Probabilistic Swaps: Theory & Applications

• Probabilistic swaps are swap contracts where probability directly shapes the outcome, integrating randomness into settlement, valuation, or payoff construction across diverse applications.
• In cryptographic contexts, protocols employ adaptor signatures and OPRFs to ensure atomic and unbiased exchanges, extending traditional atomic swaps to enable probabilistic outcomes.
• In derivatives and model-free approaches, probabilistic swaps utilize recursive expectations, switching discount rates, and aggregation identities for improved pricing accuracy and risk analysis.

Searching arXiv for recent and related papers on “probabilistic swaps” and adjacent usages. Probabilistic swaps are swap structures in which the operative object is not a purely deterministic bilateral exchange, but a payoff, settlement rule, or exchange condition that is explicitly shaped by probability, distributional functionals, or recursive valuation under uncertainty. In current arXiv usage represented here, the term spans at least two technically distinct lineages: a cryptographic lineage, where a swap embeds a fixed publicly specified probability into a trustless cross-chain exchange, and a derivatives lineage, where swap valuation or design is formulated through probabilistic discounting, distributional identities, or model-free aggregation properties (Gerhart et al., 6 May 2026).

1. Scope of the term

The expression does not denote a single universal object across literatures. Its meaning depends on the ambient formalism: cryptographic protocol design, OTC derivatives valuation, or model-free swap engineering.

Domain	Meaning	Representative paper
Blockchain protocols	Trustless exchange with a fixed probabilistic outcome	(Gerhart et al., 6 May 2026)
Interest-rate derivatives	Recursive pricing of an uncollateralized swap under switching discount rates	(Lou, 2015)
Variance and higher-moment swaps	Model-free or probabilistic swap contracts targeting distributional quantities	(Alexander et al., 2016, Bunchak et al., 18 Jun 2025)

In the cryptographic setting, the central problem is to extend atomic swaps beyond all-or-nothing settlement. In the derivatives setting, the emphasis shifts to how swap value depends on stochastic exposure, recursive discounting, or realized-variance distributions. This suggests that “probabilistic swap” is best understood as a family resemblance term: in each case, probability enters the swap’s economic meaning at the level of settlement, valuation, or payoff construction rather than as a mere ex post risk descriptor.

2. Cryptographic probabilistic swaps

In "Probabilistic Atomic Swaps for Bitcoin and Friends" (Gerhart et al., 6 May 2026), a probabilistic swap is a trustless exchange in which one party’s transfer is executed with a fixed, publicly specified probability embedded in the protocol and cannot be biased by either party. The paper motivates the primitive by identifying a limitation of ordinary atomic swaps: they guarantee that either both parties receive the agreed assets or neither party transfers, but they cannot realize exchanges whose intended outcome is itself probabilistic, such as lotteries, randomized allocation mechanisms, probabilistic cross-chain trades, or a Lightning-style payment for a fixed chance at a larger reward.

The paper formalizes the primitive through an ideal functionality. A dealer locks a larger amount, denoted $\nu$, and the counterparty pays a smaller amount, denoted $1$. The dealer must claim first. Once the dealer claims, the functionality samples $b \sim \mathrm{Bernoulli}(p)$. If $b=1$, the counterparty can claim the reward; if $b=0$, the dealer later refunds its funds after timeout. If either side remains inactive, refunds are possible after timeouts. The resulting guarantee is a form of probabilistic atomicity: after both sides commit, the realized outcome has the advertised distribution and cannot be skewed by strategic abort after early outcome revelation (Gerhart et al., 6 May 2026).

A common misconception addressed explicitly in the paper is that one can obtain the same effect by “making atomic swaps probabilistic.” The paper rejects that reduction. If one party can learn the outcome early, it can abort selectively and bias the realized odds. For that reason, the outcome must remain hidden until both sides are already committed.

3. Protocol construction and probability mechanism

The protocol combines adaptor signatures with oblivious pseudorandom functions (OPRFs) (Gerhart et al., 6 May 2026). Adaptor signatures provide atomic payment release and witness extraction. OPRFs are used to encode the winning condition as a hidden guess. The dealer samples a hidden target value $y_{\mathrm{win}}$ from a domain of size $m$, while the counterparty submits a guess $y$. The counterparty wins only if $y=y_{\mathrm{win}}$, so the baseline success probability is exactly

$p=\frac{1}{m}.$

The paper notes that rational probabilities $1$0 can be obtained by allowing $1$1 concurrent guesses over a domain of size $1$2.

