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

Anna Maltsev, Svetlana Malysheva

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 = ΣZ $ where $Z$ is a $T$ by $S$ matrix of i.i.d. light-tailed variables with mean 0 and variance 1 and $Σ$ is a diagonal matrix. In this paper, we take the randomness of $Σ$ 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 $Σ$ 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.

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 where is a by matrix of i.i.d. light-tailed variables with mean 0 and variance 1 and is a diagonal matrix. In this paper, we take the randomness of to be i.i.d. heavy tailed. We obtain an explicit formula for the empirical spectral distribution of in the particular case when the elements of 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.
Paper Structure (10 sections, 14 theorems, 101 equations, 6 figures)

This paper contains 10 sections, 14 theorems, 101 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Spectral properties of Elliptic Volatility matrix
  3. Derivation of the limiting density when $\nu=3.$
  4. Statistics of the maximal eigenvalue
  5. Proof of Lemma \ref{['lemma:nondiad_eig_to0']} for $0<\alpha<2$
  6. Proof of Lemma \ref{['lemma:nondiad_eig_to0']} for $2<\alpha<4$
  7. Comparison of EVSCE and historical data
  8. Data preparation: removing the “market mode” and re-normalisation
  9. Data analytics: comparing EVSCE and market data
  10. Appendix

Key Result

Theorem 2.1

Suppose that the entries of $\mathbf{Y}(n\times p)$ are complex random variables that are independent for each $n$ and identically distributed for all $n$ and satisfy $\mathrm{E}\left(\left|Y_{11}-\mathrm{E}\left(Y_{11}\right)\right|^2\right)=1$. Also, assume that $\mathbf{T}=\operatorname{diag}\lef

Figures (6)

  • Figure 1: Illustration of the cubic law. The tail $\overline{F}(x)= 1- F(x),$ where $F(x)$ is empirical c.d.f. of returns of the chosen stock has slope $\approx 3$ when plotted on a log-log scale. The box-whiskers plot displays the distribution of logarithms of log returns. On the left, plot for three major companies. Data from polygon.io.
  • Figure 2: Histogram of the maximal eigenvalue in a simulated EVSCE when $\sigma_i$ has Student(3) distribution. The dimensions of the matrix simulated matrix are $S=485, T=512.$ Number of simulations is $N=5000.$
  • Figure 3: Cubic law for standard deviations of return vectors at fixed time. Data taken from polygon.io
  • Figure 4: (Top left) Histograms of spectrum of simulated Student(3) EVSCE and matrix $\frac{\mathbf{X}_{cl}^*\mathbf{X}_{cl}}{T},$ and the limit obtained in Theorem \ref{['t:density']}. (Top right) Comparison of the Spectrum of $\frac{\mathbf{X}_{cl}^*\mathbf{X}_{cl}}{T},$ randomly generated EVSCE with Normally distributed $\sigma_t$'s and Marcenko-Pastur law. (Bottom) Comparison of spectrum of covariance matrix of shuffled data and similarly shuffled EVM to Marchenko-Pastur law. Data from polygon.io.
  • Figure 5: Histogram of re-normalised maximal eigenvalues: matrix $\mathbf{X}_{cl}^\prime$ were separated into 50 equal parts each part was "cleared" (in green)(data from polygon.io). Histogram of 50 re-normalised maximal eigenvalues of Student(3) EVSCE of the same size (in purple).
  • ...and 1 more figures

Theorems & Definitions (28)

  • Definition 1.1
  • Theorem 2.1
  • Remark 1
  • Lemma 2.1
  • proof
  • Corollary 2.1
  • proof
  • Theorem 2.2
  • proof : Proof of Theorem \ref{['t:density']}
  • Theorem 3.1
  • ...and 18 more