Differential Swaps: Structure & Valuation

• Differential swaps are swap contracts whose floating legs are defined through incremental differences, utilizing backward-looking averages and path-functional design.
• They incorporate collateral-currency effects and funding adjustments that materially influence pricing, especially over longer maturities.
• Advanced hedging techniques—including futures-based replication and model-free discretisation invariance—enable robust valuation and dynamic risk management.

Searching arXiv for papers on differential swaps and closely related swap-contract frameworks. Differential swaps are swap contracts whose floating legs are defined through incremental differences rather than solely through terminal levels, but the term is not used in a single universal sense across the literature. In the recent rates literature, it denotes contracts built from backward-looking averages of domestic and foreign overnight rates, with SOFR swaps as a special case (Ding et al., 9 Mar 2026). In a broader model-free derivatives literature, discretisation-invariant swap contracts are presented as a general mathematical umbrella for differential swap-style products, because their realised legs are functions of increments that aggregate cleanly in expectation (Alexander et al., 2016). In valuation theory for uncollateralized interest-rate swaps, “differential swap pricing” refers to pricing relative to a fully collateralized CCP benchmark under a discount rate that switches with the sign of the local mark-to-market (Lou, 2015). Taken together, these strands define differential swaps as path-functional swap structures whose valuation depends on how incremental payoffs, collateralization, and discounting conventions are specified.

1. Contract class and payoff architecture

The paper "Choice of Collateral Currency in Differential Swaps" defines a class of USD-referencing differential swaps whose payoff is built from backward-looking averages of overnight rates (Ding et al., 9 Mar 2026). For an accrual period $[U,T]$ with $\delta:=T-U$, the domestic and foreign overnight averages are

$$R^d(U,T):=\frac{1}{\delta}\left(\exp\left(\int_U^T r_u^d\,du\right)-1\right), \qquad R^f(U,T):=\frac{1}{\delta}\left(\exp\left(\int_U^T r_u^f\,du\right)-1\right),$$

where $r^d$ is the domestic overnight rate and $r^f$ the foreign overnight rate. These averages are backward-looking because they are $\mathcal{F}_T$-measurable but not known at time $t<T$ (Ding et al., 9 Mar 2026).

A single-period SOFR/$\text{\euro STR}$ differential swap over $[U,T]$ is settled in arrears at $T$ with payoff

$\delta:=T-U$0

where $\delta:=T-U$1 is the domestic notional, $\delta:=T-U$2 is the fixed rate, and $\delta:=T-U$3 scales the foreign floating leg (Ding et al., 9 Mar 2026). Using the exponential representation of the averages, the payoff can be rewritten as

$\delta:=T-U$4

The role of $\delta:=T-U$5 is structurally decisive. If $\delta:=T-U$6, the product reduces to a standard SOFR swap; if $\delta:=T-U$7, the contract is a differential swap; and it can also be viewed as a quanto-style swap if $\delta:=T-U$8 represents a fixed conversion factor (Ding et al., 9 Mar 2026). The same construction extends to multi-period swaps with payment dates $\delta:=T-U$9, where the $$R^d(U,T):=\frac{1}{\delta}\left(\exp\left(\int_U^T r_u^d\,du\right)-1\right), \qquad R^f(U,T):=\frac{1}{\delta}\left(\exp\left(\int_U^T r_u^f\,du\right)-1\right),$$0-th coupon is

$$R^d(U,T):=\frac{1}{\delta}\left(\exp\left(\int_U^T r_u^d\,du\right)-1\right), \qquad R^f(U,T):=\frac{1}{\delta}\left(\exp\left(\int_U^T r_u^f\,du\right)-1\right),$$1

This contract architecture places differential swaps within the family of realized-rate structures whose floating legs depend on accrued path functionals over each coupon period rather than on a single reset fixing. A plausible implication is that their valuation inherits the path dependence typical of backward-looking overnight-rate products, while their hedging inherits cross-currency structure whenever $$R^d(U,T):=\frac{1}{\delta}\left(\exp\left(\int_U^T r_u^d\,du\right)-1\right), \qquad R^f(U,T):=\frac{1}{\delta}\left(\exp\left(\int_U^T r_u^f\,du\right)-1\right),$$2 or the collateral currency differs from the cash-flow currency.

