Pricing and Hedging of Rating's Risk

We consider the problem of pricing and hedging of defaultable rating-sensitive claims. By a defaultable rating-sensitive claim, we mean a classical one broadened by a migration process and payoffs connected with changing ratings. The credit rating migrations process is modeled by F-doubly stochastic Markov chains, a broad class of processes that contains Markov chains and are fully characterized by some martingale property. We give a general formula for the form of the ex-dividend price process of defaultable rating-sensitive claims in terms of processes defining this claim and characteristics of the rating migration process. This generalizes the known results obtained for the case without rating migration (see, e.g., Bielecki, Jeanblanc, and Rutkowski [2]). Subsequently, we consider the problem of hedging general rating-sensitive claims. The topic of credit risk hedging was started by Blanchet and Jeanblanc [3] and also Belanger, Shreve and Wong [1]. We prove that by trading on the market in default-free assets and a fixed number of defaultable general rating-sensitive claims, we can replicate arbitrary rating-sensitive claims. We find appropriate martingale representations which allow us to construct replication strategies.

REFERENCES

[1] A. Belanger, S.E. Shreve and D. Wong. "A General Framework For Pricing Credit Risk" Mathematical Finance 14, No. 3, (2004), 317{350.

[2] T. Bielecki, M. Jeanblanc and M. Rutkowski. "Pricing and trading Credit default Swaps in a hazard process model" Annals of Applied Probability, Volume 18, Number 6 (2008).

[3] C. Blanchet-Scalliet and M. Jeanblanc. "Hazard rate for credit risk and hedging defaultable contingent claims." Finance and Stochastics 8 (2004) 145-159.

[4] J.Jakubowski and M. Nieweglowski. "A class of F-doubly stochastic Markov chains" Submitted in Electronic Journal of Probability.

[5] J.Jakubowski and M. Nieweglowski. "Pricing and hedging of rating sensitive claims modeled by F-doubly stochastic Markov chains" To appear in AMaMeF volume.