The main technical innovation is an atomic exchange of OPRF evaluations for payments. The dealer acts as the OPRF server. The counterparty wants the OPRF output on its blinded request, but the dealer should reveal that output only if it is paid. The dealer therefore evaluates the OPRF on the blinded input, encrypts the OPRF response under an ephemeral key $1$3, proves well-formedness of the ciphertext, and ties release of the decryption key $1$4 to its own on-chain claim using an adaptor signature. The dealer can obtain payment only by revealing $1$5, and the counterparty can decrypt the OPRF response only after extracting $1$6 from the on-chain signature. The paper explicitly notes that this atomic exchange of OPRF evaluations for payments may be of independent interest.

At a schematic level, the OPRF is based on a modified 2HashDH-style construction. The paper recalls the standard form

$1$7

and uses a witness of the form

$1$8

The reward condition is then represented by the equality $1$9. The party can claim the reward if and only if the OPRF-derived witness matches the dealer’s hidden winning witness (Gerhart et al., 6 May 2026).

The protocol is designed so that neither side can bias the outcome. The dealer does not learn the counterparty’s guess because of OPRF request privacy. The counterparty does not learn the outcome before the dealer is paid because the OPRF response is encrypted. The decryption key is extractable only from the dealer’s on-chain signature. This sequencing is the core fairness mechanism.

4. Security foundations, on-chain realization, and implementations

The paper gives a simulation-based security theorem. In summary form, if the adaptor signature scheme is adaptable and extractable, the underlying signature scheme is strongly unforgeable, the 2PC protocols for key generation and pre-signing are secure, the encryption is CPA-secure, the proof system is knowledge-sound, and the OPRF is pseudorandom and request-private, then the protocol securely realizes the ideal probabilistic swap functionality (Gerhart et al., 6 May 2026).

A notable feature is the minimal on-chain footprint. The protocol uses only standard Bitcoin scripts, such as digital signatures and timelocks, and is deployable on any blockchain that already supports atomic swaps. The script pattern is the standard signature-or-timeout form: $y_{\mathrm{win}}$0 A successful swap uses exactly four standard on-chain transactions: two funding transactions, one dealer claim transaction, and one party claim transaction. Because the construction preserves the shape of modern atomic-swap activity, the resulting transactions are described as indistinguishable from ordinary on-chain transactions, which helps preserve privacy and fungibility (Gerhart et al., 6 May 2026).

The paper presents two well-formedness-proof instantiations. The Bulletproofs-based version reports proving time about 5.6 seconds, verification time about 0.35 seconds, and proof size about 2.8 KB. The cut-and-choose version is faster for moderate parameters but has much larger proofs: at $b \sim \mathrm{Bernoulli}(p)$0, proving is about 0.27 s, verification 0.25 s, and proof size about 296 KB; at $b \sim \mathrm{Bernoulli}(p)$1, proving is about 16 s, verification about 14.5 s, and proof size about 16 MB (Gerhart et al., 6 May 2026).

The implementation section reports deployments on Bitcoin testnet4, Litecoin testnet, and a Lightning Network prototype. The cross-chain experiment includes both winning and losing cases for Bitcoin–Litecoin swaps. In the Lightning prototype, the reported timings are 21.96 s for precomputation, 0.35 s for channel setup, and 1.30 s total online HTLC execution, or about 0.43 s per round, with each Groth16 link proof having size 128 bytes (Gerhart et al., 6 May 2026).

5. Probabilistic swap pricing in OTC interest-rate derivatives

A distinct use of the concept appears in "Liability-side Pricing of Swaps and Coherent CVA and FVA by Regression/Simulation" (Lou, 2015). The paper studies an uncollateralized interest-rate swap hedged back-to-back by a CCP swap and argues that a plain back-to-back hedge is not sufficient because it leaves an open IR01 on FVA. The proposed remedy is dynamic replication through a CCP swap hedge with changing notional and a liability-side deposit or funding arrangement. The paper states that this eliminates residual IR01 leakage and yields an arbitrage-free, fully replicated pricing framework.

