Multi-scale Jump and Volatility Analysis for High-Frequency Financial Data

The wide availability of high-frequency data for many financial instruments stimulates an upsurge interest in statistical research on the estimation of volatility. Jump-diffusion processes observed with market microstructure noise are frequently used to model high-frequency financial data. Yet, existing methods are developed for either noisy data from a continuous diffusion price model or data from a jump-diffusion price model without noise. We propose methods to cope with both jumps in the price and market microstructure noise in the observed data. They allow us to estimate both integrated volatility and jump variation from the data sampled from jump-diffusion price processes, contaminated with the market microstructure noise. Our approach is to first remove jumps from the data and then apply noise-resistant methods to estimate the integrated volatility. The asymptotic analysis and the simulation study reveal that the proposed wavelet methods can successfully remove the jumps in the price processes and the integrated volatility can be estimated as accurately as in the case with no presence of jumps in the price processes. In addition, they have outstanding statistical efficiency. The methods are illustrated by applications to two high-frequency exchange rate data sets. Jianqing Fan is Frederick Moore’18 Professor of Finance, Department of Operation Research and Financial Engineering, Princeton University, Princeton, NJ 08544. Yazhen Wang is Professor, Department of Statistics, University of Connecticut, Storrs, CT 06269. Fan’s research was partially supported by the NSF grant DMS-0532370 and Wang’s research was partially supported by the NSF grant DMS-0504323. The authors thank the editor, associate editor, and two anonymous referees for stimulating comments and suggestions, which led to significant improvements in both substance and the presentation of the paper.