Fukasawa's correction is worth checking against short-dated up-and-out put marks. In the CEV test, it cuts mean pricing error from 0.09426 to 0.01248 against model prices. Those checks use Monte Carlo, plus a PDE solve for CEV. No quoted market price serves as a benchmark, and the paper leaves open whether dealer barrier quotes follow the skew relation.

Why move the barrier with expiry?

An up-and-out put pays its put payoff only while spot has never touched a barrier B above it. Hold B away from spot as maturity shrinks, and a touch becomes rare; the limiting price is the vanilla put. Fukasawa moves the barrier and strike instead: B = S0(1+√(v0θ)b) and K = S0(1+√(v0θ)k). Maturity is θ, spot variance is v0, and b > max{k,0} are fixed. Each level therefore remains a fixed number of typical diffusive moves from spot. With S0=100, B=120 and 20% vol in the numerical tests, the barrier has b=1 for one-year options and b=2 for three-month ones.

As θ goes to zero, the theorem requires joint convergence in law of the normalized terminal return, the running maximum, and Y. This last quantity measures the relative gap between instantaneous variance and the forward variance curve, scaled by θ^H, with H in (0,1/2]. Y must also be uniformly integrable. The leading term is the time-inhomogeneous Black-Scholes barrier price fitted to the forward variance curve; the first model-dependent correction has order θ^(H+1/2). The stated remainder is o(θ^(H+1/2)). With no rate or constant supplied, that statement gives no maturity at which accuracy can be expected.

ATM skew in regular models

For the regular case, H=1/2, the paper covers local and stochastic volatility models of the CEV and SABR type. Corollary 4.4 makes the correction coefficient asymptotically 2A, with A the Black-Scholes ATM skew, when the volatility shock's component orthogonal to returns is independent of the return path. The resulting price is P0 + (1+2A/√v0)P1. Here P0 is the Bachelier barrier price; P1 is the order-θ gap between Bachelier and Black-Scholes. If 0 ≥ 2A ≥ −√v0, the expression is a weighted average of their barrier prices. The Black-Scholes weight in CEV is β, although the paper calls that CEV application formal.

For regular models, the paper concludes that European option prices fix the up-and-out put price in a model-independent way, up to an o(θ) error. The claim carries Corollary 4.4's conditions: E|ηξ| finite, and η−E[ηξ]ξ independent of the terminal return and running maximum. A desk can therefore form a skew-adjusted barrier mark without calibrating a stochastic volatility model, provided those conditions hold. Rough Bergomi takes a different coefficient, a one-dimensional integral proportional to ρν√(2H). At H=0.1 its correction scales as θ^0.6.

The theorem concerns up-and-out puts only. Remark 4.6 leaves up-and-out calls and down-and-out puts as a conjecture: their digital barrier component creates a boundary layer of order (1−u)^(-1) that breaks the domination argument. For strikes above the barrier, Remark 4.3 says the γ(b,k) identity fails, while Remark 4.5 prices that case model-free anyway.

What changed in the model-price checks

All errors below are price differences on a 100 spot across 21 strikes from 80 to 120.

The results support the paper's claim that the correction "materially improves the Black-Scholes approximation." The author reports a qualification for SABR: at the four strikes from 98 to 104, Black-Scholes is closer to Monte Carlo.

The comparison is less clean than the headline errors suggest. The cruder explicit γ correction has a lower mean error than the full difference correction, 0.00674 against 0.01248 in CEV and 0.06009 against 0.07444 in SABR. On CEV maximum error, the ordering reverses: 0.01983 against 0.01747. The corrections are asymptotically equivalent, and the paper acknowledges that they can separate numerically at a one-year maturity. Two of the three tests use θ=1, far from the theorem's limiting regime. Each model also gets exactly one parameter set and one barrier. As the author notes, the CEV run is formal because absorption at zero violates the positivity assumption.

The rough Bergomi residual deserves attention. Its largest Monte Carlo standard error is 0.00649, and changing monitoring frequency shifts prices by at most 0.00005. We read the 0.10319 mean error as mostly approximation error. The paper ranks no causes: it says the remaining discrepancy includes approximation, sampling, and discretization errors. The 12,500-versus-25,000 comparison probes neither scheme discretization nor control-variate centering bias.

Continuous payoff, sampled monitoring

The theorem prices a continuously monitored barrier; Monte Carlo observes touches at 25,000 grid points. Its control variate uses B−S0=20 for centering, drawn from the continuous-monitoring identity, leaving finite-grid bias in the estimator. Halving the monitoring frequency changed CEV prices by up to 0.00477 and SABR prices by up to 0.00408, with paired standard errors near 0.0004. The author says of the SABR comparison that "these figures do not provide a rigorous bound on the remaining bias."

For SABR, those shifts are an order of magnitude below the corrected mean error of 0.07444 and do not overturn the result. CEV warrants a different yardstick. Its 0.00477 monitoring shift is about 38% of the corrected mean error of 0.01248. The PDE-to-Monte Carlo gap reaches 0.01725, or 2.60 standard errors, exceeding that mean error outright. The continuous PDE comparison, with 0.00429 mean and 0.01445 max, is therefore the better CEV check. We found no comparable continuous benchmark for SABR or rough Bergomi.

We could not test the correction ourselves. Our data contains no barrier-option contracts or quotes. Listed equity options are available only as end-of-day prices, and even minute bars cannot establish whether a continuously monitored barrier was touched. Without an observed barrier price, we have no price against which to check the correction.

The paper says ChatGPT carried out a Lean formalization of the proofs. We did not find a pointer to those files.

For a desk that believes volatility is regular (H=1/2), I would use P0 + (1+2A/√v0)P1 as a cross-check on short-dated up-and-out put marks today; under rough volatility, I would use (26). Dealer barrier quotes would change how far I trust it: A should come from the same day's vanilla surface, and I would want to see whether the near-ATM misses found in SABR reappear.