LongTail-Swap: Techniques and Benchmarks

• LongTail-Swap is a multifaceted framework that characterizes rare, long-tail events across language modeling, finance, quantum information, and decentralized systems.
• In language models, LT-Swap benchmarks rare word comprehension through paired sentence swaps, demonstrating systematic accuracy declines with decreasing word frequency.
• In finance and quantum domains, LT-Swap methods enable precise long-term swap rate calculations and robust fractional swap operations, enhancing risk management and protocol design.

LongTail-Swap (LT-Swap) refers to a family of concepts, techniques, and benchmarks across several domains—most notably in language modeling, financial mathematics, option pricing, quantum information, and decentralized finance—centered on rare-event, tail-distribution, or long-dated phenomena. In all settings, the core focus is on “tail” instances, whether rare words, extreme financial events, anomalous quantum operations, or heterogeneous preferences in asset exchanges. The term synthesizes approaches for precisely characterizing, analyzing, or robustly benchmarking systems under these long-tail conditions, providing insights that differ qualitatively from head-focused or average-case analyses.

1. Characterization and Definition of LongTail-Swap

The most recent and prominent usage of LongTail-Swap (LT-Swap) appears in the context of LLM evaluation, where it denotes a specialized benchmark designed to measure the ability of LMs to handle rare words—i.e., words in the long tail of the frequency distribution, often seen only a handful of times or once during pretraining (Algayres et al., 5 Oct 2025). LT-Swap is formulated as a test set consisting of paired sentences that isolate either the semantic or syntactic usage of rare words and evaluates models in a zero-shot setting by comparing their preferences between correct and “swapped” (incorrect) versions of these sentences.

In financial mathematics, “long-term swap rate” (also referred to as LT-Swap in financial literature) is a formal asymptotic quantity defined as the fair fixed rate of an overnight indexed swap (OIS) with an infinite sequence of cash flows (Biagini et al., 2015). Here, the term “swap” denotes a financial contract exchanging cash flows, while “long-tail” refers to statistical or time-horizon tails.

Within quantitative finance, LongTail-Swap strategies or methodologies involve pricing or risk management for deep out-of-the-money options, where payoffs and distributions are governed by heavy-tailed power laws rather than by thin-tailed, mean-reverting approximations (Taleb et al., 2019).

In distributed and decentralized systems, generalized swap protocols enable “long-tailed” trading preferences—i.e., swaps that are not uniform or one-for-one but governed by user-specific, possibly highly nonuniform, preferences over asset subsets. These LT-Swap protocols must ensure security and equilibrium even when possible trading graphs are highly irregular (Chan et al., 2022).

Finally, in quantum information, “longtail swap” (by analogy, not always using the explicit term) generalizes SWAP gates to non-integer or fractional exponents and arbitrarily distributed systems, allowing robust swap-like operations over distant or heterogeneous quantum networks (Liu et al., 2020).

2. LongTail-Swap in LLM Evaluation

The LT-Swap benchmark (Algayres et al., 5 Oct 2025) is constructed for explicit probing of LMs’ rare word abilities, addressing the limitation that prior evaluations (such as BLiMP, EwoK, or SuperGLUE) are biased toward the head of the vocabulary distribution. The benchmark deploys a corpus-specific methodology:

• Frequency Binning: Words in the pretraining corpus are frequency binned into intervals of the form $$\{[0], [2^0, 2^1[, [2^1, 2^2[, \dots, [2^n, +\infty[\}$$, with $n=9$, enabling selection of rare and never-seen (hapax) words.
• Task Construction: For each bin, sentences are generated (using Llama 3.1-405B) that each contain a rare target word. Three main subtasks are introduced:
  1. WordSwap (semantic): Pairs of correct sentences have their rare words swapped, yielding semantically anomalous sentences.
  2. InflectionSwap (syntactic/morphological): Pairs focus on inflectional forms, testing models’ knowledge of correct morphological usage given frequency constraints on base and inflected forms.
  3. AgreementSwap (syntactic agreement): Minimal pairs test subject-verb, determiner-noun, or anaphoric agreement using rare words/inflections.

