Partial differential equation pricing method for double-name credit-linked notes with counterparty risk in a reduced-form model with common shocks

In this study, we propose a partial differential equation method for pricing a double-name credit-linked note (CLN) with counterparty risk in a reduced-form framework. The default correlation among the CLN issuer and the reference entities of the double-name CLN is related to the occurrence of “common shocks,” which can simultaneously trigger defaults in some pre-specified groups of the aforementioned credit entities. By assuming that the stochastic intensities of common shocks are directly and inversely proportional to the spot interest rate, which follows a Cox-Ingersoll-Ross process, we deduce the corresponding explicit formulae for double-name CLN values. We also conducted some numerical experiments to examine how the correlated default risks among the credit entities affect the double-name CLN values and their credit valuation adjustment.