arXiv subjects

Thomas Guhr

Publications and source records attributed to Thomas Guhr.

At least 109 records · Page 6Linked to original sources

arXiv2014

Portfolio return distributions: Sample statistics with non-stationary correlations

We consider random vectors drawn from a multivariate normal distribution and compute the sample statistics in the presence of non-stationary correlations. For this purpose, we construct an ensemble of random correlation matrices and average the normal distribution over this ensemble. The resulting distribution contains a modified Bessel function of the second kind whose behavior differs significantly from the multivariate normal distribution, in the central part as well as in the tails. This result is then applied to asset returns. We compare with empirical return distributions using daily data from the Nasdaq Composite Index in the period from 1992 to 2012. The comparison reveals good agreement, the average portfolio return distribution describes the data well especially in the central part of the distribution. This in turn confirms our ansatz to model the non-stationarity by an ensemble average.

q-fin.ST↗

arXiv2014

Analysis of a decision model in the context of equilibrium pricing and order book pricing

An agent-based model for financial markets has to incorporate two aspects: decision making and price formation. We introduce a simple decision model and consider its implications in two different pricing schemes. First, we study its parameter dependence within a supply-demand balance setting. We find realistic behavior in a wide parameter range. Second, we embed our decision model in an order book setting. Here we observe interesting features which are not present in the equilibrium pricing scheme. In particular, we find a nontrivial behavior of the order book volumes which reminds of a trend switching phenomenon. Thus, the decision making model alone does not realistically represent the trading and the stylized facts. The order book mechanism is crucial.

q-fin.TR↗

arXiv2014

Distribution of the Smallest Eigenvalue in Complex and Real Correlated Wishart Ensembles

For the correlated Gaussian Wishart ensemble we compute the distribution of the smallest eigenvalue and a related gap probability.We obtain exact results for the complex (β=2) and for the real case (β=1). For a particular set of empirical correlation matrices we find universality in the spectral density, for both real and complex ensembles and all kinds of rectangularity. We calculate the asymptotic and universal results for the gap probability and the distribution of the smallest eigenvalue. We use the Supersymmetry method, in particular the generalized Hubbard-Stratonovich transformation and superbosonization.

math-ph↗

arXiv2014

Distributions of Off-Diagonal Scattering Matrix Elements: Exact Results

Scattering is a ubiquitous phenomenon which is observed in a variety of physical systems which span a wide range of length scales. The scattering matrix is the key quantity which provides a complete description of the scattering process. The universal features of scattering in chaotic systems is most generally modeled by the Heidelberg approach which introduces stochasticity to the scattering matrix at the level of the Hamiltonian describing the scattering center. The statistics of the scattering matrix is obtained by averaging over the ensemble of random Hamiltonians of appropriate symmetry. We derive exact results for the distributions of the real and imaginary parts of the off-diagonal scattering matrix elements applicable to orthogonally-invariant and unitarily-invariant Hamiltonians, thereby solving a long standing problem.

cond-mat.stat-mech↗

arXiv2013

Credit Risk and the Instability of the Financial System: an Ensemble Approach

The instability of the financial system as experienced in recent years and in previous periods is often linked to credit defaults, i.e., to the failure of obligors to make promised payments. Given the large number of credit contracts, this problem is amenable to be treated with approaches developed in statistical physics. We introduce the idea of ensemble averaging and thereby uncover generic features of credit risk. We then show that the often advertised concept of diversification, i.e., reducing the risk by distributing it, is deeply flawed when it comes to credit risk. The risk of extreme losses remain due to the ever present correlations, implying a substantial and persistent intrinsic danger to the financial system.

q-fin.RM↗

arXiv2013

Distribution of the Smallest Eigenvalue in the Correlated Wishart Model

Wishart random matrix theory is of major importance for the analysis of correlated time series. The distribution of the smallest eigenvalue for Wishart correlation matrices is particularly interesting in many applications. In the complex and in the real case, we calculate it exactly for arbitrary empirical eigenvalues, i.e., for fully correlated Gaussian Wishart ensembles. To this end, we derive certain dualities of matrix models in ordinary space. We thereby completely avoid the otherwise unsurmountable problem of computing a highly non-trivial group integral. Our results are compact and much easier to handle than previous ones. Furthermore, we obtain a new universality for the distribution of the smallest eigenvalue on the proper local scale.

