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We study the estimation of the high-dimensional covariance matrix andits eigenvalues under dynamic volatility models. Data under such modelshave nonlinear dependency both cross-sectionally and temporally. We firstinvestigate the empirical…
Financial models are studied where each asset may potentially lose value relative to any other. Conditioning on non-devaluation, each asset can serve as proper num\'eraire and classical valuation rules can be formulated. It is shown when…
We consider a large random network, in which the performance of a node depends upon that of its neighbours and some external random influence factors. This results in random vector valued fixed-point (FP) equations in large dimensional…
Price fluctuations in financial markets can be characterized by L\'evy's stable distribution, which is supported by the generalized central limit system. When the stable parameters were estimated from four different stock markets in long…
When a randomness is introduced at the level of real matrix elements, depending on its particular realization, a pair of eigenvalues can appear as real or form a complex conjugate pair. We show that in the limit of large matrix size the…
High dimensionality comparable to sample size is common in many statistical problems. We examine covariance matrix estimation in the asymptotic framework that the dimensionality $p$ tends to $\infty$ as the sample size $n$ increases.…
We extend to the matrix setting a recent result of Srivastava-Vershynin about estimating the covariance matrix of a random vector. The result can be in- terpreted as a quantified version of the law of large numbers for positive…
In this paper, we study random matrix models which are obtained as a non-commutative polynomial in random matrix variables of two kinds: (a) a first kind which have a discrete spectrum in the limit, (b) a second kind which have a joint…
This brief manuscript provides an introduction to L\'evy processes and their applications in finance as the random process that drives asset models. Characteristic functions and random variable generators of popular L\'evy processes are…
In order to pursue the issue of the relation between the financial cross-correlations and the conventional Random Matrix Theory we analyse several characteristics of the stock market correlation matrices like the distribution of…
We propose a technique for calculating and understanding the eigenvalue distribution of sums of random matrices from the known distribution of the summands. The exact problem is formidably hard. One extreme approximation to the true density…
This paper introduces a novel process for both factor and idiosyncratic volatility matrices whose eigenvalues follow the vector auto-regressive (VAR) model. We call it the factor and idiosyncratic VAR (FIVAR) model. The FIVAR model accounts…
Financial portfolio management is one of the problems that are most frequently encountered in the investment industry. Nevertheless, it is not widely recognized that both Kelly Criterion and Risk Parity collapse into Mean Variance under…
The relationship between the spectral density and free energy of a spin system is considered. The analytical expressions allowing for the calculation of the spectral density for solvable models are determined. A linear Ising model is taken…
We present a class of L\'evy processes for modelling financial market fluctuations: Bilateral Gamma processes. Our starting point is to explore the properties of bilateral Gamma distributions, and then we turn to their associated L\'evy…
We develop a stochastic volatility framework for modeling multiple currencies based on CBI-time-changed L\'evy processes. The proposed framework captures the typical risk characteristics of FX markets and is coherent with the symmetries of…
We present an overview of the broad class of financial models in which the prices of assets are L\'evy-Ito processes driven by an $n$-dimensional Brownian motion and an independent Poisson random measure. The Poisson random measure is…
We consider the problem of valuing a European option written on an asset whose dynamics are described by an exponential L\'evy-type model. In our framework, both the volatility and jump-intensity are allowed to vary stochastically in time…
We investigate the universality of singular value and eigenvalue distributions of matrix valued functions of independent random matrices and apply these general results in several examples. In particular we determine the limit distribution…
We report on the emergence of scaling laws in the temporal evolution of the daily closing values of the S\&P 500 index prices and its modeling based on the L\'evy flights in two dimensions (2D). The efficacy of our proposed model is…