We obtain a canonical representation for block matrices. The representation facilitates simple computation of the determinant, the matrix inverse, and other powers of a block matrix, as well as the matrix logarithm and the matrix exponential. These results are particularly useful for block covariance and block correlation matrices, where evaluation of the Gaussian log-likelihood and estimation are greatly simplified. We illustrate this with an empirical application using a large panel of daily asset returns. Moreover, the representation paves new ways to regularizing large covariance/correlation matrices and to test block structures in matrices.

We introduce a novel parametrization of the correlation matrix. The reparametrization facilitates modeling of correlation and covariance matrices by an unrestricted vector, where positive definiteness is an innate property. This parametrization can be viewed as a generalization of Fisher's Z-transformation to higher dimensions and has a wide range of potential applications. An algorithm for reconstructing the unique n×n correlation matrix from any vector in real space of dimensionality n(n-1)/2 is provided, and we derive its numerical complexity.

This paper provides a guide to high frequency option trade and quote data disseminated by the Options Price Reporting Authority (OPRA). First, we present a comprehensive overview of the fragmented U.S. option market, including details on market regulation and the trading processes for all 15 constituent option exchanges. Then, we review the general structure of the OPRA dataset and present a thorough empirical description of the observed option trades and quotes for a selected sample of underlying assets that contains more than 25 billion records. We outline several types of irregular observations and provide recommendations for data filtering and cleaning. Finally, we illustrate the usefulness of the high frequency option data with two empirical applications: option-implied variance estimation and risk-neutral density estimation. Both applications highlight the richer information content of the high frequency OPRA data relative to the widely used end-of-day OptionMetrics data.

We introduce a multivariate estimator of financial volatility that is based on the theory of Markov chains. The Markov chain framework takes advantage of the discreteness of high-frequency returns. We study the finite sample properties of the estimation in a simulation study and apply it to high-frequency commodity prices.

We propose a new method for generating random correlation matrices that makes it simple to control both location and dispersion. The method is based on a vector parameterization, γ=γ(C), which maps any distribution on d-dimensional Real space, where d = n(n-1)/2, to a distribution on the space of non-singular nxn correlation matrices. Correlation matrices with certain properties, such as being well-conditioned, having block structures, and having strictly positive elements, are simple to generate. We compare the new method with existing methods.

We extend the classic ”martingale-plus-noise” model for high-frequency returns to accommodate an error correction mechanism and endogenous pricing errors. It is motivated by (i) novel empirical evidence documenting that microstructure noise is interwoven with innovations to the efficient price; (ii) building a bridge between high-frequency econometrics and market microstructure models. We identify temporal pricing error corrections and noise endogeneity as complementary components driving high-frequency dynamics and inducing two separate regimes, characterized by the sign of the return serial correlation and an implied bias in realized variance estimates. We document frequent fluctuations between these regimes, which we associate with price discovery in a setting with incomplete information and learning. The model links critical concepts from high-frequency statistics and market microstructure theory, opening up new paths for volatility estimation.

We propose a novel class of multivariate Realized GARCH models that utilize realized measures of volatility and correlations. The key property of the model is a convenient parametrization of the correlation matrix that requires no additional structure to ensure positive definiteness. The correlation matrix is characterized by a vector, that can vary freely in the real vector space. A more parsimonious structure is often desired in practice, in particularly in high dimensional systems, and the framework facilitates simple and intuitive dimension reductions. We apply the model to returns of nine assets and illustrate a dimension reduction that arises from a natural block equicorrelation structure. Interestingly, we find that the empirical distribution of the transformed realized correlations is approximately Gaussian.

We extend the popular dynamic Nelson-Siegel framework by introducing time-varying volatilities in the factor dynamics and incorporating the realized measures to improve the identification of the latent volatility state. The new model is able to effectively describe the conditional distribution dynamics of a term structure variable and can still be readily estimated with the Kalman filter. We apply our framework to model the crude oil futures prices. Using more than 150 million transactions for the large panel of contracts we carefully construct the realized volatility measures corresponding to the latent Nelson-Siegel factors, estimate the model at the daily frequency and evaluate it by forecasting the conditional density of futures prices. We document that the time-varying volatility specification suggested in our model strongly outperforms the constant volatility benchmark. In addition, the use of realized measures provides moderate, but systematic gains in density forecasting.