• Filtering and Validation: Quadruplets (2 correct, 2 swapped) are filtered by in-context LLM judgments, preserving only those reliably judged as discriminative between correct and incorrect.

• Evaluation Protocol: Candidate models (GPT/BERT/hybrid) are tested zero-shot by assigning higher average log-probabilities to correct sentences, evaluated per bin and averaged across tasks.

The LT-Swap approach reveals that model accuracy drops systematically with decreasing word frequency, with pronounced differences between architectures in the long tail. For example, the spread ratio (max/min accuracy) across models reaches up to 3.8 in low-frequency (WordSwap) bins, and performance correlations with frequency are statistically quantified using Spearman $\rho$.

An example of frequency bin construction:

Bin	Frequency Range	Example Usage
0	0	“phantasmagorize”
1	[1, 2[	“to commodify”
6	[64, 128[	“to report”

All data and code are released for generating benchmarks on any English corpus, enabling both rigorous and reproducible rare word evaluation.

3. LongTail-Swap in Financial Mathematics: Long-Term Swap Rate

The long-term swap rate (“LT-Swap”) is formally defined as

$R_t = \lim_{n \to \infty} R(t, T_n)$

where $R(t, T_n)$ denotes the fixed rate of a swap (specifically an OIS) with exchange dates $T_1, T_2, \dots$, each separated by minimal spacing, starting at $T_0 = t$. The fixed leg is valued against the (discounted) sum of zero coupon bond prices. The key result (Biagini et al., 2015) is

$$R_t = \frac{1}{S_\infty(t)}, \quad \text{where } S_\infty(t) = \sum_{i=1}^\infty \delta_i P(t, T_i),$$

and $\delta_i = T_i - T_{i-1}$. This rate is always finite (if $S_\infty(t)$ converges), arbitrage-free except in degenerate cases, and cannot be strictly monotonic unless constant (Theorem 3.3).

Distinct from the long-term yield ($\ell_t$) and long-term simple rate ($L_t$), the LT-Swap rate serves as a robust alternative discounting mechanism, especially when classical rates vanish or explode. The paper systematically relates regimes of bond price asymptotics to the existence and finiteness of these rates, summarized in tables (Tables 1–3).

Closed-form expressions for $R_t$ are derived in term-structure models:

• Flesaker–Hughston model: Bond prices are given by

$P(t, T) = \frac{f(T) + g(T) M_t}{f(t) + g(t) M_t}$

For $f(t) = \exp(-\alpha t)$, $g(t) = \exp(-\beta t)$, $R_t = [f(t) + g(t) M_t] / [\delta (F + G M_t)]$, linking $R_t$ to state variable $M_t$ and parameter sums $F$, $G$.

• Linear–rational model: $P(t, T)$ expressed as

$$P(t, T) = \frac{(\varphi + \psi^\top \theta) e^{-\alpha(T-t)} + \psi^\top(X_t - \theta)e^{-(\alpha+k)(T-t)}}{\varphi + \psi^\top X_t}$$

yields $R_t$ in analytic form via closed-form sums of discount factors.

A numerical case paper benchmarks these formulas for perpetual CoCo bond pricing, showing that the LT-Swap rate enables precise decomposition between non-optional and optional (conversion feature) price components, and is calibrated in practice using bootstrapped OIS curves and high-frequency market data.

4. LongTail-Swap in Option Pricing and Risk Management

In the context of tail risk, LongTail-Swap strategies reference robust option pricing and risk management methods for deep-tail events, as presented in the extrapolative option pricing framework under power laws (Taleb et al., 2019). The methodology:

• Assumes the survival function of returns obeys $P(X > x) \sim L(x) x^{-\alpha}$ beyond a “Karamata constant” point where $L(x)$ becomes constant.
• Provides relative pricing formulas for tail options:

$$C(K_2) = C(K_1) \left( \frac{K_2}{K_1} \right)^{1 - \alpha}$$

for calls on strikes $K_2 > K_1$, and

$$C(K_2) = C(K_1) \left( \frac{K_2 - S_0}{K_1 - S_0} \right)^{1 - \alpha}$$

on returns.

