The missing vega lives on the barrier
A 69% vega error is large enough to matter on any derivatives desk. It appears in the paper's GBM plus Hull-White one-factor down-and-out call, where pathwise AAD (adjoint algorithmic differentiation) overestimates vega. At 50K paths, pathwise vega is +31.6 and the boundary correction is -12.9, leaving +18.7. The components have similar scale and opposite signs. Delta moves less: +0.714 from the pathwise estimate, a +0.125 correction, and +0.839 in total.
We could not run the paper's implementation. Its automatic route requires the AADC library and a pricing kernel connected to it. Without that stack, the available route is a hand-coded correction for an explicitly written payoff.
The failure mechanism is familiar. Differentiate P times an indicator on {g > 0}, and a delta function appears on the boundary. Since this term vanishes at almost every sample point, AAD returns the smooth contribution and misses the boundary contribution. Lakshtanov derives the missing piece analytically.
The construction rotates Z toward a direction v aligned with the gradient of g in Z-space. Z is then separated into the coordinate u and its orthogonal complement w. Along u, the one-dimensional integral has exactly one jump, located where g vanishes. Applying Leibniz to the inner integral and using the implicit function theorem produces three factors: the payoff jump, the Gaussian density at the boundary, and the parameter sensitivity of that boundary. The method never evaluates the one-dimensional integral. Per the paper, this was the breakthrough over Goloubentsev's original 2021 question about reducing the expectation's dimension.
Theorem 1 covers one indicator. Theorem 2 allows arbitrary compositions of smooth functions and indicators, matching the recursive alive/dead logic of an autocallable, provided the discontinuity surfaces are pairwise transversal.
Two operations drive the per-path cost. Newton root-finding locates the crossing along v. The pricing kernel is then replayed twice, at the crossing point plus and minus epsilon times v, to recover the jump. A screen based on |g| divided by the gradient norm, set at 20 standard deviations, removes 70 to 80% of indicators per path. Under stochastic vol, each survivor consumes 6 to 10 kernel replays. Under GBM or LMM the count is 2 because the crossing and gradients are analytic, leaving only the model evaluation needed for the jump.
The paper contains no market data anywhere.
Its first simulated benchmark is a down-and-out call under QuantLib's Black-Scholes process: S0 100, K 90, B 80, sigma 0.25, r 0.05, T 1, 50 monitoring steps, and 100K paths. The other benchmarks are a three-asset Phoenix autocallable under Heston, with 8 observation dates and 48 indicators at 10K paths, and the hybrid barrier described above. For the GBM barrier, the correction comes within 0.1 to 0.2% of analytic delta, vega and rho. The corresponding smoothing errors are -0.2%, +2.9% and -1.2%. On the autocallable, the correction agrees with bump-and-revalue to 0.4 to 0.7%.
Numbers that resist a single reading
The autocallable reference uses bump-and-revalue at 10K paths with the same seed. Elsewhere, the paper says that reaching 2% accuracy on the product takes about 20K paths for the correction and about 5M for bump-and-revalue. Read literally, a 10K-path bump estimate would have a standard error near 45%, given inverse-square-root scaling with path count. Yet the reported difference between the estimators is 0.4 to 0.7%.
Those claims do not fit comfortably together. Common random numbers could account for part of the gap, with a generous bump width accounting for more. We could not find standard errors or an explanation of how the 20K-versus-5M and 8K-versus-12M equivalences were calculated. Those ratios carry much of the paper's case.
The cost comparison raises the same problem. Forty-eight indicators with 70 to 80% screened out leave roughly ten to fourteen active per path. At 6 to 10 replays apiece, our arithmetic gives order 100 kernel replays for each corrected path, compared with one for a plain price. The paper reports 20K paths for the autocallable correction and 5M for bump-and-revalue, a 250:1 path ratio. Charging order 100 replays to every corrected path makes the cost ratio smaller than 250:1. The paper also says tape replay is 10 to 100x faster than re-simulation, which moves the comparison in the correction's favor.
We did not find a wall-clock or total-cost comparison, only replay counts, so the full calculation cannot be settled from the paper. The abstract gives a 0.1 to 4% range against references. We could not locate a 4% correction-method entry in any table; the largest deviation we found is 0.7%.
Who chooses the direction and the root?
The formula gives enough detail for an implementation. The surrounding driver leaves three decisions thinly specified.
Direction comes first. Theorem 1 requires v dotted into the gradient of g to remain positive everywhere on the boundary. During precomputation, the algorithm fixes v as the normalised gradient at Z = 0. For a curved boundary far from the origin, the required condition remains an assumption rather than something the construction guarantees. The correction integrand contains one over the directional derivative of g along v. It can blow up as the boundary approaches tangency with v, and the paper gives no variance treatment for this term.
Root selection is harder. Newton returns one crossing. Suppose an autocall trigger and a coupon barrier use the same asset on the same observation date. Their surfaces have parallel normals, allowing the line along v to cross both. Theorem 2 assumes pairwise transversality among the surfaces. Immediately before the statement, the paper supports that assumption by observing that indicators at different dates depend on different Z-coordinates. The proof uses transversality to infer exactly one jump along each line.
That argument covers different dates. It leaves two levels on the same date unresolved. The paper does not specify the arrangement of the 48 indicators. A layout of 3 assets x 8 dates x 2 levels would place two indicators on every asset-date. Nor does it explain how Newton selects the root when two boundaries share a normal. If the benchmark uses that layout, the difficult case already appears in the Heston test where the correction agrees with bump-and-revalue to 0.4 to 0.7%.
Tuning supplies the final ambiguity. The algorithm contains two constants, a screening threshold of 20 and the jump offset epsilon. The paper says the correction requires no parameter tuning, while the abstract describes the implementation as fully automatic. Both values are defensible defaults, though each remains an implementation choice. The barrier table calls its reference "Analytic". With 50 monitoring steps, the text leaves the particular analytic reference unspecified.
What we could not reproduce
These results compare Monte Carlo accuracy with analytic and bump references. They do not establish hedging P&L. Our data contains no historical contract terms or cash flows for OTC barriers and autocallables, preventing a test of whether corrected Greeks hedge real trades better than smoothed Greeks.
The comparison with smoothing
For a desk already using smoothed indicators, the paper offers a clear operational claim: the quant writes standard pricing code, while the correction driver finds and processes the discontinuities. On the well-behaved GBM barrier, the prize is the 2.9% smoothing bias in vega. On the badly behaved hybrid, it is the 69% pathwise vega error.
My reading would change with standard errors for every entry in the three tables, an explicit layout of the 48 indicators, and an account of how Newton chooses a root when two boundaries share a normal. The formula looks right. The driver remains the paper someone still has to write.