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Choice of Collateral Currency in Differential Swaps

Yining Ding, Ruyi Liu, Marek Rutkowski

Abstract

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.

Choice of Collateral Currency in Differential Swaps

Abstract

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.
Paper Structure (25 sections, 12 theorems, 136 equations, 5 figures, 3 tables)

This paper contains 25 sections, 12 theorems, 136 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Collateralised Trading Strategies and Pricing Measure
  3. Risk-free rates and related futures
  4. Differential swaps and collateral conventions
  5. collateralised futures strategies
  6. Pricing martingale measure
  7. Cross-Currency Term Structure Model
  8. Cross-currency model specification
  9. Dynamics of auxiliary discount factors
  10. Futures dynamics and the pricing martingale measure
  11. Restricted completeness
  12. Arbitrage-Free Pricing under Collateral Currency Choice
  13. Convexity corrections
  14. Pricing of swaps with full foreign collateralisation
  15. Pricing of swaps with proportional foreign collateralisation
  16. ...and 10 more sections

Key Result

Proposition 2.1

Let $(\varphi,C)$ be a self-financing collateralised futures strategy with $C=-\beta V(\varphi,C)$. Then the discounted wealth $\widetilde{V}^\beta(\varphi,C):=(B^\beta)^{-1}V(\varphi,C)$ satisfies where the discounted gains process is

Figures (5)

  • Figure 1: Closed-form par swap rate $\kappa^\star_0(T;\beta)$ for maturities $T\in\{1,2,3,5,7,10\}$ and collateralisation levels $\beta\in\{0,0.25,0.5,0.75,1\}$.
  • Figure 2: OAT $\pm10\%$ sensitivity of $\Delta\kappa^\star_0(5\mathrm{Y};1)$ (bp).
  • Figure 3: CTD: par-rate spread $\kappa^{\star,(\mathrm{EUR})}_0(T)-\kappa^{\star,(\mathrm{USD})}_0(T)$ (bp).
  • Figure 4: Gain process $G$ and hedging portfolio $X$, together with the pathwise error $X-G$ (daily rebalancing).
  • Figure 5: Normalised terminal error variance under weekly rebalancing ($T=5$, $\beta=1$).

Theorems & Definitions (35)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.1
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.1
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.2
  • ...and 25 more