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

Keywords. Absorbed diffusions; SABR model; implied-volatility wings; logarithmic tail asymptotics; moving-boundary crossing; intrinsic Riemannian distance; stochastic volatility.

2020 Mathematics Subject Classification. Primary 60J60; secondary 60F10, 60H10, 60J50, 91G20.

JEL Classification. C02, G13.

1Introduction

Absorption can alter a fixed-time logarithmic tail asymptotic even when the absorbing boundary is disjoint from the terminal target set. We establish this phenomenon for the SABR diffusion absorbed at zero: the volatility excursion that produces a large terminal value can simultaneously drive the forward toward the boundary, so paths contributing to the right tail incur an additional Brownian boundary-crossing cost. The main theorem concerns the terminal right tail of the absorbed diffusion; the implied-volatility wing then follows from a model-free transfer principle.

1.1The Problem and Henry-Labordère’s Conjectures

The SABR model of Hagan, Kumar, Lesniewski and Woodward [HKLW02] is a widely used stochastic-volatility parametrization of implied-volatility smiles in interest-rate markets and beyond:

with $F_0=f_0>0$, $a_0=\alpha >0$, $\beta \in (0,1)$, $\nu >0$ and $\rho \in (-1,1)$. Precisely, we work on a complete filtered probability space satisfying the usual conditions and carrying independent standard Brownian motions $(Z,Z^\perp )$. We set $W=\rho Z+\sqrt {1-\rho ^2}\,Z^\perp$ and

On the open state space $(0,\infty )$, $F$ is the pathwise-unique strong solution of the first equation in (1.1) up to the absorption time

and we take the continuous absorbed extension $F_t=0$ for $t\ge \zeta$. This boundary condition is part of the model: it selects the absorbing continuation among the possible post-hitting extensions of the CEV equation. Throughout, $T>0$ is a fixed maturity, $k=\ln (K/f_0)$ is the log-moneyness, and $\sigma _{\BS }(k)$ denotes the Black–Scholes implied volatility of the undiscounted call with strike $K$ and maturity $T$.

At fixed maturity, the extreme right wing probes joint rare excursions of volatility and the forward. The unrestricted hyperbolic geometry of the model suggests a correlation-independent leading coefficient, whereas the right-tail event for the absorbed process requires survival in $(0,\infty )$. The question is whether this survival constraint is negligible on the leading $k^2$ scale.

In Chapter 10 of his monograph [Hen08], Henry-Labordère writes the generator of a stochastic-volatility model as a Schrödinger-type operator on a Riemannian manifold: a Laplace–Beltrami part determined by the diffusion matrix, a connection one-form, and a scalar potential $Q$. Assuming a Gaussian bound of Li–Yau–Davies type [LY86; Dav89] for the associated heat kernel (his (10.21)), he obtains a wing formula, his (10.22), in which the extreme strikes are governed by the geodesic distance, in the unrestricted metric, from the initial point to the terminal fibre $\{F=f\}$. For SABR this metric is hyperbolic, and the resulting limit is $\nu /(1-\beta )$. Let $a_{\min }(f)$ denote the volatility coordinate of the point of the fibre $\{F=f\}$ nearest to the initial point, that is, the saddle point of the fibre integral. For SABR with $\beta <1$, the scalar potential along this saddle has the large-strike limit given in [Hen08, p. 306]:

The potential is therefore bounded along the far-right saddle. On the strength of this boundedness, Henry-Labordère poses two conjectures, which we restate in our notation, keeping his numbering.¹

The second conjecture concerns the Gaussian estimate itself.

Conjecture 1 is the proposed right-wing limit; Conjecture 2 would justify the unrestricted-distance Gaussian estimate from the boundedness of the potential at the saddle point. As the book notes, Conjecture 1 was proved for $\rho =0$ by Benaim, Friz and Lee [BFL08]. To our knowledge, the fixed-maturity correlated case $\rho \ne 0$, $\beta \in (0,1)$ has remained open (see Section 1.4). The difficulty is the gap between evaluating a coefficient at the unrestricted saddle point and controlling the Dirichlet semigroup of the diffusion killed at the boundary of its state space.

