Estimating Realized Variance and True Prices from High-Frequency Data with Microstructure Noise

The market prices and the continuous quadratic variation play critical roles in high-frequency trading. However, the microstructure noise could make the observed prices differ from the true prices and hence bias the estimates of continuous quadratic variation. Following Zhou, we assume the observed prices are the result of adding microstructure noise to the true but hidden prices. Microstructure noise is assumed to be independent and identically distributed (i.i.d.); it is also independent of true prices. Zhang et al. propose a batch estimator for the continuous quadratic variation of high-frequency data in the presence of microstructure noise. It gives the estimates after all the data arrive. This thesis proposes a recursive version of their estimator that outputs variation estimates as the data arrive. The recursive version estimator gives excellent estimates well before all the data arrive. Both real high-frequency futures data and simulation data confirm the performance of recursive estimator. When prices are sampled from a geometric Brownian motion process, the Kalman filter can produce optimal estimates of true prices from the observed prices. However, the covariance matrix of microstructure noise and that of true prices must be known for this claim to hold. In practice, neither covariance matrix is known so they must be estimated. This thesis presents a robust Kalman filter (RKF) to estimate the true prices when microstructure noise is present. The RKF does not need the aforesaid covariance matrices as inputs. Simulation results show that the RKF gives essentially identical estimates to the Kalman filter, which has access to the two above mentioned covariance matrices.