Cutting training from 195 to 36 seconds per 100 iterations on a 10-asset butterfly is the result worth taking from Kumaran's paper. The author builds his case around speed too. The transaction-cost pricing checks cover only problems already solvable on a one-dimensional finite-difference grid, using a simplified cost model. There is no hedging test.
What does the network learn?
Leland's model applies a proportional charge whenever the delta hedge is rebalanced. Rebalancing here is weekly, with a round-trip cost of k=0.005. Retaining only diagonal terms replaces each variance in the Black-Scholes PDE with σ²(1 + Le·sign Γ_ii). The Leland number Le is 0.1438. Each diagonal gamma's sign determines whether volatility rises or falls, making the equation nonlinear.
Kumaran expresses the problem as a 2BSDE: three coupled stochastic equations that track asset paths, price and delta. The delta equation's drift requires gamma and speed, the third derivative. A sine-activated ResNet with three residual blocks of width 256 produces the price and the D(D+1)/2 distinct gamma entries. The paper names the gamma output the Γ-head. Differentiating the price once gives delta. In runs using the Γ-head, differentiating that output once gives speed, avoiding a third derivative of the price.
Training penalises discrepancies between successive Euler-Maruyama steps and deviations from the terminal price, delta and gamma conditions. An iteration draws 100 fresh paths with 50 steps. The data is entirely simulated, using uncorrelated GBM assets with X0=100, σ=0.2, r=0.1 and T=1. Payoffs are equal-weight geometric basket calls struck at 100 and 90/100/110 butterflies, with D=10 and D=100. The tables report six runs. Set aside the run that passes gamma gradients through the price drift, discussed below. In the remaining five, initial-price errors range from -0.14% to -1.93%, with time-averaged RMSE no higher than 0.090.
A 5.4-fold speed gain, with path RMSE rising from 0.058 to 0.090
The D=10 butterfly provides the direct comparison between training with the head and training without it. Each 100 iterations takes 36 seconds with the head, versus 195 without. Total training takes 2.0 hours versus 15.0. The run without the head also needs more iterations, 27,600 against 19,700.
The head does worse on path RMSE and gamma. Pathwise RMSE is 0.090 with the head and 0.058 without it. Final Y0 error favours the head, at -0.79% versus -1.24%. Average the last 2,000 iterations, however, and the runs are effectively level: -0.44% and -0.40%.
Kumaran reports higher relative errors from the Γ-head on diagonal and off-diagonal entries than from the network's own Hessian. Even the Hessian of that same headed network does better.
The head buys speed.
Kumaran's Discussion makes that case explicitly, describing a drastic reduction in computation time for high-dimensional examples and presenting it as a route to scalability. The timing evidence supports the speed claim. Evidence for scalability comes only from baskets reducible to one dimension. At D=100, the author attempted no run without the head because it would have been too slow.
Wide inputs, one effective dimension
An equal-weight geometric basket of uncorrelated GBM assets is itself a GBM. Its price therefore depends on time and a single number, G. The paper acknowledges this reduction to one dimension as the reason finite differences can solve the problem. It also argues that these products pose "non-trivial challenges in terms of the change of sign curvature components and in terms of scalability to high dimensions."
The curvature argument holds up. For each constituent, diagonal gamma depends on Q = G²V_GG - (D-1)GV_G. This expression can be negative even for a plain call. The learned cash gammas on the D=10 call have negative diagonals and positive off-diagonals. So a vanilla basket does test the sign switch in Leland's model.
The dimensionality argument is less convincing. With D=100, the network accepts 101 inputs, produces 5,050 gamma entries and has 1,521,595 parameters. Yet the function it learns varies along a single direction. The exercise demonstrates that the solver can handle wide inputs and outputs; accuracy on an irreducible problem remains unshown. In his Discussion, Kumaran acknowledges the simple architectures and somewhat simple payoffs. Arithmetic baskets appear as future work. All assets have identical parameters in every run.
Below 0.1 RMSE can still mean 37% of the D=10 butterfly's friction premium
Subtract the frictionless price from the Leland price to get the friction premium. This is the part of the answer contributed by the transaction-cost machinery.
The abstract highlights time-averaged RMSE below 0.1 in every case. For the D=10 butterfly, RMSE is 0.090 across 100 test paths sampled at 51 time points. Compare that with the t=0 premium of 0.2436: the RMSE is about 37% of the premium. The network captures 91% to 95% of the premium at t=0, placing most of the error later along the paths.
The D=100 butterfly exposes a different side of the same problem. Its RMSE of 0.049 is the lowest across all runs. Among the five runs excluding gamma gradients from the price drift, however, it has the largest Y0 errors. These are -1.60% at the final iterate and -1.93% on the mean. Its premium is 2.0078 minus 1.7493, or 0.2585. The corresponding shares captured by the network are 88% and 85%.
The stopping point matters too. At its final iterate, the D=100 call has an error of -0.14% and captures 96% of the premium. Across the last 20 checkpoints, its average error is -0.78% and it captures 78%. The reported ± describes checkpoint spread within a single run. It gives no measure of variation across random seeds. Errors increase near maturity as well: call RMSE reaches about 0.13, while butterfly errors jump after t=0.9.
Before using the learned gammas to hedge
The paper describes its diagonal-only cost term as a further simplification of the full Leland expression. Its finite-difference reference solves the same simplified equation. Agreement therefore checks the numerics for that diagonal-only equation.
Pricing is Kumaran's stated objective, and he says the learned gamma components are adequate to solve the PDE. The errors leave a gap for a hedger. On the D=10 call, diagonal gamma errors are about 14% through t=0.5, increasing to about 30% near expiry. Off-diagonal errors range from about 20% to about 40%. For the D=100 call, diagonal errors remain within 40%, while off-diagonal relative errors range from 2.3 to 2.9. The author attributes the latter to true values close to zero. Both diagonal and off-diagonal errors on the D=100 butterfly remain between about 18% and 33%.
Allowing gamma gradients through the price drift leaves the D=10 butterfly priced 9.51% too high. RMSE in that run is 0.297, and the model's premium ratio reaches 213%. All other Γ-head runs block those gradients. Sensitivity of that size to a single setting calls for re-verification with any new payoff.
We could not backtest this. We know of no traded geometric-basket options on 10 or 100 stocks against which to test it. Building replicas from stock prices would test the model only against its own premium. We also did not find simulated hedging P&L in the paper.
Kumaran proposes Monte Carlo references for products such as arithmetic baskets. A run of that kind, using correlated assets and reporting spread across seeds rather than checkpoints, would change my view. Until then, the paper's solid result remains the reduction from 195 to 36 seconds per 100 iterations.