Bayesian inference on volatility in the presence of infinite jump activity and microstructure noise

Volatility estimation based on high-frequency data is important for accurate measurement and control of financial asset risks. A Levy process with infinite jump activity and microstructure noise is considered one of the simplest models for financial data at high-frequency. Utilizing this model, we propose a "purposely misspecified" posterior of the volatility obtained by ignoring the the process' jump-component. 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 make inferences for the integrated variance of general Ito semimartingales. 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.