Lina El-Jahel

Valuation of Defaultable Bonds

LINA EL-JAHEL is with Birkbeck College in London. n their seminal articles, Black and Scholes [1973] and Merton [1974] use option theory to price defaultable bonds. At the maturity date of some zerocoupon debt issue, default occurs if the value of the firm is less than the face value of the debt, and the payoff upon default is equal to the market value of the firm. Interest rates are assumed to be constant, which allows derivation of simple closed-form expressions for the price of defaultable bonds. A vast literature has used and extended this hamework; examples are Geske [1977], Ingersoll [1977a, 1977b1, and Merton [1977]. These models have failed to explain the large crel t spreads observed in practice; see, for example, Jones, Mason, and Rosenfeld [1984], Franks and Torous [1989], and IGm, Ramaswamy, and Sundaresan [1992]. The inability of these models to generate credible credit spreads has been attributed mainly to: 1) the assumption of constant interest rates; 2) the assumption that debtholders cannot force bankruptcy before the maturity of the debt; and 3) adherence to the strict absolute priority rule.2 Although Brennan and Schwartz [1980], Cooper and Mello [1991], Shmko, Tejima, and van Deventer [1993], and others have developed models with stochastic interest rates, and Black and Cox [1976] allow default to occur when the-value of the firms assets hits a lower threshold (i.e., before maturity), there has been limited success in explaining the behavior and magnitude of credit spreads. Either relaxation of all the assumptions above matters greatly, or a complete change of framework is necessary. In an effort to generate credible credit spreads, the subsequent literature on the valuation of defaultable debt has taken two approaches:

Value at Risk for Derivatives

Value at Risk is close to becoming the method of choice for assessing risk exposures faced by financial firms. The assets and liabilities in the firms risk portfolio are mapped onto a smaller set of standard factors, whose variances and correlations are given (estimated) The returns variance for the entire position is computed and the desired cutoff value for the lower tail of the portfolios returns distribution, under the typical assumption of joint lognormality, gives an estimate of the Value at Risk. But this procedure runs into large problems when derivatives with non-linear payoffs are included in the portfolio, an it also can be very difficult to implement with non-Gaussian returns. El Jahel, Perraudin, and Sellin present a new technique that greatly extends the range of assets and returns distribution that can be handled conveniently in a VaR calculation. For example, their framework easily accommodates non-linear payoffs, fat-tailed returns distributions, and stochastic volatility. It begins by computing the characteristic function for the whole portfolio, which is used to determine the first four moments of the returns distribution. Finally, an approximating distribution selected from a general family of distributions is fitted to these moments, and becomes the distribution used for VaR calculations. As the article shows, the results are distinctly more accurate than standard approaches.

Pricing defaultable bonds: a middle-way approach between structural and reduced-form models

In this paper we present a valuation model that combines features of both the structural and reduced-form approaches for modelling default risk. We maintain the cause and effect or ‘structural’ definition of default and assume that default is triggered when a state variable reaches a default boundary. However, in our model, the state variable is not interpreted as the assets of the firm, but as a latent variable signalling the credit quality of the firm. Default in our model can also occur according to a doubly stochastic hazard rate. The hazard rate is a linear function of the state variable and the interest rate. We use the Cox et al. (A theory of the term structure of interest rates. Econometrica, 1985, 53(2), 385–407) term structure model to preclude the possibility of negative probabilities of default. We also horse race the proposed valuation model against structural and reduced-form default risky bond pricing models and find that term structures of credit spreads generated using the middle-way approach are more in line with empirical observations.

Market and Model Credit Default Swap Spreads: Mind the Gap!

Structural models of default establish a relation across the fair values of various asset classes (equity, bonds, credit derivatives) referring to the same company. In most circumstances such relation is verified in practice, as different financial assets tend to move in the same direction at similar speed. However, occasional deviations from the theoretical fair values occur, especially in times of financial turmoil. Understanding how the dynamics of the theoretical fair values of various assets compares to that of their market values is crucial to a number of market participants. This paper investigates whether a popular structural model, the CreditGrades approach proposed by Finger (2002), Stamicar and Finger (2005), succeeds in explaining the dynamic relation between equity/option variables and Credit Default Swap (CDS) premia at individual company level. We find that CDS model spreads display a significant correlation with CDS market spreads. However, the gap between the two is time varying and widens substantially in times of financial turbulence. The analysis of the gap dynamics reveals that this is partly due to episodes of decoupling between equity and credit markets, and partly due to shortcomings of the model. Finally, we observe that model spreads tend to predict market spreads.

Multiple Defaults and Merton's Model

Multiple defaults and default correlations are crucial inputs in risk management, credit derivatives, and credit analysis. An extension of the structural framework to accommodate multiple defaults provides a simple and unified framework for calculating single and joint default probabilities in closed form for more than two firms. The results are useful in various financial applications.