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

Abstract

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.

Keywords:

• Stochastic term structure
• Defaultable bond
• Credit spread
• Probability of default
• Hazard rate

Acknowledgments

This work is supported by an ESRC grant, RES-000-22-0187. All errors are ours.

Notes

†r always stays positive for κ>0, μ>0 and 2κμ>σ².

† f(x_l , τ) = 0 and f(∞, τ) = 1.

‡This system of ODEs does not have a closed form solution, we solve it in a straightforward manner, using numerical methods.

§ 1 − f(x, τ)g(r, τ) is the probability that the default event is due to either the process x hitting its boundary or to the occurrence of the jump-default event before or at time T.

¶It is worth noting that the value of x/x_l is strictly greater than one, i.e. the firm is solvent when the bond is priced.

†See Geske (1977).

‡For a thorough survey see Davies and Martin (1979).

§The technique of Dubner and Abate (1968) based on Fourier series (Piessens 1969, Stehfest 1970, Zakian 1970).

† Derivations are available from the authors upon request.

† We note similar observations for the AA and A ratings. The results are available from the authors upon request.

‡ Equation (A14) is of the Ricatti type, we solve it using the Runge–Kutta method with initial conditions D(0) = 0 and D′(0) = b.

† In order to determine the weights A ₁ and A ₂ we need to take the Laplace of the boundary conditions.