As a linear superposition of translated and dilated versions of a chosen analyzing wavelet function, the wavelet transform lends itself to the analysis of underlying multi-scale structure in nonstationary time series. In this work, we use the discrete wavelet transform (DWT) to investigate scaling and search for the presence of coherent structures in financial data. Quantitative measurements are given by the DWT of the original time series and wavelet coefficient variance. We find that variations and correlations in the transform coefficients are able to indicate the presence of structure and that measurements based on the DWT allow us to observe scaling directly in the nonstationary time series. PACS. 05.45.Tp Time series analysis -05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion. -02.90.+p Other topics in mathematical methods in physics.

One of the key features of wavelet analysis is its ability to decompose non-stationary signals according to time and scale. In this work, we use discrete wavelets to analyze the influence of detrending techniques on the time-scale information structure of daily financial data. We examine the use of log returns, a linear trend and the Hodrick–Prescott (HP) filter. Quantitative measurements of information distortion are given using the mean-squared error (MSE) and correlation of the wavelet coefficients between the detrended and original data. We find that log returns and linear detrending are most distortional. We also conclude that the HP-filter is most effective, depending on appropriate selection of the filter parameter, λ, which is [Formula: see text] for the given data set.

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A new framework for portfolio diversification is introduced which goes beyond the classical mean-variance approach and portfolio allocation strategies such as risk parity. It is based on a novel concept called portfolio dimensionality that connects diversification to the non-Gaussianity of portfolio returns and can typically be defined in terms of the ratio of risk measures which are homogenous functions of equal degree. The latter arises naturally due to our requirement that diversification measures should be leverage invariant. We introduce this new framework and argue the benefits relative to existing measures of diversification in the literature, before addressing the question of optimizing diversification or, equivalently, dimensionality. Maximising portfolio dimensionality leads to highly non-trivial optimization problems with objective functions which are typically non-convex and potentially have multiple local optima. Two complementary global optimization algorithms are thus presented. For problems of moderate size and more akin to asset allocation problems, a deterministic Branch and Bound algorithm is developed, whereas for problems of larger size a stochastic global optimization algorithm based on Gradient Langevin Dynamics is given. We demonstrate analytically and through numerical experiments that the framework reflects the desired properties often discussed in the literature.