Stochastic Model of Floating Interest rate and positive research

浮动利率随机模型及实证研究

• 批准号: 09440074
• 负责人: KUSUOKA Shigeo
• 金额: $ 7.94万
• 依托单位: ATHE 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: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440074/
• 关键词: floating interest rate; interest rate model; default risk; mathematical finance; diffusion model; 確率偏微分方程式; ハザード率; 確率解析; 数値計算; マリアバン・カリキュラス; リー環; 確率テイラー展開; デリバディブ; 変動金利モデル; クレジットデリバティブ; ハザードレート; 確率変動; 確率偏微分方程

The purpose of this research was to do theoretical and positive research on stochastic partial differential equation models begun recently. We were planning to think of only government bond or LIBOR which are supposed to be riskless. However, quite recently people got interested in so-called credit derivatives concerning defaultable bonds. So we slightly changed our plan and thought of model containing default probability.First we showed the existence and uniqueness of solution to stochastic partial differential equations related to interest rate under certain conditions for boundary conditions and coefficients. Also, we gave a condition so that the solution remains in positive range.Next we did theoretical research on prices of defaultable bonds. Here we gave a counter-example for a formula which people widely believed on the relationship between hazard rate processes and conditional default probabilities. Also, we gave a formula on hazard rate process in continuous-time filtering models.In mathematical finance, practically important formula for prices are given in terms of expectations. In estimates of statistical parameters, it is important to compute such expectations precisely and rapidly. We introduced a new numerical computation method in diffusion models, which are rather restrictive but widely used, and we showed that it is quite effective theoretically. In computing hedging strategies, we need more complicated expectations, but the research of them are postponed to the future.The statistical consideration for stochastic process models will be getting important more and more. Our plan contained the construction of a theory for it. In this respect we only got an asymptotic expansion formula related to convergence of probability law of additive functionals for hypo-elliptic diffusion processes.

本研究的目的是对近年来兴起的随机偏微分方程模型进行理论和实证研究。我们计划只考虑政府债券或LIBOR，它们应该是无风险的。然而，最近人们对所谓的信用衍生品产生了兴趣，这些衍生品涉及可违约债券。因此，我们对含违约概率的模型稍微改变了一下思路和设想：首先，在一定的边界条件和系数条件下，证明了与利率相关的随机偏微分方程解的存在唯一性。其次，我们对可违约债券的价格进行了理论研究。本文给出了人们普遍认为的风险率过程与条件违约概率之间关系的一个公式的反例。在连续时间滤波模型中，我们给出了风险率过程的一个公式。在金融数学中，实际上重要的价格公式是用期望值表示的。在统计参数的估计中，精确和快速地计算这样的期望值是很重要的。本文介绍了一种新的扩散模型的数值计算方法，这是一种限制性很强但应用广泛的扩散模型，并从理论上证明了它的有效性。在套期保值策略的计算中，我们需要更复杂的期望值，但对它们的研究已经推迟到将来，随机过程模型的统计考虑将越来越重要。我们的计划包含了一个理论的建立，在这方面，我们只得到了一个与次椭圆扩散过程的可加泛函的概率律的收敛性有关的渐近展开式。

项目成果

S. Kusuoka: "Laplace Approximations for sums of independent random vectors"Probability Theory and Related Field. (発表予定).

S. Kusuoka：“独立随机向量之和的拉普拉斯近似”概率论和相关领域（待提交）。

S. Kusuoka: "Term Structure and SPDE"Advances in Mathematical Economics. vol. 2. 67-85 (2000)

S. Kusuoka：“期限结构和 SPDE”数学经济学进展。

S. Kusuoka: "Term Structure and SPDE"Advances in Mathematical Finance. 2. 67-85 (2000)

S. Kusuoka：“期限结构和 SPDE”数学金融的进展。

A. Takahashi: "An asymptotic expansion approach to pricing financial contingent claims"Asia-Pasific Financial Markets. 6. 115-151 (1999)

A. Takahashi：“金融或有债权定价的渐近扩张方法”亚太金融市场。

S. Kusuoka: "A Remark on defalt risk models"Advances in Mathematical Finance. 1. 69-82 (1999)

S. Kusuoka：“关于违约风险模型的评论”数学金融的进展。