The semi-analytic geometric hedge carries this result. After scaling and stopping, its held-out mean-squared hedging error on the arithmetic Asian is 1.703314. A learned regression correction lowers that to 1.695464, an improvement of 0.46%. The fractional kernel behind the Volterra construction receives no hedge number.
The paper combines theory with simulation: no market data, one Heston parameter set and 21 fixings on a one-year contract. We did not replicate it. None of the figures below tests the method against traded prices. The authors derive a semi-closed price and variance-optimal share count for a discretely monitored geometric Asian, then use the same contract as a control variate for the arithmetic average settled by the market. In the held-out arithmetic-Asian pricing experiment, based on 20 replications of 3,000 training and 5,000 held-out paths, the control removes 99.936% of payoff variance. The measure is one minus the ratio of adjusted-payoff variance to direct-payoff variance. The abstract describes the reduction in words and never reports the number.
The calculation
For fixing dates, the log geometric average is a weighted sum of log-prices. Under Volterra-Heston, the joint transform of that sum and the terminal log-price has an exponential-affine form. The new ingredient is its input. The Volterra-Riccati equation accepts a measure argument rather than a density on the monitoring window, allowing a discrete grid to enter as a sum of Dirac masses. The paper establishes existence, uniqueness, L^p bounds and fractional-Sobolev regularity for the measure-valued equation. It also proves stability in the kernel and the measure.
The resulting price is a contour integral. Payoffs are expressed as a + bx + cy, together with two Mellin-Bromwich integrals on the strip whose real part R lies in (0,1). Fixed-strike calls and puts use one contour measure. Floating-strike contracts use the other.
One apparatus prices four contracts.
The hedge comes from the Galtchouk-Kunita-Watanabe decomposition. When |rho| < 1, the stock-and-cash market is incomplete, and the delta projects the claim onto traded gains. The optimal share count combines the same transform blocks. Each is multiplied by beta_t(s,w) = w + s g((t,T]) + rho sigma psi_2(T-t), with the final term carrying the correlation effect. The monitoring schedule enters through g((t,T]), the mass of fixings remaining. On a discrete grid, this quantity is piecewise constant. It also enters through psi_1 and the Riccati solution psi_2. Residual variance is explicit: (1 - rho^2) sigma^2 multiplied by the time integral of E[nu_t |Q_t|^2]. Here Q_t gathers the same transform blocks, weighted by psi_2.
For completely monotone kernels, K is the Laplace transform of a positive measure. Discretising that measure produces an N-factor sum of exponentials and a finite-dimensional Markovian state. With non-singular kernels, the paper proves convergence of the lifted variance-optimal hedge to its Volterra counterpart, subject to L^2 kernel convergence and uniform bounds on the exponential sums and their derivatives.
What did the checks establish?
The Heston benchmark uses one parameter set: kappa = 1, theta = nu_0 = 0.04, sigma = 0.3, rho = -0.7, S_0 = K = 100, T = 1 and r = 0. The regular-kernel study keeps the same set except for kappa = 2. There is no market data, surface calibration or sweep over kappa, sigma or rho. The evidence instead comes from four internal checks: Monte Carlo, the constant-variance limit, a finite-difference hedge and a contour shift. These checks validate the implementation at the stated parameters. The analytic assumptions behind the pricing and hedging theorems remain necessary.
For the 21-date fixed-strike geometric call, the transform price is 4.247377. Monte Carlo with 200,000 paths produces 4.253520 and a 95% half-width of 0.025055. The 6.14e-03 gap falls inside that interval. At 2,000 steps, the estimate becomes 4.263444, with a half-width of 0.025137. The two Monte Carlo prices differ by 9.9e-03, a wider spread than the transform gap. Their intervals overlap, leaving the discretisation spread statistically unresolved, while the transform price remains inside every reported interval. Only the CIR transition is sampled exactly because the joint stock-variance path is time-discretised.
Under constant variance, the integral reproduces the analytic discrete geometric Black-Scholes price 4.36884811 to 4.55e-11. It reproduces the delta 0.48453809 to 3.04e-13. The time-zero hedge is 0.45298997 and agrees with a centred finite difference of dS_0 C_0 + (rho sigma/S_0) d nu_0 C_0. The gaps are 3.08e-04, 7.71e-05 and 1.93e-05 as h halves from 0.01 to 0.0025. Each halving gives a fourfold reduction, as expected from a second-order difference. Shifting the contour across R = 0.25, 0.50, 0.75 changes the price by under 2.0e-6.
