Mathematical analysis of a credit default swap with counterparty risks

Abstract

A credit default swap (CDS) is an exchange of premium payments for a compensation for the occurrence of a credit event. Counterparty risks refer to defaults of parties holding CDS contracts. In this paper we develop a valuation/pricing model for a CDS subject to counterparty risks. Using the Cox–Ingersoll–Ross (CIR) model for interest rate and first arrival times of Poisson processes with variable intensities for the occurrences of credit default and counterparty defaults, we derive a mathematical formulation and make a full theoretical investigation. In addition, we develop a full theory for the corresponding infinite horizon problem and establish its connection with the asymptotic long expiry behaviour of finite horizon problem. Furthermore, we establish a connection between two major frameworks for default times: the structure model approach and the intensity model approach. We show that a solution of the structure model can be obtained as the limit of a sequence of solutions of intensity models. Regarded as an important theoretical development, we remove a constraint typically imposed on the parameters of the CIR model; that is, the well-posedness (existence, uniqueness and continuous dependence of parameters) of the mathematical model holds for any empirically calibrated parameters for the CIR model.

Why do banks use credit default swaps (CDS)? A systematic review

“…Even though more research work remains to be done on the subject, we believe this is a good time to review the pieces of evidence generated through the literature thus far that reflect upon the notable performance of CDS and the important contribution they could make. A credit default swap is an exchange of two cash flows between two counterparties: (i) protection buyer; and (ii) protection seller, to trade on the bond or mortgage of a reference entity and the cash flows depend upon the credit event (an event of credit default) of an underlying asset (Chen et al., 2020). Credit default occurs when a borrower becomes unsuccessful to pay the required interest or principal amount on time.…”

A double obstacle model for pricing bi-leg defaultable interest rate swaps

“…for the problem at hand, the condition φ r ∈ L ∞ will suffice our need. It is shown in [4] that problem (2.2), supplemented by φ r ∈ L ∞ , which we call the boundary condition at the origin, is well-posed: there exists a unique solution and the solution depends continuously on the parameters (κ, β, σ ) ∈ (0, ∞) 3 .…”

“…Pricing an IRS boils down to discounting the series of future cash flows for each leg [4]. A clear way of representing the term structure of interest rate is via a discount factor curve that reflects the present values of future payments.…”

“…..Hence, we obtain a solution of (5.6). The solution satisfies s ∈C ∞ ((0, ∞)), (s − h) 2 ∈ C 1 ([0, ∞)), 0 < w r (s(t) − , t) < N and h < s(t) z(h) for all t > 0 4…”