Pricing Defaultable Bonds Using a Lévy Jump‐Diffusion Model

• Author: Ming S. Tsai,Shu L. Chiang
• DOI: http://doi.org/10.1111/irfi.12243
• Published date: 01 September 2019
• Date: 01 September 2019

Pricing Defaultable Bonds Using a

Lévy Jump-Diffusion Model

SHU L. CHIANG

†

AND MING S. TSAI

‡

†

Department of Business Management, National Kaohsiung Normal University,

Kaohsiung, Taiwan and

‡

Department of Finance, National University of Kaohsiung, Kaohsiung, Taiwan

ABSTRACT

This paper uses a reduced-form approach to derive a closed-form pricing for-

mula for defaultable bonds. The authors specify the default hazard rate as an

affine function of multiple variables which follow the Lévy jump-diffusion

processes. Because such specification allows greater flexibility in the genera-

tion of a valid probability of default, their pricing model should be more

accurate than the valuation models in traditional studies, which ignore the

jump effects. This paper also proposes a new method for estimating the

parameters in a Lévy Jump-diffusion process. The real data from the Taiwan-

ese bond market are used to illustrate how their model can be applied in

practical situations. The authors compare the pricing results for the influen-

tial variables with no jump effects, with jump magnitudes following the nor-

mal distribution, and with jump magnitudes following the gamma

distribution. The results reveal that the predictive ability is the best for the

model with the jump components. The valuation model shown in this paper

should help portfolio managers more accurately price defaultable bonds and

more effectively hedge their portfolio holdings.

Accepted: 25 August 2018

I. INTRODUCTION

The defaultable bond is frequently traded by portfolio managers who face

sophisticated investment strategies. An appropriate and effective model for pric-

ing and analyzing defaultable bonds is essential for managers engaged in port-

folio management and the analysis of hedging risks. In order to obtain a

reasonable pricing model for a defaultable bond, it must take account of two

key risks: the default risk and the interest rate risk. Because of recent financial

crises (e.g., the subprime mortgage crisis and the credit crisis in the European

bond market), researching the default risk associated with jump events in eco-

nomic circumstances (e.g., abnormal shocks to the financial markets) has

become a very important task for financial institutions. The main purpose of

this paper is to adequately capture the default risks by embedding influential

© 2018 International Review of Finance Ltd. 2018

International Review of Finance, 19:3, 2019: pp. 613–640

DOI: 10.1111/irfi.12243

variables with the jump components in our model. Then, we present a reason-

able closed-form pricing formula

1

for defaultable bonds, given these

specifications.

Two methods are commonly used to study default risks: the structural-form

approach and the reduced-form approach. Early discussion of default risks was

based on using the structural-form approach to valuating defaultable securities

(Merton 1974; Black and Cox 1976; Leland 1994). Recently, the reduced-form

approach has been widely applied for evaluating a defaultable security (Jarrow

and Turnbull 1995; Jarrow 2001; Chiang and Tsai 2010). The reduced-form

approach treats default as an unpredictable but exogenous random variable.

The time at which the default occurs is assumed to follow an exponential distri-

bution (Bielecki and Rutkowski 2002). Several studies have used the reduced-

form model to specify the default hazard rate when investigating default risk

and defaultable bond yields (Duffee 1999; Arora et al. 2005). Because it is easier

to derive a closed-form formula with the reduced-form approach than with the

structural-form approach, we chose the reduced-form approach for this paper.

Several empirical studies have found that default risk is negatively correlated

with default-free interest rate and stock return (Duffee 1999; Huang and Kong

2003; Chiang and Tsai 2010). These correlations make it difficult to simulta-

neously take account of the risks associated with these factors in valuating

defaultable bonds. In discussions of the correlations, the default hazard rate is

usually specified as an affine function of influential variables such as the inter-

est rate and the stock return (Duan and Simonato 1999; Duffie and Singleton

1999; Dai and Singleton 2000; Duffie, Pan, and Singleton, 2000; Duffie 2005;

Chiang and Tsai 2010). For obtaining a closed-form pricing formula, the pro-

cesses of the influential variables are always assumed to be Weiner processes,

that is, normally distributed in traditional research (Schwartz and Torous 1992;

Chen and Yang 1995; Chiang and Tsai 2010).

It is important to well specify the distributions of influential variables for

evaluating a defaultable bond accurately. Many researchers have argued that

the influential variables usually are not normally distributed; for example, stock

return distributions usually have fat tails and excessive kurtosis (Mandelbrot

1963) and the distributions of interest rates exhibit excess kurtosis and skew-

ness (Lekkos 1999). Various studies also have confirmed the existence of jumps

in stock returns and interest rates (Bjork et al. 1997a, 1997b; Bjork and Chris-

tensen 1999; Tauchen and Zhou 2011). For this reason, numerous authors have

added jump components to the Weiner process to realistically describe the pop-

ulation distributions of the returns of financial securities and the distributions

of interest rates (Merton 1976; Amin 1993; Jarrow and Madan 1995; Eberlein

1 Some advantages of a closed-form formula are the following: (i) it helps portfolio managers

and financial institutions accurately appraise the value of defaultable bonds; (ii) it makes

hedging analyses relatively easy; (iii) it facilitates appreciation of how sensitive security values

are to changes in relevant factors; (iv) it improves calculation efficiency; (v) it provides basic

building blocks that financial institutions can use to price complicated financial products

(Liao et al. 2008).

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