2. Aggregation-property foundations and discretisation-invariant generalizations

The paper "Model-Free Discretisation-Invariant Swap Contracts" presents a broader framework in which realised payoffs are defined by increment functions $$R^d(U,T):=\frac{1}{\delta}\left(\exp\left(\int_U^T r_u^d\,du\right)-1\right), \qquad R^f(U,T):=\frac{1}{\delta}\left(\exp\left(\int_U^T r_u^f\,du\right)-1\right),$$3 satisfying Neuberger’s aggregation property (Alexander et al., 2016). For an adapted process $$R^d(U,T):=\frac{1}{\delta}\left(\exp\left(\int_U^T r_u^d\,du\right)-1\right), \qquad R^f(U,T):=\frac{1}{\delta}\left(\exp\left(\int_U^T r_u^f\,du\right)-1\right),$$4 and a payoff function $$R^d(U,T):=\frac{1}{\delta}\left(\exp\left(\int_U^T r_u^d\,du\right)-1\right), \qquad R^f(U,T):=\frac{1}{\delta}\left(\exp\left(\int_U^T r_u^f\,du\right)-1\right),$$5, the aggregation property is

$$R^d(U,T):=\frac{1}{\delta}\left(\exp\left(\int_U^T r_u^d\,du\right)-1\right), \qquad R^f(U,T):=\frac{1}{\delta}\left(\exp\left(\int_U^T r_u^f\,du\right)-1\right),$$6

where $$R^d(U,T):=\frac{1}{\delta}\left(\exp\left(\int_U^T r_u^d\,du\right)-1\right), \qquad R^f(U,T):=\frac{1}{\delta}\left(\exp\left(\int_U^T r_u^f\,du\right)-1\right),$$7 (Alexander et al., 2016). In this framework, a swap’s floating leg is computed by summing realised increment payoffs over a monitoring partition, and if the aggregation property holds then the fair value is partition-invariant and free of discrete-monitoring bias (Alexander et al., 2016).

The paper’s main technical result is a characterization of discretisation-invariant payoffs through a second-order PDE system,

$$R^d(U,T):=\frac{1}{\delta}\left(\exp\left(\int_U^T r_u^d\,du\right)-1\right), \qquad R^f(U,T):=\frac{1}{\delta}\left(\exp\left(\int_U^T r_u^f\,du\right)-1\right),$$8

where $$R^d(U,T):=\frac{1}{\delta}\left(\exp\left(\int_U^T r_u^d\,du\right)-1\right), \qquad R^f(U,T):=\frac{1}{\delta}\left(\exp\left(\int_U^T r_u^f\,du\right)-1\right),$$9 is the Jacobian of $r^d$0, $r^d$1 is the Hessian, and $r^d$2 and $r^d$3 are the first and second derivatives of $r^d$4 with respect to $r^d$5 (Alexander et al., 2016). The paper then imposes

$r^d$6

with $r^d$7 a vector of martingale forward prices, and shows that admissible DI payoffs form a vector space: $r^d$8

Within this theory, standard realised variance $r^d$9 does not satisfy the aggregation property, whereas Neuberger’s log-variance payoff

$r^f$0

does (Alexander et al., 2016). Its fair value is

$r^f$1

with $r^f$2 the time-0 price of an OTM option with strike $r^f$3 (Alexander et al., 2016).

The connection to differential swaps is explicit in the paper’s interpretation: DI swaps are described as a general mathematical umbrella for differential swap-style contracts, because the realised leg is a function of increments, the aggregation property ensures telescoping in expectation, the contract can be priced model-free, and the hedging strategy is linked to dynamic rebalancing of fundamental traded claims (Alexander et al., 2016). This suggests that, in the most general usage, a differential swap is not restricted to a particular asset class or payoff kernel, but denotes a structural principle: the floating leg is engineered at the increment level so that pricing and replication become more robust to monitoring conventions.

