Inference and Ill-Posedness for Financial High Frequency Data

University Of Chicago, Chicago IL

Investigators

Abstract

Recent years have seen an explosion in the availability of high-frequency financial data. This has opened the possibility of estimating quantities like volatility on a daily basis with high precision. This project is concerned with the estimation of volatility and related quantities for high-frequency data, using a nonparametric "latent semi-martingale model." The existence of microstructure (statistically equivalent to measurement error) is crucial to this problem, as it substantially affects estimators. Earlier work by the investigator has found estimators that are robust to additive errors. The project is concerned with situations that call this robustness into question; in particular, when the formation of prices have a component of rounding, and when there are many (perhaps infinitely many) small jumps in the latent process. In both cases, there are several possible candidates for how to define volatility, and the separation between process and error is not always well defined. This project will explore this ill-posedness and seek to define measures of volatility that are robust to small alterations in the process structure. The project also will explore the application of such measures to risk management, forecasting, portfolio management, options trading, and regulation. This project is concerned with both estimation and with applications to financial practice and regulation. The solution proposed will rely on techniques from statistics and stochastic processes, including embedding methods, and it is expected that this research will lead to the development of new mathematical and econometric theory. Given that estimates of volatility are of interest to investors, regulators, and policymakers, the results of this research will have substantial practical implications. This award was supported as part of the fiscal year 2006 Mathematical Science priority area special competition on Mathematical Social and Behavioral Sciences (MSBS).