1.2Main Results

We measure the right tail by the probability that the terminal forward ends at or above a large strike, and compare its decay with two quadratic rates:

For $\rho \ne 0$ the rate $c_-$ exceeds $c$, and our main result shows which of the two governs the tail.

Through a model-free transfer from tail probabilities to call prices (Lemma 2.3), the decay rate determines the right wing of the implied-volatility smile.

Comparing this limit with Henry-Labordère’s prediction settles Conjecture 1.

Both corollaries follow from Theorem 1.1 and the integrability $\E F_T\le f_0$ alone. Figure 1 shows the limit as a function of $\rho$, together with the value predicted by Conjecture 1.

Our third result explains the correction geometrically. In Section 6 we write the model in Fermi coordinates $(R,q)$ of the hyperbolic plane, in which the survival domain is the coordinate half-plane $D=\{R>\Rb \}$, and identify the decay rate with the intrinsic Riemannian distance $d_D$ within $D$: for every $\rho \in (-1,1)$,

(Corollary 6.2), where $z_0$ is the initial state and $\mathcal T_k$ the terminal target set. The unrestricted distance yields the rate $c$ for every $\rho$; the restriction to $D$ changes the leading coefficient exactly when $\rho <0$. Since the limit (1.2) is finite for every $\rho$, the model shows that boundedness of the potential at the unrestricted saddle point does not suffice for the Gaussian estimate in Conjecture 2, and (1.4) is its correct replacement for absorbed SABR.

1.3Mechanism and Proof Strategy

The asymmetry between positive and negative correlation is already visible in an exact decomposition of the forward before absorption. Write $X=F^{1-\beta }/(1-\beta )$ for the Lamperti–CEV coordinate and decompose $W=\rho Z+\orth Z^\perp$, where $\orth :=\sqrt {1-\rho ^2}$. The correlated stochastic integral satisfies $\int _0^t a\,dZ=(a_t-\alpha )/\nu$ exactly, and conditionally on the volatility path the orthogonal martingale $\int _0^t a\,dZ^\perp$ is a Brownian motion $B$ run on the quadratic-variation clock $V_t=\int _0^ta_s^2\,ds$. Hence, before absorption (Lemmas 2.1 and 2.2),

A large terminal value requires a volatility excursion of logarithmic size $n=(1-\beta )k$, which costs $ck^2+o(k^2)$. For $\rho \ge 0$ the correlation term is bounded below along such an excursion, survival is cheap, and the unconstrained rate is attained. For $\rho <0$ the correlation term $\xi _t=-\frac {|\rho |}{\nu }(a_t-\alpha )$ becomes increasingly negative along the very excursion that creates the wing, and survival of $F$ forces the orthogonal Brownian motion to stay above the moving lower barrier

Along the exponential volatility path used in the lower bound, this barrier is a shifted square-root profile in the operational time $v=V_t$ (Figure 2(a)). In the logarithmic time $s=\ln v$, the process $U(s)=e^{-s/2}B(e^s)$ is a stationary Ornstein–Uhlenbeck process and the barrier becomes an essentially constant level $\ell \asymp \sqrt n$ (Figure 2(b)). Holding $U$ above $\ell$ costs $\ell ^2/8$ per unit of logarithmic time, over a window of length about $2n$. This cost, $\rho ^2c\,k^2/(1-\rho ^2)$ to leading order, is of the same order as that of the volatility excursion, and the two add up to $c_-=c/(1-\rho ^2)$. Equivalently, the effective volatility of volatility in the right wing is $\nu \orth$. The absorbing boundary thus acts on the right tail through survival conditioning, not through the mass at zero; the model-free tail-to-wing transfers [Lee04; BF09; GL14] are unaffected.

