建模高維模型對於現代統計仍然是一項重大挑戰。單維正交方法利用金融選擇權估計狀態價格密度函數(state price densities)（Teng and Liechty，2009;國科會99 -2118-的M -008 -004）的成功，誘發了我們發展從多維標的物的選擇權估計高維的狀態價格密度函數，亦即推廣單維正交模型至高維正交模型。然而，根本的統計問題仍為高維資料的模型建構。近年來，Copula方法引起了實務界與學術界廣大的迴響，Copula的架構利用邊際分佈與 Copula函數建立高維模型。在財務信貸市場，Copula被應用於建立企業之間的聯合違約事件的機率模型。Li (2000) 提出的高斯Copula架構，使的信用衍生性金融產品價格可以被快速計算，也繁榮了信貸市場。但是過於簡單化的模型，間接導致了2008年的金融危機和全球經濟衰退。一個統計的問題是，高斯Copula模型無法正確的描述實際違約事件的高維機率結構。因此，提供完善的高維機率模型是十分迫切的需要，也是本研究的另一個核心。 總而言之，這項二年期的研究計畫，第一年預計擴大單維正交模型到高維正交模型及提出一個靈活的貝氏高維機率模型。第二年的計畫則將第一年發展的高維模型統計方法應用於若干當前重要財務問題，如計算相關違約機率和最佳資產配置。 Modeling high-dimensional data remains a major challenge in modern statistics. Motivated by the success of the univariate Quadrature approach for calibrating State Price Densities using financial options (Teng and Liechty, 2009; NSC 99-2118-M-008-004), this proposal first aims at extending the Quadrature model to high-dimensional cases. However, a fundamental statistics question is the modeling for high-dimensional data. Recently, extensive studies have investigated copulas methods, which build the dependence for high-dimensional data by merging marginal distributions with a copula function. For example, the copula framework is used for modeling joint default events among companies for pricing in the credit market. Although the fast calculation for credit derivatives using Gaussian copula introduced in Li (2000) booms the credit market, this over-simplified copula mechanism indirectly leads to the 2008 financial crisis and a global economic recession. A statistics criticism is that a Gaussian copula does not properly capture the realistic dependence structure. As a result, it is an urgent need to provide promising multivariate models, and this need forms another research core of this research. In summary, this two-year proposal first intends to extend the Quadrature method to high-dimensional cases. Furthermore, this proposal aims at proposing a flexible Bayesian graphical copulas framework for modeling high-dimensional data. In the second year, this proposal will apply the preceding statistical technologies for high-dimensional modeling for several topics in financial applications, such as calculating correlated defaults and portfolio selection. 研究期間：10008 ~ 10107
