Liquidity Risk and Volatility Risk in Credit Spread Models: A Unified Approach

• DOI: http://doi.org/10.1111/eufm.12127
• Published date: 01 October 2017
• Date: 01 October 2017

Liquidity Risk and Volatility Risk in

Credit Spread Models: A Unified

Approach

Stylianos Perrakis

John Molson School of Business, Concordia University, 1455 De Maisonneuve Blvd West, Montreal

Quebec, Canada H3G 1M8

E-mail: stylianos.perrakis@concordia.ca

Rui Zhong

Chinese Academy of Finance and Development, Central University of Finance and Economics, P.R.

China, 39 South College Road, Haidian District, Beijing, P. R. of China, 100081

E-mail: ruizhong@outlook.com

Abstract

We present an integrated framework incorporating both exogenous liquidity risk in

the secondary corporate bond market and volatility risk in the dynamics of asset

value in debt rollover models. Using an innovative theoretical approach we derive

general expressions for the debt and equity values in all cases. Taking advantage of

the analytical expressions for the asset value with the constant elasticity of

variance (CEV) process, we show numerically using realistic parameter values

from empirical studies that volatility risk, together with deteriorating bond market

liquidity, decreases both debt and equity values and significantly increases the

credit spreads.

Keywords: liquidity risk, volatility risk, credit risk, structural model

JEL classification: G12,G13,G32,G3

The authors thank John Doukas (the Editor), an anonymous referee, Jan Ericsson, Nikolay

Gospodinov, Zhiguo He, Sergey Isaenko, George Jiang, Lawrence Kryzanowski, Pascal

Francois and participants at the International Symposium on Financial Engineering and

Risk Management and 2013 Mathematical Finance Days for their valuable comments.

Financial support from the Social Sciences and Humanities Research Council (SSHRC),

National Natural Science Foundation of China (NNSFC, Project No. 71501197), the RBC

Distinguished Professorship in Financial Derivatives and the Institute of Structural Finance

and Derivatives (IFSID) is gratefully acknowledged.

European Financial Management, Vol. 23, No. 5, 2017, 873–901

doi: 10.1111/eufm.12127

© 2017 John Wiley & Sons, Ltd.

1. Introduction

The credit spread of a firm is defined as the yield increment of its risky bonds over the

riskless rate. This spread primarily reflects its default risk, but also other important

factors, such as the liquidity risk of the secondary bond market, the volatility risk of a

firm’s asset and macroeconomic conditions. The default risk has been modelled

theoretically and measured empirically in a large number of studies that are known as

structural models of the firm and follow the pioneering work of Merton (1974) and Black

and Cox (1976), which in turn were inspired by the seminal Black and Scholes (1973)

model of option pricing.

Relatively less attention has been paid to the impact of liquidity risk and volatility risk

on the credit spreads. The interaction between corporate bond market liquidity and credit

spreads was documented empirically by Longstaff et al. (2005) and Chen et al. (2007).

In Ericsson and Renault (2006) bond market illiquidity plays a role only upon default and

yields a positive correlation between illiquidity and the default components of the yield

spread. Illiquidity in the secondary bond market, however, is present whenever there is

debt refinancing through rollover or other reasons, and not just upon default. Liquidity

shocks in that market are particularly serious, since the firm may not find investors

willing to refinance their debt except at a significant cost. This issue was formally studied

by Acharya, Gale and Yorulmazer (Acharya et al. 2011), He and Xiong (HX, 2012a,b)

and He and Milbradt (2014) who showed theoretically using different models that the

liquidity risk of the secondary corporate bond market amplifies the default component of

the credit spread of a firm for both exogenous and endogenous default boundaries. These

studies modeled bond market illiquidity and showed that it constitutes a ‘friction’that

comes from liquidation costs of the assets when the debt refinancing is done by collateral

borrowing, as in the stylised two-state model of AGY, or by having the equity owners

absorb the debt rollover costs in the context of the structural models of the firm of Leland

(L, 1994b) and Leland and Toft (LT, 1996), as in HX.

1

Recent empirical evidence has shown that constant asset volatility does not hold in the

real world, thus motivating the study of the impact of volatility risk on credit spreads.

2

Perrakis and Zhong (2015) and Elkamhi et al (2013) show, respectively, that state-

dependent volatility and stochastic volatility models outperform constant volatility as in

Huang and Zhou (2008) and Huang and Huang (2012) in empirically approximating

observed credit default swap (CDS) spreads. Further, in the absence of illiquidity in the

secondary corporate debt market, Perrakis and Zhong (2015) find that the endogenous

default boundary decreases if there is a negative relationship between asset volatility and

asset value, unlike what happens under constant volatility because of the option-like

properties of equity value. In reality, both liquidity risk and volatility risk are significant

determinants of the default probabilities of levered firms, which in turn play a major role

in such key economic issues as the rating of a firm’s bonds and the valuation of its credit

derivatives such as CDS.

3

However, the conflicting impact of liquidity risk and volatility

risk on the determinants of credit risk limits the capability of models incorporating

1

He and Milbradt (2014) also include frictions in the post-default bond market.

2

See Choi and Richardson (2009), Huang and Zhou (2008), Zhang et al. (2009), and Perrakis

and Zhong (2015).

3

The CDS played a major role in the 2008 financial crisis; see Stulz (2010).

© 2017 John Wiley & Sons, Ltd.

874 Stylianos Perrakis and Rui Zhong