UC Santa Cruz

Statistical models of price volatility most commonly use low-frequency (daily, weekly, or monthly) returns. However, despite their availability, two types of financial data have not been extensively studied: high-frequency data where sampling periods are on the order of seconds; and open, close, high, and low (OCHL) data which incorporate intraperiod extremes.

The first part of this dissertation focuses on the development of a filtering-based method for the estimation of volatility in high-frequency returns, which contrasts currently popular averaging-based approaches. The second part of this dissertation develops a foundational and novel method for likelihood-based estimation for bivariate OCHL, an approach unfeasible until now.

In Chapter 2, we formulate a discrete-time Bayesian stochastic volatility model for high-frequency stock-market data that directly accounts for microstructure noise, and outline a Markov chain Monte Carlo algorithm for parameter estimation. The methods described in this paper are designed to be coherent across all sampling timescales, with the goal of estimating the latent log-volatility signal from data collected at arbitrarily short sampling periods. In keeping with this goal, we carefully develop a method for eliciting priors. The empirical results derived from both simulated and real data show that directly accounting for microstructure noise in a state-space formulation allows for well-calibrated estimates of the log-volatility process driving prices.

In Chapter 3, we present and motivate the bivariate OCHL problem, enumerate the fundamental limitations of some common out-of-the-box approaches, and present a semidiscrete Galerkin numerical solver for computing the likelihood of the observed data. In addition, we prove the consistency of maximum likelihood estimates under the approximate density given by the solver.

Chapter 4 develops a closed-form, analytic solution to the OCHL likelihood problem in parameter ranges where the Galerkin solver requires near-infinite compute time and memory to produce numerically accurate results. A matching solution is also proposed to interpolate between parameter regions where neither the Galerkin nor analytic solutions are applicable. Thus, we present a method for producing likelihoods based on OCHL data over all model parameter ranges, which is a key requirement in statistical estimation algorithms. We use numerical experiments in both Chapters 3 and 4 to show the validity of our methods and demonstrate the increase in statistical power in estimating price volatility and correlation when using bivariate OCHL data.