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

• Author: Ze‐Zheng Li,Ping Li
• Published date: 01 September 2015
• DOI: http://doi.org/10.1111/eufm.12036
• Date: 01 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

E-mail: liping124@buaa.edu.cn

Ze‐Zheng Li

School of Finance, Renmin University, Beijing 100872, China

E-mail: lzzok8888@sina.com

Abstract

In this paper we use a type of dynamic copula method to characterise the dependence

structure between financial assets and price basket default swaps (BDSs). We first

employ a goodness‐of‐fit 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

find that in different time periods, the best copula fitting 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, 646–671

doi: 10.1111/eufm.12036

© 2013 John Wiley & Sons Ltd

reference entities to investors willing to assume the risk in exchange for the benefits.

Credit derivatives havebeen widely criticised and misunderstood in the wake of the global

financial 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 financial assets is an important problem

for both researchers and practitioners. It is useful in multi‐asset 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 financial 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 financial variables. He used this so‐called time‐varying

copula method to model the dependence structure between two exchange rates and

showed through empirical examples that the time‐varying copula model behaves better

than the static one. Wu and Liang (2011) incorporated time‐varying 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 goodness‐of‐fit

(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 & Poor’s 500 and NASDAQ indices;

that is, the authors dealt with two‐dimensional copulas.

The dynamic conditional correlation (DCC) model introduced by Engle (2002) is another

time‐varying 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 six‐dimensional copulas and then apply the dynamic copula method to the

pricing of a six‐dimensional BDS. There are two main approaches to pricing credit

derivatives: the structural approach initiated by Merton (1974) and the reduced‐form

approach proposed by Jarrow and Turnbull (1995). Recent related papers include those of

Brennan et al. (2009) and Bedendo et al. (2011). In pricing multi‐name credit derivatives,

a critical problem is to model the default correlation and copulas have been widely used

for this purpose. A milestone in this field is the work of Li (2000), who used a Gaussian

copula to model the default correlation and price multi‐name credit derivatives. This

© 2013 John Wiley & Sons Ltd

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