Exploring multi-resolution and multi-scaling volatility features

One of the most recent results in empirical finance is the achievement of an accuracy gain for the volatility estimates from the use of high-frequency samples. In many applications, for instance, the observed intra-day data become the best informative source of estimation for the daily volatilites. This information set, or filtration in probability terms, can in theory be optimally exploited while adopting an increasingly finer sampling rate, a fact which finds justification when assuming a stochastic characterization of the problem in terms of so-called semimartingales. These instruments are very useful for representing asset price dynamics, and in recent proposals supply realized and integrated volatility measures. The former volatility is the empirical approximation of the latter, an average of high frequency returns observed within a certain time frame. Semimartingales become suitable model tools by allowing for the quadratic variation principle to hold. This in turn means that the convergence of the realized to the integrated volatility can be verified; conversely, from both theoretical and experimental standpoints, the cumulative squared high frequency returns represent consistent estimators of the integrated volatility measure. The goal of this work is twofold: first, to show with simulations the quality of the convergence for time-based estimators, compared to that obtained when time-scale coordinate wavelet tranforms are considered. Second, to verify that special families of wavelet decompose returns and allow for multi-scaling features to be revealed, together with the possible presence of underlying nonlinear dynamics of stock index return volatility.