The central valuation rule is a switching discount rate: $b \sim \mathrm{Bernoulli}(p)$2 where $b \sim \mathrm{Bernoulli}(p)$3 is the dealer’s mark-to-market, $b \sim \mathrm{Bernoulli}(p)$4 is the dealer’s own bond or funding curve, and $b \sim \mathrm{Bernoulli}(p)$5 is the counterparty’s bond or credit curve. If the swap is a liability to the dealer, discounting uses the dealer’s own curve; if it is an asset, discounting uses the counterparty’s curve. The fair value is then expressed as a recursive risk-neutral expectation,

$b \sim \mathrm{Bernoulli}(p)$6

or, for a single swaplet with terminal payoff $b \sim \mathrm{Bernoulli}(p)$7,

$b \sim \mathrm{Bernoulli}(p)$8

Because the discount rate depends on the sign of the value being computed, the pricing problem is recursive. The paper therefore introduces a Longstaff–Schwartz least-squares regression and simulation method to decouple the switching rule. The implementation simulates the LIBOR short rate or a one-factor short rate, the risk-free or OIS rate as a spread off LIBOR, deterministic dealer and counterparty curve spreads, and the swap’s pathwise value. Two short-rate models are used for illustration: a mixed normal-lognormal short-rate model and a Black–Karasinski model (Lou, 2015).

The paper also argues for a coherent decomposition of total counterparty risk adjustment. If $b \sim \mathrm{Bernoulli}(p)$9, with $b=1$0 the risk-free value, then CVA is treated as the default-risk component and FVA as the liquidity or funding-basis component. A key numerical claim is that regression is essential, while brute-force Monte Carlo without regression decoupling produces large errors. The paper further states that common approximations based on discounting cash flows by sign or using EPE and ENE formulas can be materially inaccurate. This suggests that, in this literature, the probabilistic aspect of swaps lies not in randomized settlement but in recursive exposure-contingent valuation.

6. Distributional, model-free, and realized-variance swap constructions

A third technical lineage uses probability to characterize the payoff object being swapped. In "Model-Free Discretisation-Invariant Swap Contracts" (Alexander et al., 2016), discretisation-invariant swaps are defined by a restricted aggregation property. For a payoff $b=1$1 applied to increments of an adapted process $b=1$2, the floating leg

$b=1$3

is discretisation-invariant when

$b=1$4

The fair value is therefore independent of the monitoring partition. The paper characterizes such payoffs by a second-order PDE system and, for $b=1$5 with $b=1$6, shows that the solution set forms a vector space including linear increments in $b=1$7, quadratic forms in $b=1$8, percentage returns, and log returns. The same framework yields model-free variance, skewness, and kurtosis swaps, with fair values

$b=1$9

$b=0$0

The paper emphasizes that such contracts avoid discrete-monitoring and jump errors, and that a subclass also avoids numerical integration error over option strikes.

In "Probabilistic closed-form formulas for pricing nonlinear payoff variance and volatility derivatives under Schwartz model with time-varying log-return volatility" (Bunchak et al., 18 Jun 2025), the probabilistic element is explicit at the distributional level. Under the Schwartz one-factor model, discretely sampled realized variance is written as a weighted sum of independent noncentral chi-square random variables,

$b=0$1

with $b=0$2. Using the probability density function of this weighted sum, the paper derives closed-form analytical formulas for variance swaps, volatility swaps, and corresponding call options. The key swap strikes are

$b=0$3

The paper reports that the Laguerre-series density matches Monte Carlo histograms closely, that the closed-form swap prices match Monte Carlo estimates very well, and that truncation at $b=0$4 already yields very small error, with one reported absolute error below $b=0$5.

Two further papers delimit this lineage from opposite directions. "Probabilistic aspects of finance" (Föllmer et al., 2013) presents a pathwise variance-swap replication in which

$b=0$6

so the hedge is model-independent and “not subject to model risk.” By contrast, "Discretely sampled variance and volatility swaps versus their continuous approximations" (Jarrow et al., 2011) shows that continuous-sampling approximations are not automatically valid: for some price processes, $b=0$7 while $b=0$8. The paper therefore proves that continuous finiteness does not imply discrete finiteness and gives sufficient conditions under which $b=0$9.

Across these finance papers, the phrase “probabilistic swaps” does not refer to a randomized winner as in blockchain protocols. Instead, it refers to swaps whose fair value or payoff is defined through recursive expectations, model-free aggregation identities, or explicit probability laws for realized variance. A plausible implication is that the shared conceptual core is the replacement of deterministic swap accounting by formally specified probabilistic structure.