The Annals of Statistics

Vast volatility matrix estimation for high-frequency financial data

Yazhen Wang and Jian Zou

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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 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 the 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.

Article information

Ann. Statist. Volume 38, Number 2 (2010), 943-978.

First available in Project Euclid: 19 February 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H12: Estimation
Secondary: 62G05: Estimation 62M05: Markov processes: estimation 62P20: Applications to economics [See also 91Bxx]

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


Wang, Yazhen; Zou, Jian. Vast volatility matrix estimation for high-frequency financial data. Ann. Statist. 38 (2010), no. 2, 943--978. doi:10.1214/09-AOS730.

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