Markus Höchstötter

Estimation of α-Stable Sub-Gaussian Distributions for Asset Returns

Fitting multivariate α-stable distributions to data is still not feasible in higher dimensions since the (non-parametric) spectral measure of the characteristic function is extremely difficult to estimate in dimensions higher than 2. This was shown by Chen and Rachev (1995) and Nolan, Panorska and McCulloch (1996). α-stable sub-Gaussian distributions are a particular (parametric) subclass of the multivariate α-stable distributions. We present and extend a method based on Nolan (2005) to estimate the dispersion matrix of an α-stable sub-Gaussian distribution and estimate the tail index α of the distribution. In particular, we develop an estimator for the off-diagonal entries of the dispersion matrix that has statistical properties superior to the normal off-diagonal estimator based on the covariation. Furthermore, this approach allows estimation of the dispersion matrix of any normal variance mixture distribution up to a scale parameter. We demonstrate the behaviour of these estimators by fitting an α-stable sub-Gaussian distribution to the DAX30 components. Finally, we conduct a stable principal component analysis and calculate the coefficient of tail dependence of the prinipal components.

Analysis of stochastic technical trading algorithms

We apply the well-known CUSUM, the Girshick-Rubin, the Graversen-Peskir- Shiryaev and an improved alteration of the Brodsky-Darkovsky algorithm as trading strategies involving only mutually exclusive long positions in cash and the DAX at Xetra intraday auction prices. We select optimal pairs of fixed thresholds for up- and downmovements from a pre-defined two-dimensional grid, hence, admitting asymmetric intervals. We show that under three different scenarios for transaction costs, the improved Brodsky-Darkovsky technique not only outperforms the passive investment in the DAX but also the other three presented algorithms.

Graphical Representation of Data

Graphics, such as maps, graphs and diagrams, are used to represent large volume of data. They are necessary: • If the information is presented in tabular form or in a descriptive record, it becomes difficult to draw results. • Graphical form makes it possible to easily draw visual impressions of data. • The graphic method of the representation of data enhances our understanding. • It makes the comparisons easy. • Besides, such methods create an imprint on mind for a longer time. • It is a time consuming task to draw inferences about whatever is being presented in non–graphical form. • It presents characteristics in a simplified way. • These makes it easy to understand the patterns of population growth, distribution and the density, sex ratio, age–sex composition, occupational structure, etc.

Price calibration and hedging of correlation dependent credit derivatives using a structural model with α-stable distributions

The emergence of Credit Default Swap (CDS) indices and corresponding credit risk transfer markets with high liquidity and narrow bid–ask spreads has created standard benchmarks for market credit risk and correlation against which portfolio credit risk models can be calibrated. Integrated risk management for correlation dependent credit derivatives, such as single-tranches of synthetic Collateralized Debt Obligations (CDOs), requires an approach that adequately reflects the joint default behaviour in the underlying credit portfolios. Another important feature for such applications is a flexible model architecture that incorporates the dynamic evolution of underlying credit spreads. In this article, we present a model that can be calibrated to quotes of CDS index-tranches in a statistically sound way and simultaneously has a dynamic architecture to provide for the joint evolution of distance-to-default measures. This is accomplished by replacing the normal distribution by Smoothly Truncated α-Stable (STS) distributions in the Black/Cox version of the Merton approach for portfolio credit risk. This is possible due to the favourable features of this distribution family, namely, consistent application in the Black/Scholes no-arbitrage framework and the preservation of linear correlation concepts. The calibration to spreads of CDS index tranches is accomplished by a genetic algorithm. Our distribution assumption reflects the observed leptokurtic and asymmetric properties of empirical asset returns since the STS distribution family is basically constructed from α-stable distributions. These exhibit desirable statistical properties such as domains of attraction and the application of the generalized central limit theorem. Moreover, STS distributions fulfill technical restrictions like finite (exponential) moments of arbitrary order. In comparison to the performance of the basic normal distribution model which lacks tail dependence effects, our empirical analysis suggests that our extension with a heavy-tailed and highly peaked distribution provides a better fit to tranche quotes for the iTraxx IG index. Since the underlying implicit modelling of the dynamic evolution of credit spreads leads to such results, this suggests that the proposed model is appropriate to price and hedge complex transactions that are based on correlation dependence. A further application might be integrated risk management activities in debt portfolios where concentration risk is dissolved by means of portfolio credit risk transfer instruments such as synthetic CDOs.

Chapter 38 – A Comparison of Gaussian and Non-Gaussian Portfolio Choice Models

Publisher Summary This chapter discusses and examines the performance of Gaussian and non-Gaussian portfolio selection models. Some examples of models in the domain of attraction of stable laws are presented. The first distributional model considered is the case of the sub-Gaussian stable distributed returns. It permits a mean risk analysis quite similar to Markowitz-Tobins mean-variance one. In fact, the model admits the same analytical form for the efficient frontier but the parameters have a different meaning. The most important difference is in the way of estimating the parameters. To present the heavy tailed models that consider the asymmetry of returns, the chapter studies a three-fund separation model where the portfolios are in the domain of attraction of a stable law. A performance comparison among the sub-Gaussian, the stable three-fund separation model, and the mean-variance model is presented. The comparison holds from an ex-ante analysis on the data. The maximum expected utility of an investor is compared on different efficient frontiers considering daily data. The stable approaches present better performances than the mean-variance one.

A Comparison of Gaussian and Non-Gaussian Portfolio Choice Models

Abstract This paper discusses and examines the performance of Gaussian and non-Gaussian portfolio selection models. We analyze some allocation problems considering different distributional portfolio models. In particular, for each allocation problem, we compare the maximum expected utility obtained when the portfolios of the returns are either heavy tailed distributed or uniquely determined by the mean and the variance.