Bayesian Inference on Volatility in the Presence of Infinite Jump Activity and Microstructure Noise

TL;DR

This paper develops a Bayesian method for estimating volatility in high-frequency financial data modeled by Lévy processes with infinite jumps and microstructure noise, establishing asymptotic normality of the adjusted posterior.

Contribution

It introduces a novel adjusted Bayesian posterior for volatility that remains valid despite model misspecification and jump activity, with proven asymptotic Gaussianity.

Findings

Adjusted posterior is asymptotically Gaussian

Method accurately estimates volatility with credible intervals

Approach extends to general Itô semimartingales without microstructure noise

Abstract

Volatility estimation based on high-frequency data is key to accurately measure and control the risk of financial assets. A L\'{e}vy process with infinite jump activity and microstructure noise is considered one of the simplest, yet accurate enough, models for financial data at high-frequency. Utilizing this model, we propose a "purposely misspecified" posterior of the volatility obtained by ignoring the jump-component of the process. The misspecified posterior is further corrected by a simple estimate of the location shift and re-scaling of the log likelihood. Our main result establishes a Bernstein-von Mises (BvM) theorem, which states that the proposed adjusted posterior is asymptotically Gaussian, centered at a consistent estimator, and with variance equal to the inverse of the Fisher information. In the absence of microstructure noise, our approach can be extended to inferences of the integrated variance of a general It\^o semimartingale. Simulations are provided to demonstrate the accuracy of the resulting credible intervals, and the frequentist properties of the approximate Bayesian inference based on the adjusted posterior.