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.

中文翻译：

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具有交易对手风险的信用违约掉期数学分析

信用违约掉期 (CDS) 是为信用事件的发生而支付的保费交换。交易对手风险是指持有 CDS 合约的当事人的违约。在本文中，我们为受交易对手风险影响的 CDS 开发了一个估值/定价模型。使用 Cox-Ingersoll-Ross (CIR) 模型来研究信用违约和交易对手违约发生的不同强度的 Poisson 过程的利率和首次到达时间，我们推导出了一个数学公式并进行了全面的理论研究。此外，我们为相应的无限视界问题发展了一个完整的理论，并建立了它与有限视界问题的渐近长到期行为的联系。此外，我们在两个主要框架之间建立了默认时间的连接：结构模型法和强度模型法。我们证明了结构模型的解可以作为强度模型解序列的极限来获得。作为一项重要的理论发展，我们消除了通常施加在 CIR 模型参数上的约束；也就是说，数学模型的适定性（参数的存在性、唯一性和连续相关性）适用于 CIR 模型的任何经验校准参数。