Calibrate a one-dimensional local-volatility surface to Heston prices at rho = -1 and every call on integrated variance becomes strictly more expensive. Lucic proves the result for every K > 0 and every maturity T > 0. Gatheral's 2005 conjecture placed local volatility at the bottom of the variance-option range. In the paper's only comparison, against the one-dimensional projection of that same Heston, local volatility comes out above.
Beiglböck, Friz and Sturm had already killed the conjecture in 2011 with constructed counterexamples. The remaining question was whether a stochastic-volatility model people actually use, equipped with a Markov variance process, could reverse the ordering. This paper says Heston can, at one correlation.
The proof turns on a pathwise identity
Set rho = -1 in Heston, which gives Z = -W. The two stochastic integrals in the spot and variance equations are then the same object and can be eliminated. What remains is a pathwise identity, requiring no expectations:
v_t = v_0 + kappa*theta*t - xi*log(S_t/S_0) - beta*I^H_t, with beta = kappa + xi/2.
Conditioning on S_t reduces the identity to v_t - lambda(t,S_t) = -beta (I^H_t - E[I^H_t | S_t]), where lambda is the Dupire projection E[v_t | S_t = s]. Heston's instantaneous variance differs from the projected surface by a conditionally centered variable multiplied by minus beta.
From there, Lucic reverses the standard wrong-volatility hedge of El Karoui, Jeanblanc-Picqué and Shreve. Let V(t,s,i) denote the local-volatility value of a claim on I^LV_T, then define H = (s^2/2)V_ss + V_i. The price gap becomes a time integral of E[(v_t - lambda(t,S_t)) H]. Convexity makes i -> H(t,s,i) nondecreasing. After conditioning on S_t, the Chebyshev covariance inequality fixes the sign.
The mimicking property gives exact agreement of the means: E[I^H_T] = E[I^LV_T] = the integral of E[v_t] = theta + (v_0 - theta)e^{-kappa t}. Dominated calls with equal means yield convex order.
Strictness requires more. Appendix C supplies a localized density lower bound in addition to every ingredient used for the non-strict result. Lucic smooths the payoff into the soft call (1/m)log(1 + e^{m(i-K)}), which remains within log(2)/m of the true call. He also replaces the projected surface with smooth majorants satisfying mu + 1/k <= nu_k <= C_0(1 + |x|). A Hörmander and support-theorem argument following Herzog and Mattingly then establishes a strictly positive density for the joint law of (S_t, I^H_t) on an interior box.
Closed form stops before the trade size
The variance-swap sanity check is exact. Both models produce the same integral of E[v_t], leaving convexity to drive the entire effect rather than a difference in level. Boundary behaviour for the projected surface is explicit as well: phi_t(q) = 4a q^2 / (beta (v_0 + a t)^2) times (1 + O(q)) with a = kappa*theta. Local variance approaches zero quadratically with the gap to the barrier, while its derivative is bounded by a constant times that gap.
The paper never specifies the magnitude. Its strict-gap lower bound is beta (i_3 - i_2) q_0^2 Delta_m / (2 F_0 C_box), yet none of q_0, F_0, C_box or Delta_m is explicit. We found no numerical or Monte Carlo illustration in the paper. A replicator can verify the sign, though producing a basis-point figure requires building the full construction himself.
That construction immediately forks. The canonical lambda uses the ratio phi_t = m_t/g_t of two densities obtained through Fourier inversion of affine transforms along a Bromwich line. The tail bound C(1+|y|)^N exp(-c_0 sqrt(|y|)) justifies term-by-term differentiation. Conditional expectations determine lambda only up to Law(S_t)-null sets, and Proposition 11 selects a particular version. Anyone discretizing the model must either reproduce the inverse-transform construction or fit a surface and hope the versions coincide away from a null set.
A barrier traders cannot ignore
At rho = -1, the Heston spot stays almost surely below s*(t) = S_0 exp((v_0 + kappa*theta*t)/xi). Calibrated local variance is identically zero at that boundary and above it. Using v_0 = theta = 0.04, kappa = 2, xi = 0.5 in their formula, with inputs chosen by us, puts the cap at 1.27 times spot at one year. The European surface shared by the two calibrated models consequently values every one-year call struck above 1.27 times spot at zero.
No equity surface looks like that.
Lucic states the limitation directly. The note covers only the endpoint rho = -1, leaving rho = +1 and the whole interior open, including the question of whether strictness survives away from the endpoint. Remark 4 identifies the mechanism as the sign of beta = kappa + xi/2, with high accumulated variance reducing current variance at fixed spot, "rather than perfect correlation by itself".
I would resist carrying that interpretation very far. The identity producing beta exists because the models share one stochastic integral. The paper identifies the general-correlation counterpart as the identity underlying Broadie and Kaya's simulation scheme. In our observation rather than the paper's, that version contains an orthogonal noise term. Once it appears, v_t - lambda ceases to be a deterministic multiple of the centered I^H conditional on S_t, leaving the conditional Chebyshev step without its argument. Our own reading of calibrated equity Heston fits places rho somewhere around -0.9 to -0.5. The proof does not reach those values, and the obstruction is structural.
The paper is equally direct about two other points. It uses no well-posedness from time zero: I^LV_T always means a fixed Brunick-Shreve mimicking solution. Uniqueness in law is established only from strictly positive times. An epsilon-sandwich handles the time-zero boundary term, with its error tending to zero. Rates and dividends remain zero throughout, and neither changes the theorem.
Where verification should begin
The author discloses that the result grew out of AI-assisted exploration. ChatGPT was used to develop author-supplied ideas, while Claude was used to verify them, and he states that every argument was reconstructed and independently verified. Lemma 2, the localized density lower bound, bears the strictness claim by itself. The saddle lemma carries both theorems: it supports Proposition 10, which feeds Proposition 11's Lipschitz bound, and the well-posedness Theorem 1 also invokes it.
Can listed options test the claim?
Listed chains cannot test this result. The comparison concerns a call on integrated variance, and we have no price history for variance options or OTC variance swaps against which to test it. End-of-day listed equity and ETF chains provide an implied-volatility surface without the realized-variance leg. Separately, rho = -1 can be imposed as a model condition but can never be quoted.
The result still changes a prior. Projection smooths instantaneous variance, and conditional Jensen makes lambda(t,S_t) smaller in convex order than v_t. Once variance is integrated through time, the ordering reverses. The inversion closely resembles the VIX-market inversion documented by Guyon. Smoothing the drivers of a variance payoff does not smooth the payoff itself. I would revise my view of this counterexample's practical reach if someone extended it to a correlation that could be calibrated, or calculated the gap in variance points for a realistic parameter set.