math-ph↗

arXiv2013

Distribution of Scattering Matrix Elements in Quantum Chaotic Scattering

Scattering is an important phenomenon which is observed in systems ranging from the micro- to macroscale. In the context of nuclear reaction theory the Heidelberg approach was proposed and later demonstrated to be applicable to many chaotic scattering systems. To model the universal properties, stochasticity is introduced to the scattering matrix on the level of the Hamiltonian by using random matrices. A long-standing problem was the computation of the distribution of the off-diagonal scattering-matrix elements. We report here an exact solution to this problem and present analytical results for systems with preserved and with violated time-reversal invariance. Our derivation is based on a new variant of the supersymmetry method. We also validate our results with scattering data obtained from experiments with microwave billiards.

cond-mat.stat-mech↗

arXiv2013

Non-Stationarity in Financial Time Series and Generic Features

Financial markets are prominent examples for highly non-stationary systems. Sample averaged observables such as variances and correlation coefficients strongly depend on the time window in which they are evaluated. This implies severe limitations for approaches in the spirit of standard equilibrium statistical mechanics and thermodynamics. Nevertheless, we show that there are similar generic features which we uncover in the empirical return distributions for whole markets. We explain our findings by setting up a random matrix model.

q-fin.ST↗

arXiv2012

Supersymmetry approach to Wishart correlation matrices: Exact results

We calculate the `one-point function', meaning the marginal probability density function for any single eigenvalue, of real and complex Wishart correlation matrices. No explicit expression had been obtained for the real case so far. We succeed in doing so by using supersymmetry techniques to express the one-point function of real Wishart correlation matrices as a twofold integral. The result can be viewed as a resummation of a series of Jack polynomials in a non-trivial case. We illustrate our formula by numerical simulations. We also rederive a known expression for the one-point function of complex Wishart correlation matrices.

math.ST↗

arXiv2012

Microscopic understanding of heavy-tailed return distributions in an agent-based model

The distribution of returns in financial time series exhibits heavy tails. In empirical studies, it has been found that gaps between the orders in the order book lead to large price shifts and thereby to these heavy tails. We set up an agent based model to study this issue and, in particular, how the gaps in the order book emerge. The trading mechanism in our model is based on a double-auction order book, which is used on nearly all stock exchanges. In situations where the order book is densely occupied with limit orders we do not observe fat-tailed distributions. As soon as less liquidity is available, a gap structure forms which leads to return distributions with heavy tails. We show that return distributions with heavy tails are an order-book effect if the available liquidity is constrained. This is largely independent of the specific trading strategies.

q-fin.TR↗

arXiv2012

Identifying States of a Financial Market

The understanding of complex systems has become a central issue because complex systems exist in a wide range of scientific disciplines. Time series are typical experimental results we have about complex systems. In the analysis of such time series, stationary situations have been extensively studied and correlations have been found to be a very powerful tool. Yet most natural processes are non-stationary. In particular, in times of crisis, accident or trouble, stationarity is lost. As examples we may think of financial markets, biological systems, reactors or the weather. In non-stationary situations analysis becomes very difficult and noise is a severe problem. Following a natural urge to search for order in the system, we endeavor to define states through which systems pass and in which they remain for short times. Success in this respect would allow to get a better understanding of the system and might even lead to methods for controlling the system in more efficient ways. We here concentrate on financial markets because of the easy access we have to good data and because of the strong non-stationary effects recently seen. We analyze the S&P 500 stocks in the 19-year period 1992-2010. Here, we propose such an above mentioned definition of state for a financial market and use it to identify points of drastic change in the correlation structure. These points are mapped to occurrences of financial crises. We find that a wide variety of characteristic correlation structure patterns exist in the observation time window, and that these characteristic correlation structure patterns can be classified into several typical "market states". Using this classification we recognize transitions between different market states. A similarity measure we develop thus affords means of understanding changes in states and of recognizing developments not previously seen.

q-fin.ST↗

arXiv2011

A Random Matrix Approach to Credit Risk

We estimate generic statistical properties of a structural credit risk model by considering an ensemble of correlation matrices. This ensemble is set up by Random Matrix Theory. We demonstrate analytically that the presence of correlations severely limits the effect of diversification in a credit portfolio if the correlations are not identically zero. The existence of correlations alters the tails of the loss distribution considerably, even if their average is zero. Under the assumption of randomly fluctuating correlations, a lower bound for the estimation of the loss distribution is provided.