Turning this mechanism into a proof requires matching bounds that do not presuppose path concentration. For $\rho <0$, the upper bound on $P(k)$ (Section 3) drops the nonpositive singular drift, discretizes the volatility excursion by level-hitting times, and converts the conditional survival constraints into a finite-dimensional Cameron–Martin energy. A deterministic convex inequality (Lemma 3.7) then bounds the total cost from below; its key step, Lemma 3.6, shows that the part of the Gaussian cost that is linear in the displacement is minimized by deferring the displacement for as long as the barrier allows. The matching lower bound (Section 4) combines a change-of-measure tube for volatility with a Brownian path that enters and stays above the induced barrier, followed by a bootstrap control of the singular drift. For $\rho \ge 0$ (Section 5), a terminal-only upper bound is paired with a lower-bound construction based on scale-invariant doubling and $O(\ln n)$ dyadic level-raising steps. These arguments use elementary probability and finite-dimensional convexity rather than a sample-path large-deviation principle, heat-kernel estimates or spectral asymptotics.

1.4Relation to Prior Work

The problem lies at the intersection of several established theories, but not inside any one of their proved regimes. Among the results we located, summarized in Table 1, none treats the fixed-maturity extreme right wing of correlated SABR with $\beta \in (0,1)$. The closest results concern the uncorrelated $\beta <1$ wing, the moment-explosion regime at $\beta =1$, asymptotics of survival probabilities, small-time geodesic expansions, and the left wing generated by the atom at zero. In particular, Paulot [Pau15] cautions that small-time geodesic expansions are valid only as long as the geodesic does not reach the boundary; the present result is a fixed-maturity, extreme-strike manifestation of the same boundary obstruction.

1.5Scope and Organization

Before turning to the proofs, we record precisely what is proved. All limits are taken for fixed $T>0$, $\beta \in (0,1)$ and $\rho \in (-1,1)$. We do not prove uniformity in these parameters, a quantitative remainder, results at $\rho =\pm 1$ or $\beta \in \{0,1\}$, or a left-wing theorem. No Dirichlet heat-kernel estimate is used or proved; Remark 6.3 describes what such an estimate would have to provide. For $\rho <0$ we state a conjectural $k^{3/2}$ correction (Conjecture 8.1), supported by formal calculations only; no proved result depends on it. The numerical results of Section 7 illustrate the theorems and play no role in the proofs. Selected deterministic steps have also been formalized in Lean; see the Supplementary Material section.

Section 2 collects the reductions: the Lamperti decomposition, conditioning on volatility, and the transfer to implied volatility. Sections 3 and 4 prove the upper and lower bounds for $\rho <0$; Section 5 treats $\rho \ge 0$ and completes the proof of Theorem 1.1. Section 6 proves the intrinsic-distance identification, Section 7 presents numerical illustrations, and Section 8 discusses the subleading correction and neighbouring parameter regimes. Appendix A records measurability facts and Appendix B the numerical methods.

2Preliminaries

The proofs rest on two reductions. The Lamperti–CEV transform isolates the correlation term and lets us represent the orthogonal martingale in its quadratic-variation time; a model-free argument converts logarithmic tail rates into implied-volatility limits.

Throughout, $\beta \in (0,1)$, $\nu >0$, $T>0$ and $\rho \in (-1,1)$ are fixed, $\orth =\sqrt {1-\rho ^2}$, and

so that $\ln m=n+O(1)$. Constants $c_1,c_2,\dots$ depend only on the model parameters and on the auxiliary parameters fixed in the current section, never on $k$; their values may change from section to section. Upper and lower bounds always refer to $P(k)$.

2.1The Lamperti Decomposition

The first reduction is a change of coordinates. In the Lamperti coordinate the component driven by the volatility Brownian motion integrates exactly, leaving an orthogonal Gaussian martingale and a nonpositive singular drift.

We write

for the process with the singular drift removed.

2.2Conditioning on Volatility: Operational Time

Conditioning on the volatility path completes the first reduction: the orthogonal martingale then becomes a Brownian motion run on its quadratic-variation clock.

The jointly measurable versions and the kernel construction are recorded in Appendix A. We call $v=V_t$ the operational time, write $t(\cdot )$ for the inverse quadratic-variation clock of the volatility path, and set

so that $M_t=B(V_t)$. By Lemma 2.2, conditionally on $\sigma (a)$, $B$ is a standard Brownian motion on $[0,V_T]$. “Conditionally on the volatility path” always means evaluation under the kernel $\mathsf K$ for $\Prob _a$-almost every path; the operational times, thresholds and events entering such evaluations are measurable functions of the path.

