Jean-David Fermanian

Some Statistical Pitfalls in Copula Modeling for Financial Applications

In this paper we discuss some statistical pitfalls that may occur in modeling cross-dependences with copulas in financial applications. In particular we focus on issues arising in the estimation and the empirical choice of copulas as well as in the design of time-dependent copulas.

Hedging default risks of CDOs in Markovian contagion models

We describe a replicating strategy of CDO tranches based upon dynamic trading of the corresponding credit default swap index. The aggregate loss follows a homogeneous Markov chain associated with contagion effects. Default intensities depend upon the number of defaults and are calibrated onto an input loss surface. Numerical implementation can be carried out thanks to a recombining tree. We examine how input loss distributions drive the credit deltas. We find that the deltas of the equity tranche are lower than those computed in the standard base correlation framework. This is related to the dynamics of dependence between defaults.

Sensitivity analysis of VaR and Expected Shortfall for portfolios under netting agreements

In this paper, we characterize explicitly the first derivative of the Value at Risk and the Expected Shortfall with respect to portfolio allocation when netting between positions exists. As a particular case, we examine a simple Gaussian example in order to illustrate the impact of netting agreements in credit risk management. We further provide nonparametric estimators for sensitivities and derive their asymptotic distributions. An empirical application on a typical banking portfolio is finally provided.

Nonparametric Estimation of Copulas for Time Series

We consider a nonparametric method to estimate copulas, i.e. functions linking joint distributions to their univariate margins. We derive the asymptotic properties of kernel estimators of copulas and their derivatives in the context of a multivariate stationary process satisfactory strong mixing conditions. Monte Carlo results are reported for a stationary vector autoregressive process of order one with Gaussian innovations. An empirical illustration containing a comparison with the independent, comotonic and Gaussian copulas is given for European and US stock index returns.

Time-dependent copulas

For the study of dynamic dependence structures, the authors introduce the concept of a pseudo-copula, which extends Pattons definition of a conditional copula. They state the equivalent of Sklars theorem for pseudo-copulas. They establish the asymptotic normality of nonparametric estimators of pseudo-copulas under strong mixing assumptions, and discuss applications to specification tests. They complement the theory with a small simulation study on the power of the proposed tests.

An Overview of the Goodness-of-Fit Test Problem for Copulas

We review the main “omnibus procedures” for goodness-of-fit (GOF) testing for copulas: tests based on the empirical copula process, on probability integral transformations (PITs), on Kendall’s dependence function, etc., and some corresponding reductions of dimension techniques. The problems of finding asymptotic distribution-free test statistics and the calculation of reliable p-values are discussed. Some particular cases, like convenient tests for time-dependent copulas, for Archimedean or extreme-value copulas, etc., are dealt with. Finally, the practical performances of the proposed approaches are briefly summarized.

The limits of granularity adjustments

We provide a rigorous proof of granularity adjustment (GA) formulas to evaluate loss distributions and risk measures (value-at-risk) in the case of heterogenous portfolios, multiple systematic factors and random recoveries. As a significant improvement with respect to the literature, we detail all the technical conditions of validity and provide an upper bound of the remainder term for finite portfolio sizes. Moreover, we deal explicitly with the case of general loss distributions, possibly with masses. For some simple portfolio models, we prove empirically that the granularity adjustments do not always improve the infinitely granular first-order approximations. This stresses the importance of checking some conditions of regularity before relying on such techniques. Smoothing the underlying loss distributions through random recoveries or exposures improves the GA performances in general.

On the Stationarity of Dynamic Conditional Correlation Models

We provide conditions for the existence and the unicity of strictly stationary solutions of the usual Dynamic Conditional Correlation GARCH models (DCC-GARCH). The proof is based on Tweedies (1988) criteria, after having rewritten DCC-GARCH models as nonlinear Markov chains. Moreover, we study the existence of their finite moments.

Combining Cumulative Sum Change-Point Detection Tests for Assessing the Stationarity of Univariate Time Series: Combined change-point tests for stationarity

We derive tests of stationarity for univariate time series by combining change-point tests sensitive to changes in the contemporary distribution with tests sensitive to changes in the serial dependence. The proposed approach relies on a general procedure for combining dependent tests based on resampling. After proving the asymptotic validity of the combining procedure under the conjunction of null hypotheses and investigating its consistency, we study rank-based tests of stationarity by combining cumulative sum change-point tests based on the contemporary empirical distribution function and on the empirical autocopula at a given lag. Extensions based on tests solely focusing on second-order characteristics are proposed next. The finite-sample behaviors of all the derived statistical procedures for assessing stationarity are investigated in large-scale Monte Carlo experiments and illustrations on two real data sets are provided. Extensions to multivariate time series are briefly discussed as well.

Asymptotic Total Variation Tests for Copulas

We propose a new platform of goodness-of-fit tests for copulas, based on empirical copula processes and nonparametric bootstrap counterparts. The standard Kolmogorov–Smirnov type test for copulas that takes the supremum of the empirical copula process indexed by orthants is extended by test statistics based on the empirical copula process indexed by families of Ln disjoint boxes, with Ln slowly tending to infinity. Although the underlying empirical process does not converge, the critical values of our new test statistics can be consistently estimated by nonparametric bootstrap techniques, under simple or composite null assumptions. We implemented a particular example of these tests and our simulations confirm that the power of the new procedure is oftentimes higher than the power of the standard Kolmogorov–Smirnov or the Cramer–von Mises tests for copulas.