A second-order biased scheme beats two exact schemes and one low-bias scheme when monitoring becomes frequent. Exact simulation of the Ornstein-Uhlenbeck stochastic volatility model incurs its per-step cost on every monitoring date. For a weekly Asian option, that means 52 payments. He and Kwok's splitting scheme costs almost nothing per step, while its discretization bias falls automatically as h = T/M declines with rising M.
At M = 52, the trade goes their way.
What happens inside the split?
The model is Schöbel-Zhu: log spot has instantaneous volatility that follows a mean-reverting OU process and is correlated with returns. The difficult quantity is integrated variance over a step, conditional on endpoint volatilities and integrated volatility. Li and Wu derived its Laplace transform, though pathwise inversion is slow. Zeng et al. inverted a conditional characteristic function through Hilbert transforms and an interpolation table. Choi represented the OU bridge with a Karhunen-Loève sine series, turning integrated volatility and integrated variance into linear and quadratic forms in independent normals. Brignone and Sgarra instead matched the first two conditional moments of integrated variance to an inverse Gaussian. Their scheme is low-bias instead of exact.
He and Kwok avoid that quantity. They split the system so the correlated component, dX = rho*sigma dW2 and dsigma = xi dW2, integrates exactly by Itô as (rho/2xi)(sigma_{t+h}^2 - sigma_t^2 - xi^2 h). Integrated variance disappears from this component. In the other sub-step, volatility remains deterministic along the OU drift, making its variance contribution Q(v,h) an explicit integral.
Strang-Marchuk composition uses a half step, a full step and another half step. Proposition 1 then gives an update requiring two independent standard normals and elementary functions. The claimed global error is O(h^2).
The variate counts make the comparison plain. Splitting draws 2qNM normals for N paths, M monitoring dates and q substeps per interval. Brignone-Sgarra uses 4NM normals plus NM uniforms. Choi requires (6+L)NM normals when L series terms are retained.
Our option data consists of end-of-day prices and Greeks. It contains no bid-ask quotes or historical marks for bespoke barrier, Asian or corridor contracts. We have not run the paper's scheme against a market, and no figure below comes from our own testing.
One parameter set, four products
The shared calibration is S0 = K = 100, T = 1, sigma0 = theta = 0.2, kappa = 4, xi = 0.1, rho = -0.7 and r = 0.09531, following Li and Wu and Choi. N ranges from 1,000 to 100,000 paths. Each case is replicated R = 200 times, and RMSE is measured against a reference value.
With q = 1, splitting delivers the best accuracy against CPU for three of the four products. The weekly arithmetic Asian call has a reference value of 7.141068705, obtained from Zhang et al.'s recursion quadrature. For the weekly down-and-in barrier call at B = 90, the reference is 1.568940412. The weekly corridor variance swap, with L = 80 and U = 120, has a reference fair strike of 0.034198980391.
The European call breaks the pattern. Its reference value is 13.214920018, based on high-precision Fourier inversion of the Riccati characteristic function of Schöbel and Zhu. Splitting requires q = 8 to match the inverse-Gaussian and Karhunen-Loève schemes. The authors report persistent bias at q = 4, causing RMSE to decline more slowly with N.
Among the four methods, the Hilbert scheme is reported as least competitive in the European and Asian tests, despite carrying the heaviest implementation burden. On the barrier, the paper places it alongside the inverse-Gaussian and Karhunen-Loève schemes at large N. Reported CPU times omit one-time setup. The paper identifies its own Hilbert interpolation table as one of those exclusions, and He and Kwok constructed that table using 2048 Fourier terms with frequency spacing 0.1. Even with that favorable exclusion, Hilbert finishes last in the European and Asian tests.
Three gaps in the ranking
The first gap concerns the size of the win. No table reports RMSE or CPU figures in the text; the four comparisons appear only in plots. Readers can see the ordering, though the margin remains unclear. Variate counts put splitting at 2NM against 4NM normals plus NM uniforms for the inverse-Gaussian scheme. That suggests a factor of two or three. The paper adds that related computations may cost more than the sampling itself, leaving those counts as a floor.
Second, xi = 0.1 and kappa = 4 produce a quiet volatility process. The numerical section uses only this one parameter set. Accuracy for a second-order scheme at q = 1 when vol-of-vol is 0.1 does not establish the same accuracy at 0.5. The convergence argument is deferred to a working paper, Hu et al., 2026. With no empirical convergence-order table reported, the O(h^2) exponent remains an assertion here.
Third comes the barrier benchmark. The authors state that 1.568940412 comes from their own implementation of Choi's scheme, using 5 million paths and L = 48, with standard error 9.62e-4. A competing method therefore serves as the reference. Since the paper prints no RMSE column, we cannot compare that standard error directly with the ranked errors. The authors disclose this clearly, but it still restricts what the barrier panel can establish. Implementation language and hardware are also unstated, the familiar problem in cross-author CPU comparisons.
Why traded exotics remain out of reach
Schöbel-Zhu vanilla prices are available through Fourier inversion, which the paper uses for its European benchmark. The faster simulator earns its keep on path-dependent payoffs. Those are precisely the contracts we cannot mark. End-of-day prices and Greeks, without bid-ask quotes, allow tests of listed vanilla relative value and simulated exotic pricing accuracy. They cannot provide exotic pricing error against traded prices, while spread and execution cost must be modelled.
The mechanism holds up. Frequent monitoring favors a biased scheme while charging exact schemes at every date, and the split is chosen so the correlated leg collapses to one Itô identity. One table would change my view of the ranking: RMSE against CPU at xi = 0.5, kappa = 1 and T = 2, with q = 1 still shown on the axis.