3. Pricing under collateralization and collateral-currency choice

A central result of (Ding et al., 9 Mar 2026) is that the collateral currency can matter even when all cash flows are domestic. The paper separates the cash-flow currency, which is USD for the swap payoff, from the collateral currency, which may be USD or EUR. If collateral is posted in a foreign currency, then

$r^f$4

where $r^f$5 is collateral in foreign units and $r^f$6 is the FX rate quoted as domestic currency per unit of foreign currency (Ding et al., 9 Mar 2026). The collateral remuneration rate is specified as

$r^f$7

for domestic collateral and

$r^f$8

for foreign collateral.

Under proportional collateralization,

$r^f$9

with $\mathcal{F}_T$0 the hedger’s wealth and $\mathcal{F}_T$1 the collateralization level, the effective funding rate becomes

$\mathcal{F}_T$2

(Ding et al., 9 Mar 2026). Hence $\mathcal{F}_T$3 corresponds to funding at $\mathcal{F}_T$4, $\mathcal{F}_T$5 to funding at $\mathcal{F}_T$6, and intermediate $\mathcal{F}_T$7 to partial collateralization. The paper’s pricing relation is

$\mathcal{F}_T$8

where $\mathcal{F}_T$9 is the fictitious collateral/funding account defined by

$t<T$0

For proportional foreign collateralization, the single-period differential swap price is decomposed as

$t<T$1

with

$t<T$2

$t<T$3

$t<T$4

(Ding et al., 9 Mar 2026). These expressions show explicitly that collateral currency enters valuation through the mix of domestic and foreign bond factors.

The numerical study in (Ding et al., 9 Mar 2026) is designed to isolate collateral effects by setting

$t<T$5

so that the payoff is a pure SOFR swap and any foreign dependence comes only from collateralization. The paper finds that the par swap rate $t<T$6 increases with $t<T$7, that the effect is small for short maturities but material at long maturities, and that for full foreign collateralization ($t<T$8) the shift reaches several basis points (Ding et al., 9 Mar 2026). Reported par-rate shifts relative to $t<T$9 are $\text{\euro STR}$0 bp at 1Y, $\text{\euro STR}$1 bp at 2Y, $\text{\euro STR}$2 bp at 3Y, $\text{\euro STR}$3 bp at 5Y, $\text{\euro STR}$4 bp at 7Y, and $\text{\euro STR}$5 bp at 10Y (Ding et al., 9 Mar 2026). A one-at-a-time $\text{\euro STR}$6 perturbation study for a 5Y swap identifies FX volatility, domestic-FX correlation $\text{\euro STR}$7, and domestic short-rate volatility as the most important sensitivities (Ding et al., 9 Mar 2026).

Under a multi-currency CSA with collateral choice in $\text{\euro STR}$8, the EUR-collateralized par curve lies uniformly above the USD-collateralized curve in the baseline, and the gap widens with maturity (Ding et al., 9 Mar 2026). The paper interprets this as a cheapest-to-deliver effect. This suggests that collateral optionality is not an operational afterthought but an embedded state-dependent funding feature of the contract.

4. Futures-based replication and hedging structure

The hedging methodology in (Ding et al., 9 Mar 2026) is built around exchange-traded futures on backward-looking overnight averages. The domestic and foreign futures rates are

$\text{\euro STR}$9

identified as SOFR futures and $[U,T]$0 futures, respectively (Ding et al., 9 Mar 2026). A futures trading strategy is written as $[U,T]$1, where $[U,T]$2 and $[U,T]$3 denote positions in SOFR and $[U,T]$4 futures and $[U,T]$5 denotes the cash/funding position.

The self-financing condition is

$[U,T]$6

where foreign futures gains in domestic currency are represented by

$[U,T]$7

The quadratic covariation term $[U,T]$8 is essential because daily settlement of futures and FX conversion generate extra drift and convexity effects (Ding et al., 9 Mar 2026).

