pseudo random numbers. It is known that a kind of balanced point set is better than random numbers with respect to precisions, called (t,m,s)-nets or (t,s)-sequences.We have studied to construct and analyze such point sets. When the strength t=3, 4, we proved a (t,m,s)-net is equivalent to a configuration, called Theta-configuration, in a finite projective space. Also we proved that a (3,m,s)-net is equivalent to an orthogonal array.There are sequences called Hickrnel sequences which are proposed recently. We define the sequences using polynomials over a finite field and proved it is a (t, s)-sequence.The most uniformly distributed sequences are (0,s)-sequences in (t,s)-sequences. We used to choose parameters randomly. We found a method to choose good parameters. This method is called"De-randomization". Also we proposed an approach using i-binomial property. We analyzed properties of de-randomizations.We studied and improved an ordinal pseudo random number generator, called Mersenne twister, which have very long period. And also we studied many basic combinatorial problems which may have a relation to constructions of (t,m,s)-nets or (t,s)-sequences

在金融工程中计算期权衍生产品的价格时，需要进行高维积分。对于有序伪随机数的计算问题，通常采用蒙特卡罗方法。已知有一类平衡点集在精度上优于随机数，称为（t，m，s）-网或（t，s）-序列，我们研究了这类点集的构造和分析。当强度t=3，4时，我们证明了（t，m，s）-网等价于有限射影空间中的一个构形，称为θ-构形。证明了（3，m，s）-网与正交表是等价的。利用有限域上的多项式定义了这类序列，并证明了它是（t，s）-序列，最均匀分布的序列是（t，s）-序列中的（0，s）-序列。我们过去常常随机选择参数。我们找到了一种方法来选择好的参数。这种方法被称为“去随机化”。此外，我们提出了一种方法，利用i-二项式性质。分析了去随机化的性质，研究并改进了一种具有很长周期的有序伪随机数发生器Mersenne twister。同时，我们还研究了许多与（t，m，s）-网或（t，s）-序列的构造有关的基本组合问题

项目成果

K.Ozawa, M.Jimbo, S.Kageyama and S.Mejza: "Optimality and efficiency of incomplete split-block designs"Metrika. (掲載予定).

K.Ozawa、M.Jimbo、S.Kageyama 和 S.Mejza：“不完整分块设计的最优性和效率”Metrika（待出版）。

T.Adachi, M.Jimbo and S.Kageyama: "Combinatorial structure of group divisible designs without α-resolution classes in each group"Discrete Mathematics. (掲載予定).

T.Adachi、M.Jimbo 和 S.Kageyama：“每组中没有 α 分辨率类的组可分设计的组合结构”离散数学（待出版）。

S.Tezuka: "Toward Derandomization of Owen's Random Scrambling"Dagstuhl Seminar on Algorithm and Complexity for Continuous Promblem. 17 (2002)

S.Tezuka：“走向欧文随机扰乱的去随机化”达格斯图尔连续问题算法和复杂性研讨会。

S.Kageyama and Y.Miao: "Existence of nested designs with block size five"Journal of Statistical Planning and Inference. 94. 249-254 (2001)

S.Kageyama 和 Y.Miao：“块大小为 5 的嵌套设计的存在”统计规划与推理杂志。

M.Mueller, M.Jimbo: "Consecutive positive detecable matrices and group testing for consecutive positives"Discrete Math.. (印刷中). (2004)

M.Mueller、M.Jimbo：“连续阳性可检测矩阵和连续阳性的组测试”离散数学（2004 年出版）。