The Right Wing of the SABR Smile with an Absorbing Boundary: Resolution of Conjectures of Henry-Labordère

Paper: arXiv 2610.04209

Authors: Zeyu Cao, Shaosai Huang

Abstract

We determine the leading logarithmic asymptotics of the fixed-maturity right tail of the SABR model with $β\in(0,1)$ and absorption at zero, for every correlation $ρ\in(-1,1)$. If $P(k)$ is the probability that the terminal forward is at least $f_0e^k$, then $-k^{-2}\ln P(k)\to(1-β)^2/(2ν^2T(1-(ρ\wedge0)^2))$ as $k\to\infty$. For $ρ\ge0$ this is the rate predicted by the unrestricted hyperbolic geometry. For $ρ<0$ it is strictly larger: along the volatility excursion that produces the tail, survival forces the noise orthogonal to volatility to stay above a moving square-root barrier, at an additional cost of the same order $k^2$. Consequently the Black-Scholes implied volatility tends to $ν\sqrt{1-(ρ\wedge0)^2}/(1-β)$: Henry-Labordère’s conjectured wing limit $ν/(1-β)$ holds for $ρ\ge0$ and fails for every $ρ<0$. For his second conjecture, whose unrestricted-distance Gaussian estimate does not hold for $ρ<0$, we also identify the correct replacement for this model: at the level of tail probabilities, the rate is given by the squared intrinsic Riemannian distance, within the survival domain, from the initial state to the terminal target set. The proofs are direct, combining a stopped Lamperti transform, a conditional time change, Gaussian moving-barrier bounds and change-of-measure constructions, and use no heat-kernel estimates.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents highly complex mathematical derivations to resolve long-standing conjectures in the SABR model. While the theoretical contribution is significant, it lacks any empirical validation or backtesting, focusing purely on the mathematical resolution.

Research Flowchart

  flowchart TD
    A[Research Goal: Determine Leading Logarithmic Asymptotics of SABR Right Tail with Absorption] --> B{Key Methodology: Lamperti Transform, Conditional Time Change, Gaussian Moving-Barrier Bounds, Change-of-Measure}
    B --> C(Inputs: SABR Model Parameters (β, ν, ρ, T), Probability P(k) for terminal forward)
    C --> D[Computational Process: Derivation of Asymptotic Formulas for -k^-2 ln P(k) and Implied Volatility]
    D --> E{Key Finding 1: Asymptotics of -k^-2 ln P(k) derived for all ρ. Different rates for ρ >= 0 vs. ρ < 0 due to survival cost.}
    E --> F{Key Finding 2: Implied Volatility tends to ν√(1-(ρ∧0)²)/(1-β). Henry-Labordère's conjecture holds for ρ >= 0, fails for ρ < 0.}
    F --> G[Key Finding 3: For ρ < 0, tail probability rate is given by squared intrinsic Riemannian distance within survival domain.]

Cited References (26)

1. Modern SABR Analytics (2019) — link
2. The principle of not feeling the boundary for the SABR model (2018) — link
3. The survival probability of the SABR model: asymptotics and application (2017) — link
4. Functional Analytic (Ir-)Regularity Properties Of Sabr-Type Processes (2017) — link
5. On the probability of hitting the boundary for Brownian motions on the SABR plane (2016) — link
6. Mass at zero in the uncorrelated SABR model and implied volatility asymptotics (2015) — link
7. Asymptotics of implied volatility to arbitrary order (2014) — link
8. The large-maturity smile for the SABR and CEV-heston models (2013) — link

0 of 26 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.