Romain Allez

Individual and collective stock dynamics: intra-day seasonalities

We establish several new stylized facts concerning the intra-day seasonalities of stock dynamics. Beyond the well-known U-shaped pattern of the volatility, we find that the average correlation between stocks increases throughout the day, leading to a smaller relative dispersion between stocks. Somewhat paradoxically, the kurtosis (a measure of volatility surprises) reaches a minimum at the open of the market, when the volatility is at its peak. We confirm that the dispersion kurtosis is a markedly decreasing function of the index return. This means that during large market swings, the idiosyncratic component of the stock dynamics becomes sub-dominant. In a nutshell, the early hours of trading are dominated by idiosyncratic or sector-specific effects with little surprises, whereas the influence of the market factor increases throughout the day, and surprises become more frequent.

Invariant beta ensembles and the Gauss-Wigner crossover.

We define a new diffusive matrix model converging toward the β-Dyson Brownian motion for all β is an element of [0,2] that provides an explicit construction of beta ensembles of random matrices that is invariant under the orthogonal or unitary group. For small values of β, our process allows one to interpolate smoothly between the Gaussian distribution and the Wigner semicircle. The interpolating limit distributions form a one parameter family that can be explicitly computed. This also allows us to compute the finite-size corrections to the semicircle.

Principal Regression Analysis and the index leverage effect

We revisit the index leverage effect, that can be decomposed into a volatility effect and a correlation effect. We investigate the latter using a matrix regression analysis, that we call ‘Principal Regression Analysis’ (PRA) and for which we provide some analytical (using Random Matrix Theory) and numerical benchmarks. We find that downward index trends increase the average correlation between stocks (as measured by the most negative eigenvalue of the conditional correlation matrix), and makes the market mode more uniform. Upward trends, on the other hand, also increase the average correlation between stocks but rotates the corresponding market mode away from uniformity. There are two time scales associated to these effects, a short one on the order of a month (20 trading days), and a longer time scale on the order of a year. We also find indications of a leverage effect for sectorial correlations as well, which reveals itself in the second and third mode of the PRA.

Rotational Invariant Estimator for General Noisy Matrices

We investigate the problem of estimating a given real symmetric signal matrix C from a noisy observation matrix M in the limit of large dimension. We consider the case where the noisy measurement M comes either from an arbitrary additive or multiplicative rotational invariant perturbation. We establish, using the replica method, the asymptotic global law estimate for three general classes of noisy matrices, significantly extending previously obtained results. We give exact results concerning the asymptotic deviations (called overlaps) of the perturbed eigenvectors away from the true ones, and we explain how to use these overlaps to “clean” the noisy eigenvalues of M. We provide some numerical checks for the different estimators proposed in this paper and we also make the connection with some well-known results of Bayesian statistics.

From Sine kernel to Poisson statistics

We study the Sine beta process introduced in Valko and Virag, when the inverse temperature beta tends to 0 . This point process has been shown to be the scaling limit of the eigenvalues point process in the bulk of beta -ensembles and its law is characterised in terms of the winding numbers of the Brownian carrousel at different angular speeds. After a careful analysis of this family of coupled diffusion processes, we prove that the Sine-beta point process converges weakly to a Poisson point process on the real line . Thus, the Sine-beta point processes establish a smooth crossover between the rigid clock (or picket fence) process (corresponding to

Random Matrices in Non-confining Potentials

\beta=\infty

The continuous Anderson hamiltonian in dimension two

) and the Poisson process.

Eigenvector dynamics: general theory and some applications

We consider invariant matrix processes diffusing in non-confining cubic potentials of the form

\beta

V_a(x)= x^3/3 - a x, a\in \mathbb {R}