Time‐Varying Asset Volatility and the Credit Spread Puzzle

• DOI: http://doi.org/10.1111/jofi.12765
• Author: DU DU,REDOUANE ELKAMHI,JAN ERICSSON
• Date: 01 August 2019
• Published date: 01 August 2019

THE JOURNAL OF FINANCE •VOL. LXXIV, NO. 4 •AUGUST 2019

Time-Varying Asset Volatility and the Credit

Spread Puzzle

DU DU, REDOUANE ELKAMHI, and JAN ERICSSON∗

ABSTRACT

Most extant structural credit risk models underestimate credit spreads—a shortcom-

ing known as the credit spread puzzle. We consider a model with priced stochastic

asset risk that is able to fit medium- to long-term spreads. The model, augmented by

jumps to help explain short-term spreads, is estimated on firm-level data and identi-

fies significant asset variance risk premia. An important feature of the model is the

significant time variation in risk premia induced by the uncertainty about asset risk.

Variousextensions are considered, among them optimal leverage and endogenous de-

fault.

STRUCTURAL CREDIT RISK MODELS HAVE MET WITH SIGNIFICANT difficulties in aca-

demic research. First, attempts to empirically implement models on individual

corporate bond prices have failed.1Second, efforts to calibrate models to ob-

servable moments including historical default rates and Sharpe ratios have

been unable to match average credit spreads levels (the credit spread puzzle;

Huang and Huang (2012)). Finally, models have been unable to jointly explain

dynamics of credit spreads and equity volatilities (Huang and Zhou (2008)).

An important recent insight is that model improvements are likely to

come from modeling risk premia rather than default probabilities (Chen,

∗Du Du is with the City University Hong Kong. Redouane Elkamhi is with the Rotman School of

Management at the University of Toronto. Jan Ericsson is with Desautels Faculty of Management

at McGill University. Min Jiang contributed to an earlier version of this paper. Ericsson has ben-

efited from financial support from a Desmarais Faculty Scholarship, from the Institut de Finance

Math´

ematique de Montr´

eal, as well as SSHRC. Du has benefited from financial support by the

National Natural Science Foundation of China (No. 71720107002). The paper has benefited from

comments by seminar participants at Case WesternReserve University, HEC Montreal, Hong Kong

University of Science and Technology, University of Houston, University of Iowa, Konstanz Uni-

versity, National University of Singapore, and University of Toronto and conference participants

at the 9th International Paris Finance Meeting 2011, ITAM 2012, Tremblant Risk Management

2012, Risk Management Conference NUS 2011, SKK, IFSID 2012, Montreal, and Affi 2016, Li`

ege.

Special thanks to David Bates, Peter Christoffersen, Sudipto Dasgupta, Jin Chuan Duan, Adlai

Fisher (discussant), Santiago Garc´

ıa Verd´

u, Kris Jacobs, Chanik Jo, Aytek Malkhozov, Franck

Moraux (discussant), Raunaq Pungaliya, Tom Rietz, Sheikh Sadik, Ashish Tiwari, Anand Vijh,

Hao Wang, and Hao Zhou (discussant). We thank Evan Zhou for exceptional research assistance.

The authors have read the Journal of Finance’s disclosure policy and have no conflicts of interest

to declare.

1See Jones, Mason, and Rosenfeld (1984) and Eom, Helwege, and Huang (2004).

DOI: 10.1111/jofi.12765

1841

1842 The Journal of Finance R

Collin-Dufresne, and Goldstein (2009, CCG)). Building on this insight, we de-

velop a structural model with time-varying priced asset volatility to explain

levels and dynamics of both credit spreads and equity volatilities. Our first

contribution is to show that, in calibrations, a reasonable unlevered asset

variance risk premium (AVRP) allows our model to match spread levels for

medium and longer maturities without difficulty. Our second contribution is

to estimate a stochastic volatility jump-diffusion (SVJ) model on firm-level

data for default swap spreads and equity volatility. We identify an economi-

cally significant AVRP and find that modeling priced stochastic asset volatility

strongly improves the model’s ability to not only account for time variation in

equity volatility, but also explain the time series of default swap spread term

structures.

Recent empirical work on default swap spreads provides evidence suggestive

of an important role for stochastic and priced asset volatility in credit risk

modeling.2Although a compelling extension to a class of models that has been

around for almost 40 years, stochastic volatility (SV) has not garnered much

attention in the credit risk literature. We present semi–closed-form (up to a

Fourier Inversion) solutions to debt and equity prices in a stochastic asset

volatility framework where default is triggered by a default boundary, as in

Black and Cox (1976), Longstaff and Schwartz (1995), and Collin-Dufresne and

Goldstein (2001).3In a model with SV, different aspects of variance dynamics

influence credit spreads. However, by far, the strongest effect arises from the

market price of asset volatility risk.