The paper proves a restricted completeness result: if the discounted price process depends only on the first two Brownian drivers, then the market is replicable using SOFR and $[U,T]$9 futures (Ding et al., 9 Mar 2026). In that case there is a unique hedging strategy with positions

$T$0

where $T$1 is the martingale representation of the discounted price.

For the single-period swap under full foreign collateralization $T$2, the replicating strategy has

$T$3

and hedge ratios

$T$4

$T$5

(Ding et al., 9 Mar 2026). For constant proportional collateralization $T$6, the replicating strategy satisfies

$T$7

and the foreign-futures hedge remains necessary whenever $T$8 (Ding et al., 9 Mar 2026).

The hedging experiments reported in (Ding et al., 9 Mar 2026) show that daily rebalancing tracks the target price very closely, that using domestic futures only leaves a residual hedging error under foreign collateralization, and that adding foreign futures removes almost all of the residual error. For a 5Y swap under $T$9 with weekly rebalancing, the paper reports

$\delta:=T-U$00

This indicates that domestic hedging removes most risk, but about $\delta:=T-U$01 of unhedged variance remains if foreign futures are omitted (Ding et al., 9 Mar 2026). A plausible implication is that collateral-currency choice creates a hedge dimension that cannot be neutralized by domestic rates instruments alone.

5. Liability-side pricing, switching discount rates, and coherent valuation adjustments

A different use of differential swap terminology appears in "Liability-side Pricing of Swaps and Coherent CVA and FVA by Regression/Simulation" (Lou, 2015). There, an uncollateralized swap is valued relative to a fully collateralized CCP swap by dynamic replication with a time-varying hedge notional chosen to eliminate open IR01 (Lou, 2015). The paper’s central valuation rule is the state-dependent discount rate

$\delta:=T-U$02

where $\delta:=T-U$03 is the fair value of the uncollateralized swap from the dealer’s perspective, $\delta:=T-U$04 is the dealer’s own effective funding or bond curve, and $\delta:=T-U$05 is the counterparty’s effective curve (Lou, 2015).

The pricing formula is

$\delta:=T-U$06

or, for a terminal payoff $\delta:=T-U$07,

$\delta:=T-U$08

If the short-rate driver $\delta:=T-U$09 follows

$\delta:=T-U$10

then the uncollateralized swap value solves the switching-rate PDE

$\delta:=T-U$11

with terminal condition given by the swaplet payoff (Lou, 2015).

Within this framework, the “differential” aspect is the valuation difference between the risk-free / CCP-swap price and the liability-side risky price under switching discounting (Lou, 2015). The paper writes

$\delta:=T-U$12

and insists that CVA and FVA should be computed coherently from the same replication model rather than as separate add-ons (Lou, 2015). The recursive character of the problem comes from the fact that the discount-rate switch depends on the sign of $\delta:=T-U$13 itself. To solve this recursion, the paper adapts a Longstaff–Schwartz least-squares simulation scheme in which the fitted continuation value is used to determine the local switching rule (Lou, 2015).

The numerical findings reported in (Lou, 2015) include a decline in the fair ATM swap rate as the counterparty spread widens, a CCP hedge ratio that falls below $\delta:=T-U$14 in stressed cases, and bid/ask spreads that emerge because the switching-discount PDE is position asymmetric. The paper also reports that finite-difference and regression/simulation results match very closely—differences of only a few hundredths of a basis point—while brute-force Monte Carlo without regression can be off by several basis points or more (Lou, 2015). This suggests that differential swap pricing, in the liability-side sense, is fundamentally a nonlinear pricing problem driven by endogenous discounting.

6. Relation to variance-style contracts and adjacent realized-payoff swaps

Although (Lorig et al., 2012) is a paper on variance swaps rather than on differential swaps in the rates sense, it states explicitly that its framework is directly relevant to differential-style swaps because valuation is driven by the quadratic variation of $\delta:=T-U$15 under a default-adjusted time-changed semimartingale (Lorig et al., 2012). The paper modifies the usual variance swap payoff so that realized variance is accumulated only up to the pre-default time, with floating leg

$\delta:=T-U$16

and fair strike

$\delta:=T-U$17

(Lorig et al., 2012). The factor $\delta:=T-U$18 excludes default from the realized leg, preventing the quadratic variation from blowing up at default (Lorig et al., 2012).