Convergence is clean for the regular kernel K(t) = (1 - e^{-t})/t. Its L^2 error declines from 3.16e-03 with two factors to 1.94e-07 with 256 factors. Price and hedge errors use the 256-factor result as their reference. At 128 factors, those errors are 7.42e-09 and 3.73e-09, compared with 4.06e-05 and 2.03e-05 at two. Eight to sixteen factors already deliver price errors near 1e-6. On common Brownian paths against a 128-factor reference, the RMSE for a 20-interval discrete hedging gain falls from 4.71e-04 at four factors to 1.38e-06 at 64.
As the trading grid is refined, hedging MSE decreases monotonically from 5.538 at five intervals to 0.645 at 200. The frozen-state proxy is 0.518308. No transaction costs are charged anywhere in this study. Because the proxy freezes the state at time zero, the remaining gap combines discretisation error with proxy error, as the authors acknowledge.
Rough hedging stays out
Volterra-Heston is motivated by the fractional kernel, K(t) = t^{H-1/2}/Gamma(H+1/2), with H below one half. This kernel is infinite at zero, whereas a finite sum of positive exponentials stays finite. Custers, Friesen and Karbach conducted a prespecified sweep and failed to meet their own admission threshold of 5% relative L^2(0,T) kernel error. Their best result at alpha = 0.6 was 19.1%. At alpha = 0.75, nearer the non-rough side, it was 1.44%. They consequently report no fractional-kernel pathwise hedge results and state explicitly that adding factors cannot transfer the regular-kernel evidence.
I respect that disclosure, and it should determine how the numerical results are read. The abstract confines its numerical evidence to a regular non-Markovian kernel, leaving the rough-volatility framing open to question. The theory includes rough kernels. The reported numbers cover Heston and a regular completely monotone kernel. Compare our review of a rough Hawkes-Heston microfoundation.
Arithmetic hedging barely needs the learned correction
Across 20 replications of 3,000 training and 5,000 held-out paths, lambda is fitted on the independent training sample. The pricing half-width drops from 0.043844 to 0.001186. The CV-minus-direct difference is 0.002203 against a half-width of 0.043892, so the two estimators agree.
Hedging gives a less flattering result. Both regressions use one basis that includes the geometric hedge ratio as a covariate. A third strategy simply scales the geometric hedge. All three are evaluated on the same held-out paths. Direct backward regression on the centred arithmetic payoff records MSE 3.548026. Scaling and stopping the semi-analytic geometric hedge gives 1.703314. Regression on the low-variance payoff residual, added to the geometric hedge, gives 1.695464.
The paper presents the CV-versus-direct comparison as a 52.2% reduction. The authors attribute it to the regression target rather than the strategy class. Against the geometric-only hedge, the contrast is -0.007850, equivalent to a 0.46% improvement. It is statistically resolved and economically nothing. The authors make the same point: the geometric hedge "accounts for nearly all of the benefit in this benchmark, while the learned correction is small but statistically resolved." The per-step OLS result keeps the continuation target fixed, and the authors leave error propagation across dates open. Because the cross-term in decomposition (60) does not vanish, we read the control-variate case for hedging as motivation rather than proof.
Small samples expose the learned layer. Direct-regression MSE is 13.66 with 500 training paths and reaches 2.27 only at 10,000. The control-variate result remains near 1.70 throughout.
Limits of our test
The Asian contracts required for a direct test are OTC. We have no observed prices, quotes or transaction records for them, which prevents a backtest of entry pricing and tradability. The paper contains no market data, and we have no FX price data either. Our test therefore substitutes synthetic Asian liabilities hedged with US equity ETFs and their listed end-of-day vanilla options. Nobody has shown that the mechanism survives this substitution. End-of-day option inputs allow daily rebalancing to be tested. Intraday rebalancing and intraday surface calibration remain untestable. Any Volterra-Heston calibration and Riccati-Volterra solve comes from our own code, so the exercise tests our factor discretisation rather than the paper's specification against traded Asian quotes.
Nothing here is measured against market prices, leaving misspecification untested.
I would use the geometric-Asian transform to control arithmetic Monte Carlo and treat the semi-analytic geometric hedge as a starting delta. Evidence on the rough side would change my view if a factor construction resolved the short-time boundary layer and passed a kernel-error gate anywhere near 5% at alpha = 0.6, the open problem identified by the authors.