Analytical valuation of vulnerable derivative claims with bilateral cash flows under credit, funding and wrong-way risk

Abstract

We study the problem of valuing and hedging a vulnerable derivative claim with bilateral cash flows between two counterparties in the presence of asymmetric funding costs, defaults and wrong way risk (WWR). We characterize the pre-default claim value as the solution to a non-linear Cauchy problem. We show an explicit stochastic representation of the solution exists under a funding policy which linearises the Cauchy PDE. We apply this framework to the valuation of a vulnerable equity forward and show it can be represented as a portfolio of European options. Despite the complexity of the model, we prove the forward's value admits an analytical formula involving only elementary functions and Gaussian integrals. Based on this explicit formula, numerical analysis demonstrates WWR has a significant impact even under benign assumptions: with a parameter configuration less punitive than that representative of Archegos AM default, we find WWR can shift values for vulnerable forwards by 100bps of notional, while peak exposures increase by 25% of notional. This framework is the first to apply to contracts with bilateral cash flows in the presence of credit, funding and WWR, resulting in a non-linear valuation formula which admits a closed-form solution under a suitable funding policy.