Absorbed SABR's right-wing limit shrinks when ρ<0

An absorbed SABR smile with β in (0,1) and negative vol-forward correlation has a lower right-wing limit than ν/(1−β). Using Henry-Labordère's conjecture as an extrapolation anchor puts it too high. Cao and Huang prove the correction for the absorbed model.

Where does the conjecture break?

Henry-Labordère's 2008 monograph conjectured a SABR right-wing limit of ν/(1−β) at every correlation. Benaim, Friz and Lee proved the claim at ρ=0. Cao and Huang settle it for β in (0,1) when the forward is absorbed at zero: the conjecture holds for ρ≥0 and fails for every ρ<0.

Define P(k) as the chance that the forward finishes at or above f0·e^k. For ρ<0, −ln P(k)/k² converges to c₋. Its value, c₋ = (1−β)²/(2ν²T(1−ρ²)), exceeds the unconstrained rate c = (1−β)²/(2ν²T) by the factor 1/(1−ρ²). A model-free lemma then gives the implied-vol limit ν√(1−ρ²)/(1−β), independent of α, f0 and T.

For the authors' reference values ν=0.6 and β=0.5, the conjectured asymptote is 1.2.

At ρ=−0.6, the proved limit is 1.2 × 0.8 = 0.96.

Survival changes the tail rate

The Lamperti coordinate X = F^(1−β)/(1−β) makes the boundary mechanism visible. Until absorption, X consists of its initial value X0, a correlation term (ρ/ν)(a_t−α), a conditionally Brownian term and a nonpositive drift. Conditional on the volatility path, that Brownian term is scaled by √(1−ρ²) and runs on the integrated-variance clock.

Reaching a large terminal forward calls for a volatility excursion of log size n = (1−β)k. At ρ<0, the excursion pulls the correlation term downward in proportion to volatility. A path headed for the strike is also headed toward zero.

Survival therefore requires the orthogonal noise to clear a barrier that rises like a square root in variance time. Change to log time and the problem becomes an Ornstein-Uhlenbeck process kept above a roughly constant level ℓ. The cost is about ℓ²/8 per unit of log time across a window of about 2n. It has order k², matching the volatility excursion's cost; together they produce c₋. For ρ≥0, the correlation term remains bounded below during the excursion. Survival is cheap and the rate remains c.

The authors give a geometric account too. To leading order, the tail rate is the squared intrinsic Riemannian distance within the survival domain, divided by 2T. Unrestricted hyperbolic distance yields c at every ρ. For ρ<0, its cheapest route follows a line outside that domain. The proofs rely on level-hitting skeletons, Gaussian convex-set bounds and change-of-measure tubes, without heat-kernel estimates.

A distant limit

The authors put the numerical gap plainly: "At these strikes the asymptotic regime is still distant". With β=0.5 and ρ=0, implied vol reaches about 0.87 at k=12, while its limit is 1.2. A strike at k=12 is about 160,000 times the forward.

Their deterministic variational problems agree with the predicted rates through k=160. Coefficients estimated using the pair (80,160) are each within 1.5% of the prediction. At those values, survival-to-terminal ratios are 1.55 for β=0.5 and 1.57 for β=0.7, versus a predicted 1.5625.

Monte Carlo through k=12 displays the correlation asymmetry, although the authors warn that Table 2 cannot be read as a rate. At β=0.5, the ρ=+0.6 spread relative to ρ=0 has second differences within 0.15 of zero over k=2 to 6. The ρ=−0.6 spread has second differences between 0.7 and 0.9 across the same range, exceeding the 0.39 implied by the k² term alone. The authors suggest a k^(3/2) correction for the excess. They call it conjectural and say this window cannot separate it.

The paper supplies no quantitative remainder. For a trader, there is a proved limit, a snapshot of 0.87 against 1.2 at k=12, and an unproved k^(3/2) correction. There is no proved convergence rate.

What can quoted ETF strikes tell us?

We cannot trade interest-rate forward options, so our test is our own construction using end-of-day options on liquid US ETFs. Each day we fit SABR with fixed β to the inner strikes of each expiry, reserving the outermost quoted calls. We then extrapolate the wing with ν/(1−β) as one anchor and ν√(1−(ρ∧0)²)/(1−β) as the other, using fitted ρ and ν. Both are scored against the held-out quotes in vol points and as a fraction of the bid-ask spread.

Those strikes are nowhere near k=12. The exercise asks whether the corrected limit makes a better extrapolation anchor at quoted strikes. ETF dynamics differ from those of an absorbed CEV (constant-elasticity-of-variance) forward, and listed ETF options have American exercise while the theorem addresses European calls. Its limit offers no pricing guarantee for ETF options. We are running the test and will report its numbers separately.

The boundary convention matters

The paper makes absorption a model choice, selecting the absorbing continuation from the possible CEV extensions. That choice drives the correction. Normal SABR at β=0 has no absorbing boundary and has a wing limit of ν for every ρ. As β↓0, the authors' formula retains √(1−ρ²): the approaching models still absorb at F=0. The change of state space explains the gap. A desk using another boundary convention, or quoting from the Hagan expansion, prices a different object.

At ρ=0 the limit is C¹ but not C². Its second derivative is −ν/(1−β) on the left and 0 on the right. For ρ=−0.2, √0.96 ≈ 0.98 gives a haircut of about 2%, so noise in a fitted ρ near zero barely shifts the anchor. At ρ=−0.6, the haircut is 20%.

The authors state the result for fixed T. It excludes ρ=±1 and β∈{0,1}, and gives no left-wing theorem.

The proof changes the far-wing number when ρ is materially negative. Whether that number pays at strikes a desk can quote remains open; our ETF run asks a narrower question about listed ETF options.