Change Analysis for the Dependence Structure and Dynamic Pricing of Basket Default Swaps

Published date01 September 2015
Date01 September 2015
Change Analysis for the Dependence
Structure and Dynamic Pricing of
Basket Default Swaps
Ping Li
School of Economics and Management, Beihang University, Beijing 100191, China
ZeZheng Li
School of Finance, Renmin University, Beijing 100872, China
In this paper we use a type of dynamic copula method to characterise the dependence
structure between nancial assets and price basket default swaps (BDSs). We rst
employ a goodnessoft test and a binary segmentation procedure to analyse the
change of dependence structure between the obligations underlying a BDS, then present
a numerical example to demonstrate the change analysis and BDS pricing process. We
nd that in different time periods, the best copula tting the data is not the same;
therefore the tranche spreads of the BDS are also different. We also compare our results
with those obtained from static copulas and dynamic Gaussian copulas. The results
show that the static copula and dynamic Gaussian copula methods underestimate the
spreads for riskier tranches and overestimate those for less risky tranches.
Keywords: change analysis, basket default swap, dynamic copula, dependence structure
JEL classification: G12, G13, G17
1. Introduction
In recent years, credit derivatives have become a major tool for transferring and hedging
credit risk. Instruments such as credit default swaps (CDSs), collateralised debt
obligations (CDOs) and basket default swaps (BDSs) can transfer the credit risk of
This work was supported by the National Natural Science Foundation of China (No.
71271015, 70971006, 70831001), and was done while Ping Li was visiting Columbia
University and the University of South Carolina. We acknowledge Steven Kou and Zhiliang
Ying of Columbia University and Hong Yan of the University of South Carolina for their
helpful discussions. We are also very grateful to the anonymous referee, the EFM journal
editor John Doukas and the English editor Joelma Nascimento for their great work on
improving the quality of the paper.
European Financial Management, Vol. 21, No. 4, 2015, 646671
doi: 10.1111/eufm.12036
© 2013 John Wiley & Sons Ltd
reference entities to investors willing to assume the risk in exchange for the benets.
Credit derivatives havebeen widely criticised and misunderstood in the wake of the global
nancial crisis; however, their important role in credit risk hedging and management
should not be overlooked. CDOs and BDSs are the two most important credit risk
management instruments, but they are also the most complicated, since the dependence
structure and joint distribution between the underlying obligations must be determined.
Determining the dependence structure between nancial assets is an important problem
for both researchers and practitioners. It is useful in multiasset derivative pricing, portfolio
selection, and portfolio risk management. Linear correlation has traditionally been used to
model dependence because of its simplicity in calculation and understanding, but it is
satisfactory only within the Gaussian or elliptical frameworks (Embrechts et al., 1999).
Increasingly, copulas have been widely used to model dependence because of their
advantages in characterising nonlinear and tail dependence and constructing multidimen-
sional distribution functions. However, most of the related literature is based on static
copulas, among which static Gaussian copulas are especially common (e.g., Li, 2000).
However, most nancial datasets cover long periods and economic factors can
therefore induce changes in the dependence structure. There are two ways to consider the
change of dependence structure when employing copulas: One is to consider the change
of copula parameters while the copula family keeps the same and the other one is to
consider the change of copula family. Patton (2006) considered the change of copula
parameters by assuming that the dependence measure is a function of the conditional
volatilities of the underlying nancial variables. He used this socalled timevarying
copula method to model the dependence structure between two exchange rates and
showed through empirical examples that the timevarying copula model behaves better
than the static one. Wu and Liang (2011) incorporated timevarying copulas into a range
based volatility model to describe the dependence structures and volatility of stock and
bond returns. Guégan and Zhang (2010) considered changes in both the copula family and
the copula parameters. They used tests based on conditional copulas and goodnessoft
(GOF) to determine the type of change and then applied their approach to compute the risk
measures VaR and ES for a portfolio of Standard & Poors 500 and NASDAQ indices;
that is, the authors dealt with twodimensional copulas.
The dynamic conditional correlation (DCC) model introduced by Engle (2002) is another
timevarying model used for correlations and dependence structures. This model is a new
kind of multivariate GARCH model that allows the correlation matrix to vary with time.
Cappiello et al. (2006) subsequently presented the asymmetric generalised DCC model to
study conditional asymmetries in correlation dynamics and applied their model to analyse the
behaviour of international equities and government bonds. The DCC and asymmetric
generalised DCC models are good but are built on the assumption of normality. In contrast,
the dynamic copula model is not restricted by the assumption of marginal distributions.
In this paper we analyse the change of copulas, as Guégan and Zhang (2010) did, but we
deal with sixdimensional copulas and then apply the dynamic copula method to the
pricing of a sixdimensional BDS. There are two main approaches to pricing credit
derivatives: the structural approach initiated by Merton (1974) and the reducedform
approach proposed by Jarrow and Turnbull (1995). Recent related papers include those of
Brennan et al. (2009) and Bedendo et al. (2011). In pricing multiname credit derivatives,
a critical problem is to model the default correlation and copulas have been widely used
for this purpose. A milestone in this eld is the work of Li (2000), who used a Gaussian
copula to model the default correlation and price multiname credit derivatives. This
© 2013 John Wiley & Sons Ltd
Dependence Structure and Dynamic Pricing of Basket Default Swaps 647

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