q-fin.RM↗

arXiv2010

Collective versus Single--Particle Motion in Quantum Many--Body Systems: Spreading and its Semiclassical Interpretation

We study the interplay between collective and incoherent single-particle motion in a model of two chains of particles whose interaction comprises a non-integrable part. In the perturbative regime, but for a general form of the interaction, we calculate the spectral density for collective excitations. We obtain the remarkable result that it always has a unique semiclassical interpretation. We show this by a proper renormalization procedure which allows us to map our system to a Caldeira-Leggett--type of model in which the bath is part of the system.

quant-ph↗

arXiv2010

On the Eigenvalue Density of Real and Complex Wishart Correlation Matrices

Wishart correlation matrices are the standard model for the statistical analysis of time series. The ensemble averaged eigenvalue density is of considerable practical and theoretical interest. For complex time series and correlation matrices, the eigenvalue density is known exactly. In the real case, however, a fundamental mathematical obstacle made it forbidingly complicated to obtain exact results. We use the supersymmetry method to fully circumvent this problem. We present an exact formula for the eigenvalue density in the real case in terms of twofold integrals and finite sums.

math-ph↗

arXiv2010

Statistical causes for the Epps effect in microstructure noise

We present two statistical causes for the distortion of correlations on high-frequency financial data. We demonstrate that the asynchrony of trades as well as the decimalization of stock prices has a large impact on the decline of the correlation coefficients towards smaller return intervals (Epps effect). These distortions depend on the properties of the time series and are of purely statistical origin. We are able to present parameter-free compensation methods, which we validate in a model setup. Furthermore, the compensation methods are applied to high-frequency empirical data from the NYSE's TAQ database. A major fraction of the Epps effect can be compensated. The contribution of the presented causes is particularly high for stocks that are traded at low prices.

q-fin.ST↗

arXiv2010

Impact of the tick-size on financial returns and correlations

We demonstrate that the lowest possible price change (tick-size) has a large impact on the structure of financial return distributions. It induces a microstructure as well as it can alter the tail behavior. On small return intervals, the tick-size can distort the calculation of correlations. This especially occurs on small return intervals and thus contributes to the decay of the correlation coefficient towards smaller return intervals (Epps effect). We study this behavior within a model and identify the effect in market data. Furthermore, we present a method to compensate this purely statistical error.

q-fin.ST↗

arXiv2010

Compensating asynchrony effects in the calculation of financial correlations

We present a method to compensate statistical errors in the calculation of correlations on asynchronous time series. The method is based on the assumption of an underlying time series. We set up a model and apply it to financial data to examine the decrease of calculated correlations towards smaller return intervals (Epps effect). We show that this statistical effect is a major cause of the Epps effect. Hence, we are able to quantify and to compensate it using only trading prices and trading times.

q-fin.ST↗

arXiv2010

Strain in Semiconductor Core-Shell Nanowires

We compute strain distributions in core-shell nanowires of zinc blende structure. We use both continuum elasticity theory and an atomistic model, and consider both finite and infinite wires. The atomistic valence force-field (VFF) model has only few assumptions. But it is less computationally efficient than the finite-element (FEM) continuum elasticity model. The generic properties of the strain distributions in core-shell nanowires obtained based on the two models agree well. This agreement indicates that although the calculations based on the VFF model are computationally feasible in many cases, the continuum elasticity theory suffices to describe the strain distributions in large core-shell nanowire structures. We find that the obtained strain distributions for infinite wires are excellent approximations to the strain distributions in finite wires, except in the regions close to the ends. Thus, our most computationally efficient model, the finite-element continuum elasticity model developed for infinite wires, is sufficient, unless edge effects are important. We give a comprehensive discussion of strain profiles. We find that the hydrostatic strain in the core is dominated by the axial strain-component, $\varepsilon_{\z \z}$. We also find that although the individual strain components have a complex structure, the hydrostatic strain shows a much simpler structure. All in-plane strain components are of similar magnitude. The non-planar off-diagonal strain-components ($\varepsilon_{\x \z}$ and $\varepsilon_{\y \z}$) are small but nonvanishing. Thus the material is not only stretched and compressed but also warped. The models used can be extended for study of wurtzite nanowire structures, as well as nanowires with multiple shells.

cond-mat.mes-hall↗