• Arbitrage constraints are enforced via restrictions on $\alpha$ and derivative inequalities, ensuring consistency with Breeden–Litzenberger and no negative densities.
• This pricing framework, contingent only on tail index $\alpha$ (estimable via the Hill estimator) and an anchor market price, sidesteps the finiteness-of-variance assumption and provides a natural tool for “LongTail-Swap” option strategies, volatility surface extrapolation, and stress-testing extreme event scenarios.

5. LongTail-Swap in Quantum Information

The quantum LongTail-Swap concept generalizes SWAP and controlled-SWAP (Fredkin) gates to fractional exponents $(\mathrm{SWAP})^{1/m}$, with implementation schemes for solid-state qubits using single photons and quantum dots (Liu et al., 2020). Key results include:

• Deterministic realization of $(\mathrm{SWAP})^{1/m}$ and controlled-$(\mathrm{SWAP})^{1/m}$ gates without ancilla qubits, using optical phase control (quarter- and half-wave plates).
• The gate parameter $m$ smoothly tunes the swap fraction, allowing for fine control over entanglement and distributed quantum communication—essentially extending the swap operation to long-range or long-tailed quantum networks.
• Demonstrated average fidelities exceed 99% and efficiencies over 96% under current experimental parameters (high $Q$-factors, strong cavity coupling), confirming the feasibility of “LongTail-Swap” quantum operations at scale.

6. LongTail-Swap Protocols in Decentralized Asset Exchanges

LT-Swap-type protocols in decentralized asset trading are formalized via graphical game-theoretic models of asset swaps with user-specified preferences over subsets of incoming and outgoing assets (Chan et al., 2022). Innovations and properties include:

• The preference relation space is defined as a partially ordered set (poset) over all possible outcome pairs $(A_\mathrm{in}(v), A_\mathrm{out}(v))$ for each party, subject to monotonicity (more incoming, less outgoing $\Rightarrow$ preferred).
• Existence of a safe, live, and strong Nash equilibrium protocol is characterized via subgraph selection: a swap system admits such a protocol if and only if a spanning subgraph exists that is piecewise strongly connected, outcome-dominant, and minimal (no proper subgraph strictly dominates).
• The decision problem (SwapAtomic) for verifying the existence of such protocols is shown to be both NP-hard (by reduction from 3SAT) and $\Sigma_2^P$-complete (by reduction from quantified Boolean formulas).
• These protocols accommodate LT-Swap settings where parties have heterogeneous, long-tailed portfolios and complex, user-specific trade requirements.

Setting	LT-Swap Concept	Core Focus
Language Modeling	Rare word benchmark	Generalization to tail vocabulary
Financial Mathematics	Long-term swap rate	Asymptotic swap evaluation
Option Pricing	Tail-based ratio heuristics	Extrapolation under power laws
Quantum Information	Fractional/Cross-node swaps	Nonlocal swap gate realizations
Decentralized Finance	Preference-based asset swaps	Complex heterogeneous exchanges

7. Implications, Open Directions, and Public Resources

LT-Swap methodologies highlight significant performance gaps between conventional and tail-focused systems across domains:

• In language modeling, architectural and data regime choices have disproportionately large effects on tail-word abilities (Algayres et al., 5 Oct 2025).
• In financial mathematics and option pricing, failure to model or price using long-tail rates or distributions underestimates risk and misprices extreme-event derivatives (Biagini et al., 2015, Taleb et al., 2019).
• Protocol design for LT-Swap systems in decentralized finance remains computationally complex, though feasible for tractable instances (Chan et al., 2022).
• Quantum schemes demonstrate that tail-robust, long-range swapping is feasible and resource-efficient (Liu et al., 2020).

The LT-Swap benchmark, codebase, and generators are publicly available for linguistic evaluation at https://github.com/facebookresearch/lt-swap (Algayres et al., 5 Oct 2025). Financial mathematics and decentralized systems approaches are mathematically formalized, providing concrete algorithms and analytic regimes for practitioners.

Ongoing research directions include expanding LT-Swap methods to other languages and modalities, further quantifying head/tail discrepancies, and developing more computationally efficient protocols in settings with highly heterogeneous, long-tailed asset or operation distributions.