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Mathematical analysis of a credit default swap with counterparty risks

Part of: Partial differential equations Mathematical finance

Published online by Cambridge University Press:  09 September 2019

XINFU CHEN ,
PENG HE ,
JING LIU Open the ORCID record for JING LIU [Opens in a new window]  and
SHUAI ZHAO
XINFU CHEN
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA15260, USA emails: xinfu@pitt.edu; jil156@pitt.edu
PENG HE
Affiliation:
Model Risk Management Group, PNC Financial Service Group, Pittsburgh, PA15222, USA email: peh33@pitt.edu
JING LIU
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA15260, USA emails: xinfu@pitt.edu; jil156@pitt.edu
SHUAI ZHAO
Affiliation:
Model Risk Management Group, U.S. Bank Financial Services Company, Richfield, MN55433, USA email:shz40@pitt.edu
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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.

Keywords

Credit default swapcounterparty riskstructure modelintensity modelCIR model

MSC classification

Primary: 91G40: Credit risk
Secondary: 35A02: Uniqueness problems 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems) 35A01: Existence problems
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 5 , October 2020 , pp. 737 - 762
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Copyright
© Cambridge University Press 2019

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Footnotes

†

This research was partially supported by the National Science Foundation grant DMS–1516344.

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