Pricing credit-risky bonds and spread options modelling credit-spread term structures with two-dimensional Markov-modulated jump-diffusion

Abstract

The relationship between company hazard rates and the business cycle becomes more apparent after a financial crisis. To address this relationship, a regime-switching process with an intensity function is adopted in this paper. In addition, the dynamics of both interest rates and asset values are modelled with a Markov-modulated jump-diffusion model, and a 2-factor hazard rate model is also considered. Based on this more suitable model setting, a closed-form model of pricing risky bonds is derived. The difference in yield between a risky bond and risk-free zero coupon bond is used to model a term structure of credit spreads (CSs) from which a closed-form pricing model of a call option on CSs is obtained. In addition, the degree to which the explicit regime shift affects CSs and credit-risky bond prices is numerically examined using three forward-rate functions under various business-cycle patterns.

Keywords:

• Markov-modulated jump-diffusion model
• Credit spread
• Business cycle

Acknowledgements

We would like to thank the associate editor and the anonymous referee for helpful comments and suggestions. We also thank Professor Shih-Kuei Lin for his useful suggestions regarding the Matlab code. The authors are indebted to the Ministry of Science and Technology of the Republic of China, Taiwan, for their financial support of this research under [grant number NSC 100-2410-H-264-004-].

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ Most CS data are associated with US Treasury bonds.

² Both the GARCH-jump (generalized autoregressive conditional heteroscedasticity-jump) and SV-jump (stochastic volatility-jump) models can be used to modify the shortcomings of the Black–Scholes model. Duan et al. (2006) developed the pricing of options under the GARCH-jump model, in which the interest rate is assumed to be constant. Liang et al. (2010) priced European options under the SV-jump model with the exponential Vasicek model, in which the interest rate follows a one-factor model. We adopt the MMJDM-HJM model because it has the merit of mathematical tractability under the two-factor hazard rate model. In addition, the RS default intensity model enabled capturing satisfactorily certain critical market features and economic behaviours.

³ Regarding equation (6(6) ), the reader may refer to Elliott et al. (2005).

⁴ The details can be found in Rabinovitch (1989).

⁵ The details can be found in Chiarella and Sklibosios (2003).

⁶ The settings of equations (22a) and (22b) indicate that the spot rate dynamic is a Markovain representation (Chiarella and Sklibosios 2003).

⁷ Because of limited space, the paper presents only the instance of f₁. The numerical results of other instances can be obtained from the authors.

⁸ The details can be found in Raible (2000).

⁹ The details can be found in Schoutens (2003).

Additional information

Funding

This work was financially supported by the Ministry of Science and Technology of the Republic of China Taiwan [grant number NSC 100-2410-H-264-004-].