To illustrate this point, we replicate the Huang and Huang (2012)andChen

Collin-Dufresne and Goldstein (2009) calibrations. We confirm that, in the

absence of stochastic asset risk, our model replicates the credit spread puzzle.

Without a risk premium on asset variance, the model faces the same problem

that past studies have struggled with—it is hard to generate sufficiently high

spreads. The presence of stochastic variance per se does not help sufficiently in

2Huang and Zhou (2008) test a broad set of structural models and show the models’ inability

to fit the dynamics of credit default swap prices and equity volatilities. In particular, they find

that the models have difficulty generating sufficient time variation in equity volatility, suggesting

that an extension allowing for stochastic asset volatility is desirable. Zhang, Zhou, and Zhu (2009)

perform an empirical study of the effect of volatility and jumps on default swap prices. Their results

also point to the importance of modeling time-varying volatility. Further evidence is provided in

Wang, Zhou, and Zhou (2013), who show that not only are equity variance levels important for the

price of default protection, but the associated risk premium is also a key determinant of firm-level

credit spreads. Given this evidence, financial leverage would have to be the sole source of variation

in stock return volatility for asset volatility to be constant, as it is assumed to be in the majority

of structural credit risk models. Instead, recent empirical work by Choi and Richardsson (2016)

documents time variability in unlevered asset risk.

3Heston (1993) provides a closed-form solution for the price of a European option with Cox,

Ingersoll, and Ross’s (1985) dynamics for the variance. Fouque et al. (2003,2004,2011) introduce

perturbation techniques to address SV in a variety of option pricing settings. Related to our work,

Fouque et al. (2006) use a slow variation asymptotic approximation to a free boundary problem

with Gaussian variance dynamics to study defaultable securities. In the Internet Appendix, we

discuss these and other methodologies. The Internet Appendix may be found in the online version

of this article.

Time-Varying Asset Volatility and the Credit Spread Puzzle 1843

doing so. However, for reasonable parameter values governing the AVRP, our

model has no difficulty matching medium- to long-term spread levels. Because

short-term spreads remain hard to explain, consistent with previous work, we

introduce a jump component to our asset value dynamics using a specification

similartoPan(2002) that allows for a risk premium on jump-size uncertainty.4

To better understand this extension to our model, we conduct a calibration

exercise for a representative firm as we do for the SV case. We fix Sharpe ratios

and total volatility in our calibration. Note that introducing priced jumps affects

the amount of variance risk the model will bear—this would not matter if jump

and volatility risk were substitutes, but they influence different parts of the

term structure. We find a combination that does well in explaining spreads

across the term structure while allowing the model to fit both long- and short-

term default probabilities.

In addition to conducting calibration exercises for representative firms, we

run firm-by-firm estimations. In particular, we estimate firm-specific variance

risk premia using time series of default swap term structures and option-

implied volatilities for a cross section of firms with a variety of industry and

credit rating characteristics. The estimations require identification of the dy-

namics of asset values, volatilities, and jumps, all of which are unobservable.

We estimate physical and risk-adjusted firm value, volatility, and jump dynam-

ics using a simulated maximum likelihood (ML) methodology.

Tothe best of our knowledge, we provide the first estimates of firm-level asset

variance and jump dynamics together with their risk premia in the literature.

We obtain a broad range of estimates for the AVRP that are negative and sta-

tistically significant for a large majority of firms. The implied mean equity vari-

ance risk premium (EVRP) is −16.7%, with 5% and 95% percentiles of −44%

and −1.5%, respectively. The average jump intensity estimate across firms is

1%, with a cross-sectional variation of 0.1% to 4.6%, which is plausible given

the variation in credit ratings in our sample. The risk-adjusted jump size is

significant for the majority of firms, with an average of −81% of asset value.5

It is important to note that, in addition to explaining credit spread levels, our

model fits the level of equity volatility. Likelihood ratio tests show that our SV

model strongly dominates the nested constant volatility case.

To benchmark the level of our estimates, we translate them into an EVRP

that we compare with extant literature. The closest paper to our specification

is Pan (2002), who produces equity index-level estimates. Given the diversifi-

cation provided by an index, estimates for the index variance risk premium

4Since the advent of reduced-form credit models (see, e.g., Duffie and Singleton (1999)) that

are based on jumps-to-default, it has been noted that diffusion-based models generate insufficient

short-term spreads. Duffie and Lando (2001) find that jumps-to-default arise endogenously in a

structural model when asset values cannot be observed precisely and that this helps generate

significant short-term spreads. Collin-Dufresne, Goldstein, and Yang (2012) introduce jumps into

a structural model for credit index spreads for precisely this reason.

5In the Internet Appendix, we report on the estimation of the nested SV case and find that

the inclusion of jumps does not significantly change the estimates of the parameters of the volatil-

ity dynamics.