Choice of Collateral Currency in Differential Swaps

The role of collateral in derivative pricing has evolved beyond credit risk mitigation, particularly following the global financial crisis, when funding costs and basis spreads became central to valuation practices. This development coincided with the transition from the London Interbank Offered Rate (LIBOR) to risk-free rates (RFRs) and the increasing standardization of collateralised trading. We study the valuation and hedging of a class of differential swaps referencing backward-looking averages of overnight rates, with SOFR swaps appearing as a particular instance. The focus is on the impact of the collateral currency. Extending earlier results Ding et al. [Math. Finance 36 (2026), pp.~180--202], we allow the collateral account to be denominated in a currency different from that of the contractual cash flows and derive explicit pricing and hedging strategies using a futures-based replication approach. We show that the choice of collateral currency can have a non-trivial effect on both valuation and risk management. In particular, foreign-currency collateral can introduce additional risk exposures even when contractual cash flows are entirely denominated in the domestic currency. Numerical study demonstrates that collateral effects can lead to significant valuation adjustments and therefore need to be properly incorporated in modern multi-currency modelling frameworks.