Abstract
Volatility estimation based on high-frequency data is important for accurate measurement and control of financial asset risks. A Lévy 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 Itô 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.
Highlights
In the past decade, jumps have played an increasingly important role in asset price modeling
Our main result is a Bernstein-von Mises Theorem for the adjusted posterior for the volatility parameter, which shows that the proposed posterior is asymptotically normal and centered at a consistent estimator, and with variance shrinking at rates n−1/2 and n−1, respectively, depending on whether a microstructure noise is incorporated or not in the model
Asymptotic normal approximation N from Bernstein-von Mises (BvM) Theorem highest posterior density (HPD) interval based on posterior πn(θ) = πn(θ + [J]n) Equal-tail credible interval based on posterior πn(θ) = πn(θ + [J]n) Frequentist central limit theorem (CLT) for threshold estimator and variance from [37]
Summary
Jumps have played an increasingly important role in asset price modeling. The variance or volatility, is not affected by the jumps, and is modeled by a simple Gaussian process, for which Bayesian inference can be more obtained Based on this observation, one plausible idea to tackle the problem is to ignore the nuisance parameters in the nonparametric part of the process, replace the nuisance parameters in the parametric part by their consistent estimators, and construct a posterior only for the parameter of interest. Our main result is a Bernstein-von Mises Theorem for the adjusted posterior for the volatility parameter, which shows that the proposed posterior is asymptotically normal and centered at a consistent estimator, and with variance shrinking at rates n−1/2 and n−1, respectively, depending on whether a microstructure noise is incorporated or not in the model. The proofs and further technical details appear in the Appendix
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