The Asymmetric t-Copula with Individual Degrees of Freedom d-fine GmbH Christ Church University of Oxford A thesis submitted for the degree of MSc in Mathematical Finance Michaelmas 2012 Abstract This thesis investigates asymmetric dependence structures of multivariate asset returns. Evidence of such asymmetry for equity returns has been reported in the literature. In order to model the dependence structure, a new t-copula approach is proposed called the skewed t-copula with individ- ual degrees of freedom (SID t-copula). This copula provides the flexibility to assign an individual degree-of-freedom parameter and an individual skewness parameter to each asset in a multivariate setting. Applying this approach to GARCH residuals of bivariate equity index return data and using maximum likelihood estimation, we find significant asymmetry. By means of the Akaike information criterion, it is demonstrated that the SID t-copula provides the best model for the market data compared to other copula approaches without explicit asymmetry parameters. In addition, it yields a better fit than the conventional skewed t-copula with a single degree-of-freedom parameter. In a model impact study, we analyse the errors which can occur when mod- elling asymmetric multivariate SID-t returns with the symmetric multi- variate Gauss or standard t-distribution. The comparison is done in terms of the risk measures value-at-risk and expected shortfall. We find large deviations between the modelled and the true VaR/ES of a spread posi- tion composed of asymmetrically distributed risk factors. Going from the bivariate case to a larger number of risk factors, the model errors increase. Contents 1 Introduction 1 2 Asymmetric t-copula approach 4 2.1 Copulas . 4 2.2 The Gauss copula . 5 2.3 The standard t-copula . 6 2.4 The skewed t-copula with individual degrees of freedom (SID t-copula) 7 2.5 Simulation of the SID t-copula . 9 2.6 Calibration of the SID t-copula . 11 3 Application of the SID t-copula to bivariate equity returns 15 3.1 Univariate equity index returns . 15 3.2 Tail dependence . 17 3.3 Calibration to market data . 17 4 Modelling the risk of asymmetric multivariate returns 24 4.1 Simulation of SID t-distributed returns . 24 4.2 Results for d = 2 ............................. 25 4.3 Results for d > 2 ............................. 31 5 Summary and Outlook 34 References 37 A Generalized hyperbolic skew Student-t distribution 39 B SID t-copula density 41 C ARMA(r,m)-GARCH(p,q) model 43 D Binomial statistical error 44 E Notes on numerics 45 i Chapter 1 Introduction Copulas have become the subject of intense research in the field of statistics over the last decades. A key benefit of copulas lies in the separation of the dependence structure between stochastic variables and their marginal distributions, thus being a more general concept than multivariate distributions where dependence and the univariate distribution are inextracably linked. Besides allowing for more flexible modelling approaches, copulas overcome well-known limitations and pitfalls of linear correlation approaches by providing more general measures of dependence like rank correlations [1]. An important field of application of the copula concept is financial risk manage- ment. However, for reasons of convenience, mainly the Gauss copula and the stan- dard Student-t copula with a single degree-of-freedom parameter are used in practice. These two copulas belong to the class of elliptical copulas and are the unique copulas of the related multivariate distributions, namely the multivariate Gauss distribution and the multivariate Student-t distribution [2]. They are symmetric in the sense that they do not have a parameter which captures asymmetric features with respect to the upper and lower tail of the joint probability distribution. For example, the asymp- totic tail dependence is in both cases symmetric, and goes to the same finite value for the standard Student-t copula and to zero for the Gauss copula [2]. Among the variety of copulas and multivariate distributions which have been studied in connection with risk management, the t-copula and the multivariate t- distribution belong to the most intensively studied besides the Gaussian counter- parts. There is strong empirical evidence showing that the Gaussian assumption for modelling multivariate financial return data is questionable (see [3] for an overview). The probably most important advantage of the t-distribution over the Gaussian dis- tribution is that it embodies important stylized features like heavy tails while being analytically and numerically relatively tractable. In fact, the t-copula and distribu- tion comprises the Gauss copula and distribution since the latter are recovered from the former by letting the degree-of-freedom parameter go to infinity. Various generalizations of the standard t-copula have been proposed. The skewed t-copula