The Copula-GARCH Model: Application to Variable Annuity Guarantee Valuations on Multiple Assets

Copula-GARCH模型在多資產標的變額年金評價上的應用

宣葳(Wei Hsuan) ； 張士傑(Shih-Chieh Chang)

《風險管理學報》 20卷2期 (2018/12) Pp. 131-164

摘要

The premise of effective risk management is the ability to delineate the probabilistic relationship between multiple underlying assets and resources and to derive quantitative indicators which reflect the current status of the system to be controlled. Developments of modern computing technologies have enabled the transition from traditional simplistic models to full-fledged stochastic ones with real-world considerations; multivariate probability models that can faithfully characterize their elements are in ever greater need. The copula is a such multivariate model useful to high-dimensional statistical applications as one is allowed to estimate the distribution of random vectors by estimating marginals and copula separately. Here we review the essence of the copula-GARCH model and the associated statistical tests. As an illustration we are able to show rigorously that the comovement of the monthly S&P500 and S&P600 indices is best described by a certain copula-GARCH model and subsequently apply this probability model for the evaluation of corresponding variable annuity product.

關鍵字

copula ； GARCH ； empirical process ； goodness-of-fit test ； variable annuity

並列摘要

有效的風險管理前提在於推估各種資產間的機率關係，並計算反映系統狀態的各種定量指標的能力。現代計算技術的進步使得更符合實際、不須過份簡化的多變量機率模型運用變為可能，而copula正是如此的多變量機率模型。我們利用一系列基於無母數統計與經驗過程理論的穩健統計檢定方法，針對給定S&P500與S&P600指數時間序列選擇並匹配最適copula-GARCH模型，進而推估變額年金保證價值。

並列關鍵字

Copula ； GARCH ； 經驗過程 ； 適合度檢定 ； 變額年金

參考文獻

Bai, J. (2003). Testing parametric conditional distributions of dynamic models. The Review of Economics and Statistics, 531-549.

Billingsley, P. (1999). Convergence of probability measures. New York: John Wiley & Sons.

Chan, N.-H., Chen, J., Chen, X., Fan, Y., & Peng, L. (2009). Statistical inference for multivariate residual copula of GARCH models. Statistica Sinica, 53-70.

Chen, X., & Fan, Y. (2006). Estimation and model selection of semiparametric copula-based multivariate dynamic models under copula misspecification. Journal of Econometrics, 125-154.

Chen, X., & Fan, Y. (2006). Estimation of copula-based semiparametric time series models. Journal of Econometrics, 307-335.