Svetlozar T. Rachev

Modeling asset returns with alternative stable distributions

In the 1960s Benoit Mandelbrot and Eugene Fama argued strongly in favor of the stable Paretian distribution as a model for the unconditional distribution of asset returns. Although a substantial body of subsequent empirical studies supported this position, the stable Paretian model plays a minor role in current empirical work. While in the economics and finance literature stable distributions are virtually exclusively associated with stable Paretian distributions, in this paper we adopt a more fundamental view and extend the concept of stability to a variety of probabilistic schemes. These schemes give rise to alternative stable distributions, which we compare empirically using S&P 500 stock return data. In this comparison the Weibull distribution, associated with both the nonrandom-minimum and geometric-random summation schemes dominates the other stable distributions considered-including the stable Paretian model.

Different Approaches to Risk Estimation in Portfolio Theory

Some new performance measures may be regarded as alternatives to the most popular criterion for portfolio optimization, the Sharpe ratio. Analysis of some allocation problems here takes into consideration portfolio selection models based on different risk perceptions and sample paths of the final wealth process for each allocation problem. One new performance ratio seems to be suitable for some optimization problems, but we need a thorough classification of the set of performance measures that would be ideal for large classes of financial optimization problems.

A characterization of random variables with minimum L 2 -distance

A complete characterization of multivariate random variables with minimum L2 Wasserstein-distance is proved by means of duality theory and convex analysis. This characterization allows to determine explicitly the optimal couplings for several multivariate distributions. A partial solution of this problem has been found in recent papers by Knott and Smith.

Desirable Properties of an Ideal Risk Measure in Portfolio Theory

This paper examines the properties that a risk measure should satisfy in order to characterize an investors preferences. In particular, we propose some intuitive and realistic examples that describe several desirable features of an ideal risk measure. This analysis is the first step in understanding how to classify an investors risk. Risk is an asymmetric, relative, heteroskedastic, multidimensional concept that has to take into account asymptotic behavior of returns, inter-temporal dependence, risk-time aggregation, and the impact of several economic phenomena that could influence an investors preferences. In order to consider the financial impact of the several aspects of risk, we propose and analyze the relationship between distributional modeling and risk measures. Similar to the notion of ideal probability metric to a given approximation problem, we are in the search for an ideal risk measure or ideal performance ratio for a portfolio selection problem. We then emphasize the parallels between risk measures and probability metrics, underlying the computational advantage and disadvantage of different approaches.

Maximum likelihood estimation of stable Paretian models

Stable Paretian distributions have attractive properties for empirical modeling in finance, because they include the normal distribution as a special case but can also allow for heavier tails and skewness. A major reason for the limited use of stable distributions in applied work is due to the facts that there are, in general, no closed-form expressions for its probability density function and that numerical approximations are nontrivial and computationally demanding. Therefore, Maximum Likelihood (ML) estimation of stable Paretian models is rather difficult and time consuming. Here, we study the problem of ML estimation using fast Fourier transforms to approximate the stable density functions. The performance of the ML estimation approach is investigated in a Monte Carlo study and compared to that of a widely used quantile estimator. Extensions to more general distributional models characterized by time-varying location and scale are discussed.

Diagnosing and treating the fat tails in financial returns data

Abstract We address an open and important question regarding the nature of the fat tails found in financial-return data, which has been raised by Ghose and Kroner [Journal of Empirical Finance, 2 (1995) 225]. These authors find that two classes of models used for modeling financial returns, namely the independent and identically distributed (iid) stable Paretian and the GARCH assumption, have several features in common, with the latter being preferred. We advocate models that combine the two allegedly disjoint paradigms, i.e., GARCH processes driven by stable Paretian innovations, and investigate some of their theoretical and small-sample properties. Finally, we demonstrate the plausibility of the new models for several exchange-rate series involved in the Asian crisis.

Stationarity of stable power-GARCH processes

We present conditions for strict stationarity of power-GARCH processes whose innovations are described by a heavy-tailed and possibly asymmetric stable Paretian distribution. The results generalize those of Bougerol and Picard (J. Econom. 52 (1992) 115), who derived analogous conditions for standard, i.e., power-two, GARCH processes with finite-variance innovations.

Multivariate Geometric Stable Laws

The paper summarizes recent advances in the theory of geometric stable (GS) distributions. The results presented include parametrizations, characterizations, mixture representations, properties, asymptotic and convergent series expansions of densities and distribution functions, moments and tail behavior, simulation, and estimation.

Stable GARCH models for financial time series

Abstract Generalized autoregressive conditional heteroskedasticity (GARCH) models having normal or Student-t distributions as conditional distributions are widely used in financial modeling. Normal or Student-t distributions may be inappropriate for very heavy-tailed times series as can be encountered in financial economics, for example. Here, we propose GARCH models with stable Paretian conditional distributions to deal with such time series. We state conditions for stationarity and discuss simulation aspects.

Stable modeling of value at risk

The value-at-risk (VAR) measurements are widely applied to estimate exposure to market risks. The traditional approaches to VAR computations-the variance-covariance method, historical simulation, Monte Carlo simulation, and stress-testing-do not provide satisfactory evaluation of possible losses. In this paper, we analyze the use of stable Paretian distributions in VAR modeling.