Eigenvalue statistics of Elliptic Volatility Model with power-law tailed volatility

Abstract

In this paper we study an ensemble of random matrices called Elliptic Volatility Model, which arises in finance as models of stock returns. This model consists of a product of independent matrices $X = \Sigma Z $ where $Z$ is a $T$ by $S$ matrix of i.i.d. light-tailed variables with mean 0 and variance 1 and $\Sigma$ is a diagonal matrix. In this paper, we take the randomness of $\Sigma$ to be i.i.d. heavy tailed. We obtain an explicit formula for the empirical spectral distribution of $X^*X$ in the particular case when the elements of $\Sigma$ are distributed as Student's t with parameter 3. We furthermore obtain the distribution of the largest eigenvalue in more general case, and we compare our results to financial data.