Matching European option prices leaves realized-variance prices wide open. Lucic proves that in one corner of Heston, the Dupire local-volatility model charges strictly more for every positive-strike variance call at every maturity. Gatheral's conjecture predicted the opposite ordering.
What is being priced
The Heston model is dS/S = sqrt(v) dW, dv = kappa(theta - v) dt + xi sqrt(v) dZ, with correlation rho between the two Brownian motions. Its integrated variance, I^H_T, is the time integral of v. The comparison uses the one-dimensional local-volatility model with coefficient lambda(t,s) = E[v_t | S_t = s] and accumulated variance I^LV_T.
Brunick and Shreve establish the existence of a mimicking diffusion with the same one-time marginals. The Kurtz and Stockbridge corollary reprinted as Corollary 5 gives another route. Those marginals deliver identical vanilla prices at every expiry. The means match as well: E[I^H_T] = E[I^LV_T] = the integral of E[v_t] over [0, T], with E[v_t] = theta + (v_0 - theta)e^{-kappa t}.
Theorem 3 gives the reversal. For every K > 0 and every T > 0, E[(I^H_T - K)+] < E[(I^LV_T - K)+]. Any strictly convex payoff integrable under both models has the same strict ordering. At K <= 0, the call values coincide because their means are equal.
Gatheral's conjecture, restated on p. 155 of his book, placed local volatility at the minimum variance-option value among calibrated models. Beiglböck, Friz and Sturm had already disproved it through constructed counterexamples. Lucic puts the counterexample inside a standard stochastic-volatility model whose variance process is Markovian.
Perfect negative correlation supplies the sign
At rho = -1, set Z = -W. Eliminating the shared stochastic integral from log-spot and the variance equation gives a pathwise identity:
v_t = v_0 + kappa theta t - xi log(S_t/S_0) - beta I^H_t, with beta = kappa + xi/2.
Conditioning on spot then yields v_t - lambda(t,S_t) = -beta (I^H_t - E[I^H_t | S_t]). The discrepancy between the models becomes a conditionally centered multiple of accumulated variance.
The El Karoui, Jeanblanc-Picqué and Shreve identity prices the P&L from hedging with the wrong volatility. It expresses the price gap as a time integral of E[(v_t - lambda(t,S_t)) H(t,S_t,I^H_t)]. For a convex payoff, H increases with the accumulated-variance argument. Insert the pathwise identity and every term becomes a conditional covariance between a variable and a nondecreasing function of that same variable. Chebyshev fixes the sign.
The paper attributes that sign to beta > 0: at fixed spot, high accumulated variance depresses current variance. Perfect correlation supplies the reduction to a scalar rather than the economic mechanism itself. Instantaneous variance has the reverse convex ordering, since Jensen makes lambda(t,S_t) = E[v_t | S_t] smaller in convex order than v_t. Integration flips it.
A projection you can compute
The construction is unusually explicit. At rho = -1, Q_t = v_t + beta I^H_t equals A(t,S_t), so conditioning on spot is equivalent to conditioning on Q_t. The projection lambda is a ratio of two densities. Both come from inverting an explicit affine transform of the CIR process along a vertical inversion contour, subject to the tail bound C(1+|y|)^N exp(-c_0 sqrt(|y|)). Lemma 6 and Proposition 7 give the density ratio directly, avoiding differentiation of a fitted implied-volatility surface.
The state-space geometry is explicit too. Almost surely, spot remains below the deterministic barrier s*(t) = S_0 exp((v_0 + kappa theta t)/xi). At and above that barrier, lambda is identically zero. Below it, lambda is strictly positive and C-infinity in spot. Its approach to the boundary is quadratic: phi_t(q) = 4a q^2 / (beta (v_0 + a t)^2) (1 + O(q)) with a = kappa theta.
Proposition 11 makes the canonical version continuous below the barrier, where the spot density is strictly positive. Any continuous fit therefore agrees with it, leaving approximation error as the discretization risk. Proposition 11(iv) gives local Lipschitz regularity in spot, which Theorem 12 uses to prove well-posedness. The result starts only from strictly positive times. Uniqueness in law is proved only there, so I^LV_T denotes a fixed mimicking solution.
The missing magnitude
The proof does not produce a usable estimate of the effect. For the smoothed call Phi_m, the gap has the lower bound beta (i_3 - i_2) q_0^2 Delta_m / (2 F_0 C_box), after choosing the box 0 < i_1 < i_2 < K < i_3 < i_4 arbitrarily in the interior. Removing the smoothing sends Delta_m to Delta_K and restores the call inequality.
The constants have three separate sources. Lemma 2 supplies q_0, the localized density floor obtained from the Hörmander and support-theorem argument. F_0 bounds the jointly continuous spot density from above. C_box is the uniform linear-growth bound for the smooth majorants. None of the three is explicit.
The sign is proved. The magnitude remains open.
We found no numerical illustration or simulated check anywhere in the paper. Whether plausible (kappa, theta, xi, v_0) make the difference worth a basis point or a vol point is an empirical question. Rates and dividends are zero throughout, and the paper does not discuss relaxing either assumption. Discretely sampled realized variance, the contract convention, is also untreated.
The boundary case
The author explicitly leaves rho = +1 and the entire interior -1 < rho < 1 open. Our own reading of calibrated equity fits places rho nearer -0.9 to -0.5. The theorem therefore lives on a boundary. Spot is a deterministic function of (t, v_t, I^H_t), while calibrated local variance vanishes beyond a known barrier. This is a knife-edge model with a degeneracy absent from fitted models.
The paper's defence rests on the feedback coefficient beta = kappa + xi/2, which is positive for any rho. Perfect correlation only converts the generator discrepancy into a conditionally centered scalar. Nothing in the paper proves that the ordering persists away from rho = -1, and the author states the limitation plainly.
The paper also discloses that AI-assisted exploration led to the result. ChatGPT was used during development, Claude during verification, and the author reconstructed every argument. Four appendices cover a saddle-point transfer lemma and a localized density argument. Readers should allow time for them.
We could not test any of this.
Our data consists of end-of-day listed US equity and ETF option chains. It contains no prices for variance options or OTC realized-variance claims, while the comparison depends entirely on two such prices. The condition rho = -1 is a model assumption, not an observable that permits a sort. We have encountered the same obstacle where a priced quantity lacked a spanning instrument: /articles/the-correlation-rotation-premium-that-no-traded-instrument-spans.
The result worth paying for is the one the paper declines to provide: the same inequality at rho = -0.7, or one simulated number showing how much of the convex-order gap remains after the barrier degeneracy disappears. Until then, this is a clean proof that Markovian projection can increase variance-option value inside a model practitioners use, established at exactly the one correlation value, rho = -1, that nobody calibrates to.