Counterparty risk valuation of <i>k</i> th-to-default credit-linked notes with contagion risk

We evaluate the counterparty risk of kth-to-default credit-linked notes (CLN) with contagion risk under a Markov chain model. Assume that the interest rate follows a Cox-Ingersoll-Ross (CIR) process and both internal and external risks are taken into account in the context of contagion risk. We characterize the value of the kth-to-default CLN as a unique solution to a system of partial differential equations (PDE), each of which is associated with a default or shock realization state. Furthermore, the credit value adjustment (CVA) of the note with counterparty risk is also derived. Numerical simulations are conducted to show the effects of the correlated default risk on the kth-to-default CLN values and their CVAs.