Marek Rutkowski (U. of South Wales)

Pricing and Hedging of Convertible Bonds with Credit Risk

In our works [3]-[6], we attempt to shed more light on mathematical modeling of convertible bonds, thus continuing the previous research presented, for instance, in Andersen and Bu®um [1], Ayache et al. [2], Davis and Lischka [7], Kallsen and KÄuhn [8], and Kwok and Lau [9].

In [3], we consider the problem of the decomposition of a convertible bond into a bond component and an option component. This decomposition is indeed well established in the case of an `exchange option', when the conversion can only occur at maturity, and there are no put or call clauses. However, it was not previously studied in the general case of a defaultable convertible bond with call and/or put covenants.

In [4], we specify the valuation results for a defaultable game option (in particular, a convertible bond) to the context of default risk model based on the hazard process. The approach is based on the reduction of the information flow from the full ¯ltration to the reference ¯ltration. Our main existence result for hedging strategies in a hazard process set-up can be informally stated as follows: under the assumption that a related doubly re°ected BSDE admits a solution under some risk-neutral measure, the state-process multiplied by the default indicator process is the minimal super-hedging price up to a sigma martingale cost process.

The associated hedging strategies are subsequently analyzed by means of a martingale decomposition of a solution to the related doubly reflected BSDE. It is worth stressing that these decompositions are by no means arti¯cial. On the contrary, they arise naturally in the context of a Markovian framework, which is studied in some detail in the follow-up paper [5]. Under a rather general speci¯cation of the in¯nitesimal generator of a driving Markov factor process, we develop in [5] the variational inequality approach to pricing and hedging of a defaultable game option.

In [6], we consider a Markovian diffusion set-up with default. In this model, we show that a doubly reflected BSDE related to the convertible security has a solution, and we provide the related super-hedging strategy. Moreover, we characterize the price of a convertible security in terms of a viscosity solution of the associated variational inequality and we prove the convergence of a suitable approximation scheme.

References
[1] Andersen, L. and Buffum, L.: Calibration and implementation of convertible bond models. Journal of Computational Finance 7 (2004), 1-34
[2] Ayache, E., Forsyth, P. and Vetzal, K.: Valuation of convertible bonds with credit risk. The Journal of Derivatives, Fall 2003.
[3] Bielecki, T.R., Crepey, S., Jeanblanc, M. and Rutkowski, M.: Arbitrage pricing of defaultable game options with applications to convertible bonds. Forthcoming in Quantitative Finance.
[4] Bielecki, T., Crepey, S., Jeanblanc, M. and Rutkowski, M.: Valuation and hedging of defaultable options in a hazard process model. Submitted.
[5] Bielecki, T.R., Crepey, S., Jeanblanc, M. and Rutkowski, M.: Defaultable options in a Markovian intensity model of credit risk. Forthcoming in Mathematical Finance.
[6] Bielecki, T.R., Crepey, S., Jeanblanc, M. and Rutkowski, M.: Convertible bonds in a defaultable diffusion model. Submitted.
[7] Davis, M. and Lischka, F.: Convertible bonds with market risk and credit risk. In: Applied Probability, R. Chan et al., eds., American Mathematical Society/International Press, 2002, pp. 45-58.
[8] Kallsen, J. and KÄuhn, C.: Convertible bonds: ¯nancial derivatives of game type. In: Exotic Option Pricing and Advanced Levy Models, A. Kyprianou et al., eds., Wiley, 2005, pp. 277-288.
[9] Kwok, Y. and Lau, K.: Anatomy of option features in convertible bonds. Journal of Futures Markets 24 (2004), 513-532.