Multiscale Analysis of Stock Index Return Volatility

We present a study where wavelet approximation techniques and some related computational algorithms are applied to non-stationary high frequency financial times series. Wavelets represent a novel instrument as far as concerned applications in the finance setting, but have a great relevance in many domains, from physics to statistics. Thus, while one goal of the paper is to compare the numerical performance of global and local function optimizers, another goal is to try to show that ad hoc wavelet-based function dictionaries are very useful for financial modeling through signal decomposition and approximation. Detecting the latent dependence features which are typically found in high frequency financial returns is particularly important for the scope of proposing models which are able to achieve reliable results in parameter estimation and pointwise function prediction. We show that by pre-processing data with wavelet dictionaries we effectively account for hidden periodic components, whose discovery allows to attain and improve the feature extraction power. We refer to sparse approximation through the Matching Pursuit algorithm, thus handling the negative effects of covariance non-stationarity at very high frequencies.