Applying free random variables to random matrix analysis of financial data. Part I: The Gaussian case

Abstract

We apply the concept of free random variables to doubly correlated (Gaussian) Wishart random matrix models, appearing, for example, in a multivariate analysis of financial time series, and displaying both inter-asset cross-covariances and temporal auto-covariances. We give a comprehensive introduction to the rich financial reality behind such models. We explain in an elementary way the main techniques of free random variables calculus, with a view to promoting them in the quantitative finance community. We apply our findings to tackle several financially relevant problems, such as a universe of assets displaying exponentially decaying temporal covariances, or the exponentially weighted moving average, both with an arbitrary structure of cross-covariances.Portfolio theory, Power laws, Statistical physics, Risk measures, Random walks, Options pricing, Random matrix theory,

A Random Matrix Approach to Dynamic Factors in Macroeconomic Data

“…This has been done in detail in [7,8], so we will only accentuate the main results here, referring the reader to the original papers for an exact explanation. The idea is that one uses twice the cyclic property of the trace (which permits cyclic shifts in the order of the terms), and twice the FRV multiplication law (8) (to break the N -transforms of products of matrices down to their constituents), in order to reduce the problem to solving the uncorrelated Wishart ensemble (1/T ) Y T Y.…”

Comparison of eigeninference based on one- and two-point Green's functions

“…From preliminary numerical studies done by us, we got relatively similar results for the eigen-inference, with slight advantage of the G-estimators method. It is not puzzling, since the conformal mapping we use [11,26] is closely related G-estimators. Simple comparison is however not easy, since G-estimator method [22] requires the knowledge of probabilities p i and infers the values on the unknown eigenvalues only, whereas analytic method infers both sets of values of unknown probabilities and spectrum.…”

“…Taking into account the importance of the "spiked" events, we plan to extend our analysis in the future for the case of unusual N scaling of both probabilities and eigenvalues, and to present the results of such analysis in future publication. [6] (valid in the limit when dimensions N, T tend to infinity while the ratio is r = N/T is kept fixed), one can reduce the problem of inference to surprisingly simple relation [11] N c (z) = rzN A (rz)N B (z)…”

“…Consequently, random matrix analysis of huge samples has entered several new domains. In economy and financial engineering, twinned papers [7,8] launched massive response for examining the role of spectral properties of covariance matrix for portfolio optimization [9][10][11]. In telecommunication, papers [12,13] opened new theoretical and practical possibilities for Multiple Output Multiple Input wireless systems [14,15].…”

Compound Real Wishart and q-Wishart Matrices

“…When B is positive-definite, Ref. [10] interprets W as a sample covariance under correlated sampling. Similar interpretation for the complex case appears in Ref.…”