Vast volatility matrix estimation for high-frequency financial data. (English) Zbl 1183.62184

Wang, Yazhen; Zou, Jian

Ann. Stat. 38, No. 2, 943-978 (2010).

Summary: High-frequency data observed on the prices of financial assets are commonly modeled by diffusion processes with micro-structure noise, and realized volatility-based methods are often used to estimate the integrated volatility. For problems involving a large number of assets, the estimation objects we face are volatility matrices of large size. The existing volatility estimators work well for a small number of assets but perform poorly when the number of assets is very large. In fact, they are inconsistent when both the number, \(p\), of the assets and the average sample size, \(n\), of the price data on the \(p\) assets go to infinity.
This paper proposes a new type of estimators for the integrated volatility matrix and establishes asymptotic theory for the proposed estimators in a framework that allows both \(n\) and \(p\) to approach to infinity. The theory shows that the proposed estimators achieve high convergence rates under a sparsity assumption on the integrated volatility matrix. The numerical studies demonstrate that the proposed estimators perform well for large \(p\) and complex price and volatility models. The proposed method is applied to real high-frequency financial data.

MSC:

62P05	Applications of statistics to actuarial sciences and financial mathematics
91B25	Asset pricing models (MSC2010)
62G20	Asymptotic properties of nonparametric inference
65C60	Computational problems in statistics (MSC2010)
91G70	Statistical methods; risk measures
62H12	Estimation in multivariate analysis
62G05	Nonparametric estimation
62M05	Markov processes: estimation; hidden Markov models

Keywords:

convergence rate; diffusion; integrated volatility; matrix norm; micro-structure noise; realized volatility; regularization; sparsity; threshold

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