The same paper emphasizes the decomposition

$\delta:=T-U$19

with default excluded by the factor $\delta:=T-U$20 (Lorig et al., 2012). It then notes that this is the same kind of decomposition one would use to price differential swaps, corridor variance swaps, or other realized-volatility contracts that pay on realized path functionals rather than terminal levels (Lorig et al., 2012). In that setting, the underlying is modeled as

$\delta:=T-U$21

where $\delta:=T-U$22 is a scalar Feller diffusion with local stochastic volatility and state-dependent killing/default intensity, and $\delta:=T-U$23 is a Lévy subordinator (Lorig et al., 2012).

A further connection arises from the paper’s market-implied time-change result. Once the background diffusion $\delta:=T-U$24 is fixed, the Laplace exponent $\delta:=T-U$25 of the subordinator can be recovered nonparametrically from European call/put prices via

$\delta:=T-U$26

using the transition density and Breeden–Litzenberger (Lorig et al., 2012). The paper describes this as a robustness feature: once the background diffusion is fixed and the market-implied subordinator is inferred from option smiles, the swap value is essentially pinned down without parametric assumptions on the time-change (Lorig et al., 2012).

This variance-swap literature does not define differential swaps directly, but it shows that the same analytical ingredients recur across realized-payoff contracts: pathwise accumulation, decomposition into continuous and jump components, robustness to monitoring conventions, and calibration from option-implied information.

7. Conceptual synthesis and scope of the term

The cited literature supports three non-identical but connected meanings of differential swaps. First, in overnight-indexed rates markets, they are contracts with floating legs of the form

$\delta:=T-U$27

with SOFR swaps as the special case $\delta:=T-U$28 (Ding et al., 9 Mar 2026). Second, in the model-free DI framework, differential swap-style products are swaps whose realised legs are functions of increments satisfying an aggregation property, so that fair values are invariant to the monitoring partition and may be free of jump and discrete-monitoring errors (Alexander et al., 2016). Third, in liability-side pricing, differential swap valuation is the coherent price difference between an uncollateralized swap and its CCP benchmark under a switching discount rule tied to the local sign of the mark-to-market (Lou, 2015).

These meanings are compatible at the level of structure rather than nomenclature. Each emphasizes that the economically relevant object is not simply a terminal payoff, but a sequence of pathwise differentials or locally signed cash-flow adjustments. Each also treats replication as central: futures-based replication in the collateral-currency setting (Ding et al., 9 Mar 2026), dynamic hedging with forwards, options, and power-log contracts in the DI setting (Alexander et al., 2016), and dynamic CCP hedging plus funding/deposit replication in liability-side pricing (Lou, 2015). The variance-swap literature further indicates that differential-style valuation naturally extends to realized-volatility and realized-jump functionals once default and monitoring conventions are specified carefully (Lorig et al., 2012).

A common misconception is that collateral should matter only for credit mitigation, not for valuation when contractual cash flows are domestic. The foreign-collateral results of (Ding et al., 9 Mar 2026) directly contradict that view: foreign-currency collateral can introduce additional risk exposures even when contractual cash flows are entirely denominated in the domestic currency. Another misconception is that uncollateralized swap valuation can be handled by discounting positive and negative cash flows separately on different curves; (Lou, 2015) argues instead that the switch must be based on the swap’s local PV $\delta:=T-U$29, not on raw cash-flow sign. In the model-free literature, a further misconception is that standard realized variance is inherently the canonical swap payoff; (Alexander et al., 2016) shows that $\delta:=T-U$30 lacks the aggregation property and is therefore exposed to discrete-monitoring and jump errors that DI payoffs are designed to avoid.

Differential swaps, in this combined sense, are therefore best understood as a family of swap constructions in which pricing, hedging, and economic interpretation are driven by increment-level design, collateral and funding conventions, and the precise way realized path functionals aggregate over time.