A model to price puttable corporate bonds with default risk

Journal of the Academy of Business and Economics, Jan, 2004 by David Wang

ABSTRACT

This paper presents a mode/for pricing puttable corporate bonds that are subject to default risk. The mode/incorporates three essential ingredients in the pricing of defaultable puttable bonds. stochastic interest rate, default risk, and put provision. The stochastic interest rate is modeled as a square-root diffusion process. The default risk is modeled as a constant spread, with the magnitude of this spread impacting the probability of a Poisson process governing the arrival of the default event. The put provision is modeled as a constraint on the value of the bond in the finite difference scheme. This paper can be used both as a benchmark for models for pricing puttable corporate bonds that are subject to default risk and as a direction for future research.

1. INTRODUCTION

The pricing of defaultable securities has been of interest in the academic and practitioner literature for some time. The standard theoretical paradigm for pricing defaultable securities is the contingent claims approach pioneered by Black and Scholes (1973). Much of the literature follows Merton (1974) by explicitly linking the risk of a firm's default to the variability in the firm's asset value. Although this line of research has proven very useful in addressing the qualitatively important aspects of pricing defaultable securities, it has been less successful in practical applications. The lack of success owes to the fact that firms' capital structures are typically quite complex and priority rules are often violated. In response to these difficulties, an alternative modeling approach has been pursued in a number of papers, including Madan and Unal (1994), Jarrow and Turnbull (1995), Duffle and Singleton (1999). At each instant, there is some probability that a firm defaults on its obligation. This is called the instantaneous probability of default. The processes of both this probability and the recovery rate determine the value of default risk. Although these processes are not formally linked to the firm's asset value, there is presumably some underlying relation, thus Duffle and Singleton describe this alternative approach as a reduced-form model (Duffee, 1999). This paper is an effort to develop one such model for pricing puttable corporate bonds that are subject to default risk.

2. MODEL

I derive the pricing model for defaultable bonds by adopting the reduced-form approach by Duffle and Singleton (1999) and the replicating-portfolio approach by Neftci (2000).

2.1 Reduced-Form Approach

Reduced-form approaches directly assume that defaultable bonds can be valued by discounting at a default-adjusted interest rate. Specifically, I fix some defaultable discount bond that, in the event of no default, pays a face value X at maturity time T. I take as given an arbitrage-free setting in which all securities are priced in terms of some short-term interest rate process r and equivalent martingale measure Q. Under this risk-neutral probability measure, I let h denote the hazard rate for default (i.e., the instantaneous probability of default) at time t and let L denote the loss rate (i.e., the expected fractional loss in the market value) if default were to occur at time t, conditional on the information available up to time t. Under technical conditions, this defaultable discount bond can be priced as if it were default-free by replacing the usual short-term interest rate process rwith the default-adjusted short-term interest rate process:

(1) R = r hL.

That is, the price at time 0 of the defaultable discount bond is:

(2) [B.sub.0] = [E.sup.Q.sub.0][exp(-[[instegral].sup.T.sub.0][R.sub.t]dt],

where [E.sup.Q.sub.0] denotes risk-neutral, conditional expectation at date 0. This is natural, in that hL is the risk neutral mean-loss rate of the defaultable discount bond due to default. Discounting at the default-adjusted short-term interest rate R therefore accounts for both the probability and timing of default, as well as for the effect of losses on default. A key feature of Equation (2) is that, assuming the risk neutral mean-loss rate process hL being given exogenously, standard term-structure models for default-free debt are directly applicable to defaultable debt by parameterizing R instead of r.

2.2 Replicating-Portfolio Approach

I assume that the default-adjusted term structure R fits a Cox, Ingersoll, and Ross (CIR)-style model (1985). The model is extended to defaultable bonds by assuming a constant risk neutral mean-loss rate hL. Specifically, assume two stochastic differential equations (SDEs) describing the dynamics of two defaultable discount bond prices, B(t, [T.sub.1])and B(t, [T.sub.2]), with maturities [T.sub.1] and [T.sub.2]. The bond prices are driven by the same Wiener process, z To simplify the notation, I write:

(3) [B.sup.1] = B(t, [T.sub.1]),

(4) [B.sup.2] = B(t, [T.sub.1]).

The bond prices are postulated to have the following dynamics:

(5) d[B.sup.1] = [[mu].sub.1] ([B.sup.1], t) [B.sup.1] dt [[sigma].sub.1] ([B.sup.1], t) [B.sup.1] dz,