approach in Ref. [4] generalizes the standard t-copula in order to model asymmetric correlation and tail dependence in multidimensional return data. This was motivated by the empirical evidence of asymmetric multivariate equity or eq- 1 uity/FX portfolio returns which were shown to be more strongly correlated in bearish than in bullish markets [5, 6, 7]. However, only a single degree-of-freedom parameter can be calibrated in the approach proposed in [4]. The limitation of the standard t- copula to a single degree-of-freedom parameter for all risk factors, on the other hand, was relieved by the grouped t-copula with a common degree-of-freedom parameter for pre-specified groups of risk factors [8] or even individual degree-of-freedom pa- rameters for every risk factor [9]. However, in these two approaches the t-copula is symmetric. This thesis focuses on the investigation how asymmetric correlations can be mod- elled in terms of a suitably chosen and calibrated t-copula. To this end, we propose a new approach called the skewed t-copula with individual degrees of freedom (SID-t copula) which essentially is a combination of the t-copulas used in [4] and [9]. With the SID t-copula, we will not only be able to describe asymmetric dependence in multivariate asset returns explicitly, but can also investigate other versions of the t-copula, symmetric or asymmetric, which are contained as sub-models in the SID t-copula, i.e. which are recovered by putting restrictions on the parameters of the SID t-copula. Value-at-risk (VaR) and expected shortfall (ES) are two standard measures to quantify risk [2, 10]. Both refer to quantiles of the profit and loss distribution of a risky asset or portfolio. It is not only of theoretical but also practical interest to know the error one makes when using symmetric multivariate distributions to model asymmetric returns. We address the issue in this thesis in a Monte Carlo model study which not only treats the bivariate case but also comprises calculations for more than two risk factors. We conclude the introduction by giving an outline of the chapters of this thesis. In Chapter 2, a brief introduction to the concept of copulas is given with the main definitions and theorems. Using the stochastic representation, the SID t-copula is constructed by extending the representation of the standard t-copula. Some examples of different copulas are simulated and discussed in terms of the bivariate dependence structure before concluding the chapter with a section on the calibration of the SID t-copula. In Chapter 3 the SID t-copula is applied to bivariate stock index return data. Rather than taking the distributions of log-returns directly as marginal input for the copula, we will follow standard practice and apply our copula model to ARMA- GARCH-filtered residual returns. The residuals are obtained from a fit of the empiri- cal data under some distributional assumption, which we choose in our case to be the same marginal distribution as the one implied by the multivariate SID t-distribution. After inspecting the univariate distribution of the residual returns, the empirical tail dependence is analysed for signs of asymmetry. Eventually the SID t-copula is cali- brated to the joint returns of the two German stock indexes DAX and TECDAX. Chapter 4 contains a Monte Carlo study of the model errors in terms of value-at- risk and expected shortfall which occur when SID-t multivariate distributed returns are modelled using symmetric multivariate distributions, namely the Gaussian or the symmetric t-distribution. Rather than fitting the copulas, this analysis is entirely in the context of multivariate distributions. VaR and ES are calculated for various 2 spread positions and the deviations between the true SID-t VaR/ES and the approxi- mative Gaussian and symmetric t-VaRs/ES’s are discussed. While most of the study deals with bivariate spread positions, we also consider spread positions composed of up to eight risk factors. The last chapter summarizes the main results and conclusions of this thesis and gives an outlook on possible further research on the topic. 3 Chapter 2 Asymmetric t-copula approach 2.1 Copulas For the definition of a copula, we restrict ourselves to the bivariate case and summarize the main properties of d-dimensional copulas which are important for the present work. For the generalization to more than two dimensions, proofs, and detailed discussions of a number of further properties, we refer to standard textbooks on the topic by Nelsen [11] and McNeil et al. [2]. Following Nelsen [11], a function C : [0, 1]2 −→ [0, 1] , (u, v) 7−→ C(u, v) (2.1) with the properties 1. For all u, v ∈ [0, 1] it holds: C(u, 0) = C(0, v) = 0 and C(u, 1) = u and C(1, v) = v 2. For all u1, u2, v1, v2 ∈ [0, 1] with u1 ≤ u2 and v1 ≤ v2 it holds: C(u2, v2) − C(u2, v1) − C(u1, v2) + C(u1, v1) ≥ 0 is called a (bivariate) copula (function).