Optimal Convergence Trading

This article examines arbitrage investment in a mispriced asset when the mispricing follows the Ornstein-Uhlenbeck process and a credit-constrained investor maximizes a generalization of the Kelly criterion. The optimal differentiable and threshold policies are derived. The optimal differentiable policy is linear with respect to mispricing and risk-free in the long run. The optimal threshold policy calls for investing immediately when the mispricing is greater than zero with the investment amount inversely proportional to the risk aversion parameter. The investment is risky even in the long run. The results are consistent with the belief that credit-constrained arbitrageurs should be risk-neutral if they are to engage in convergence trading.Comment: 16 pages, no figure

On the Markus-Spielman-Srivastava inequality for sums of rank-one matrices

We extend the result of Markus, Spielman, and Srivastava about the sum of rank-one symmetric random matrices to the case when the isotropy assumption on the random matrices is relaxed.Comment: 4 page

Statistical properties of zeta functions' zeros

The paper reviews existing results about the statistical distribution of zeros for the three main types of zeta functions: number-theoretical, geometrical, and dynamical. It provides necessary background and some details about the proofs of main results.Comment: 40 pages, 1 figur

On the largest Lyapunov exponent for products of Gaussian matrices

The paper provides a new integral formula for the largest Lyapunov exponent of Gaussian matrices, which is valid in the real, complex and quaternion-valued cases. This formula is applied to derive asymptotic expressions for the largest Lyapunov exponent when the size of the matrix is large and compare the Lyapunov exponents in models with a spike and no spikes.Comment: 15 pages, 4 figures, accepted to the Journal of Statistical Physic

Variation of word frequencies in Russian literary texts

We study the variation of word frequencies in Russian literary texts. Our findings indicate that the standard deviation of a word's frequency across texts depends on its average frequency according to a power law with exponent $0.62,$ showing that the rarer words have a relatively larger degree of frequency volatility (i.e., "burstiness"). Several latent factors models have been estimated to investigate the structure of the word frequency distribution. The dependence of a word's frequency volatility on its average frequency can be explained by the asymmetry in the distribution of latent factors.Comment: 17 page

Consistent Estimation of Pricing Kernels from Noisy Price Data

If pricing kernels are assumed non-negative then the inverse problem of finding the pricing kernel is well-posed. The constrained least squares method provides a consistent estimate of the pricing kernel. When the data are limited, a new method is suggested: relaxed maximization of the relative entropy. This estimator is also consistent. Keywords: $\epsilon$-entropy, non-parametric estimation, pricing kernel, inverse problems.Comment: 13 page

Lattice Option Pricing By Multidimensional Interpolation

This note proposes a method for pricing high-dimensional American options based on modern methods of multidimensional interpolation. The method allows using sparse grids and thus mitigates the curse of dimensionality. A framework of the pricing algorithm and the corresponding interpolation methods are discussed, and a theorem is demonstrated that suggests that the pricing method is less vulnerable to the curse of dimensionality. The method is illustrated by an application to rainbow options and compared to Least Squares Monte Carlo and other benchmarks.Comment: 12 pages, tables omitte

On Fluctuations of Riemann's Zeta Zeros

It is shown that the normalized fluctuations of Riemann's zeta zeros around their predicted locations follow the Gaussian law. It is also shown that fluctuations of two zeros, $\gamma _{k}$ and $\gamma _{k+x},$ with $x\sim(\log k)^{\beta}$, $\beta >0$, for large $k$ follow the two-variate Gaussian distribution with correlation $(1-\beta)_{+}$.Comment: 25 page

Lecture Notes on Free Probability

Lecture notes for a graduate course lectured at Stanford University.Comment: 100 page

On the Chernoff bound for efficiency of quantum hypothesis testing

The paper estimates the Chernoff rate for the efficiency of quantum hypothesis testing. For both joint and separable measurements, approximate bounds for the rate are given if both states are mixed and exact expressions are derived if at least one of the states is pure. The efficiency of tests with separable measurements is found to be close to the efficiency of tests with joint measurements. The results are illustrated by a test of quantum entanglement.Comment: 15 pages, no figure