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

Source 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.