2.3Transfer to Implied Volatility

The second reduction is model-free: once the logarithmic tail rate is known, no further SABR structure is needed to identify the right-wing limit. The forward $F$ is a nonnegative local martingale, hence a supermartingale, so $\E F_T\le f_0$; this is all the integrability we use. Write $\mathcal C(k):=\E (F_T-f_0e^k)^+$ for the undiscounted call price.

By Lemma 2.3, Theorem 1.1 implies Corollary 1.2, and hence Corollary 1.3; the positivity $P(k)>0$ follows from the lower-bound constructions of Sections 4 and 5. Sections 3 to 5 prove Theorem 1.1.

3Upper Bound for Negative Correlation

We begin with the upper bound on $P(k)$ for negative correlation. This half of Theorem 1.1 must rule out every cheaper way of producing the tail, including paths that raise the orthogonal component before the barrier requires it.

Conditionally on volatility, removing the nonpositive drift turns survival into a Brownian moving-barrier event. We discretize the volatility excursion by level-hitting times, convert each conditional event into a convex quadratic energy, bound the combined volatility and boundary-crossing cost from below, and sum over the resulting configurations. Throughout this section $\rho \in (-1,0)$.

3.1Drift Removal

The singular drift is nonpositive before absorption. Dropping it enlarges the joint survival-and-terminal event and isolates the Gaussian constraints that determine the upper bound.

3.2A Gaussian Convex-Set Bound

At the level-hitting times introduced below, the relaxed survival and terminal requirements become linear inequalities for a Gaussian vector. The following projection estimate converts their conditional probability into a Cameron–Martin energy.

Apply the lemma to $B$ at $\sigma (a)$-measurable operational times $0<v_1<\dots <v_J\le V_T$ with thresholds $(h_j)$ and a terminal threshold $H$ at $V_T$. Put $v_0=0$, $v_{J+1}=V_T$, $\ell _j=h_j$ for $1\le j\le J$, and $\ell _{J+1}=H$. For $s\ge 0$ define the extended-real increment energy

The constraint set is a nonempty intersection of half-spaces, and the Markov structure of the Brownian covariance identifies its squared Mahalanobis distance with the moving-barrier quadratic program

the Cameron–Martin energy of the cheapest interpolation through the constrained nodes. In the applications below $H>0$ for all sufficiently large $k$, so the constraint set does not contain the origin. If $v_J=V_T$, the zero-increment convention forces $x_{J+1}=x_J$; this amounts to merging the coincident nodes with the single threshold $\max (h_J,H)$, and Lemma 3.3 is applied after this merge, so that its covariance matrix is positive definite. The same formula with no intermediate nodes covers the terminal-only case $J=0$.

3.3Level-Hitting Skeleton

To retain sharp costs with finitely many constraints, we discretize the increase of log-volatility by successive level-hitting times. The same skeleton controls the available quadratic variation, the induced survival thresholds, and the probability of completing each level crossing quickly.

Fix $\eta \in (0,\tfrac 12)$ and set $A_j:=\alpha e^{j\eta }$, $\tau _j:=\inf \{t: a_t\ge A_j\}$ (so $\tau _0=0$), $J^*:=\max \{j:\tau _j\le T\}$, $\Delta t_j:=\tau _j-\tau _{j-1}$ and $v_j:=V_{\tau _j}$. Along this skeleton the levels control the available quadratic variation, and survival becomes a finite list of threshold conditions at the nodes.

Conversely, crossing a level quickly is costly for the volatility, as the reflection principle shows.

3.4Deferred Displacement and the Energy Inequality

The preceding reductions leave a deterministic question: can a path bring its combined volatility and Gaussian cost below $c_-k^2$, by redistributing the volatility increase or by raising the Gaussian coordinate before the barrier requires it? The next two lemmas show that it cannot. The key input is an elementary summation-by-parts bound for the part of the Gaussian cost that is linear in the displacement.

With this bound for the linear part, the combined volatility and Gaussian energy can be bounded below uniformly over all allocations of time and quadratic variation.