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

We study the problem of valuing a vulnerable derivative with bilateral cash flows between two counterparties in the presence of funding, credit and wrong-way risks, and derive a closed-form valuation formula for an at-the-money (ATM) forward contract as well as a second order approximation for the general case. We posit a model with heterogeneous interest rates and default occurrence and infer a Cauchy problem for the pre-default valuation function of the contract, which includes ab initio any counterparty risk - as opposed to calculating valuation adjustments collectively known as XVA. Under a specific funding policy which linearises the Cauchy problem, we obtain a generic probabilistic representation for the pre-default valuation (Theorem 1). We apply this general framework to the valuation of an equity forward and establish the contract can be expressed as a continuous portfolio of European options with suitably chosen strikes and expiries under a particular probability measure (Theorem 2). Our valuation formula admits a closed-form expression when the forward contract is ATM (Corollary 2) and we derive a second order approximation in moneyness when the contract is close to ATM (Theorem 3). Numerical results of our model show that the forward is more sensitive to funding factors than credit ones, while higher stock funding costs increase sensitivity to credit spreads and wrong-way risk.