Examining Applications of Fourier Transforms to Financial Data and Covariance Estimation

Fourier transforms project functions and signals onto a space of orthogonal trigonometric functions. The transform preserves all the information contained in a function and gives insight into the spectral components, or different frequency components, that make up the function. As a result, Fourier trans- forms have been useful in many fields of engineering, mathematics, statistics and finance. This paper will discuss some potential new uses of Fourier transforms in financial time series analysis. First, we show that traditional autoregressive models omit information that is captured by a Fourier transform. We then apply spectral decomposition to obtain better parameter estimates for an autoregressive process with a new estimation technique. For a certain set of parameters, the spectral decomposition estimator is more accurate and more precise than a traditional maximum likelihood estimator. We conclude with a discussion on extending this method to covariance estimation.