  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.
