High-frequency data for volatility modelling

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The covariance matrix of asset returns is important for a wide range of individuals. Academics use estimates of the covariance matrix to test asset-pricing theories. Risk managers use the matrix to construct measures such as "value at risk." Corporate managers require accurate measures of covariances for hedging strategies. Last but not least, portfolio managers use the covariance matrix in designing tracking strategies where the return on their portfolio is designed to closely follow the return on a benchmark portfolio. But, covariance matrix, can be difficult to estimate in the context of portfolio optimization when the investable universe consists of a large number of assets. For example, in the Markowitz model of mean-variance optimization, an unconstrained covariance matrix with n assets necessitates the estimation of n(n + 1)/2 elements, which quickly becomes unmanageable as n grows, and even if feasible would often result in optimal asset allocation weights that have undesirable properties, such as extreme long and short positions. Various approaches have been proposed in the literature to deal with this problem. One approach consists in imposing some further structure on the covariance matrix to reduce the number of parameters to be estimated, typically in the form of a factor model. One of the most popular methods for analysing large cross-sectional data sets is factor analysis. Some of the most influential economic theories, e.g. the arbitrage pricing theory of Ross (1976) are based on factor models.