Learned stratification improves variance reduction 5% to 16% over benchmarks

A desk pricing the paper's five-year CMS range accrual could match crude Monte Carlo's standard error using about 18% of the paths. Arandjelović and Cheridito obtain that result with a learned three-dimensional stratification factor, reporting variance reduction of 5.448, their highest figure. Before building it, a desk needs to separate the neural network's contribution to that 5.448 from the gain available by stratifying on any sensible factor.

What the factor does

Stratification groups paths into bins according to a low-dimensional factor Y. The estimator draws a fixed number of paths in each bin, then weights the bin means by their probabilities. Under proportional allocation, each bin receives draws in proportion to its probability. As the bins become finer, scaled variance approaches Var(G) minus Var(E[G|Y]). The aim is therefore to choose Y so that the conditional mean payoff varies as much as possible. The authors measure this with the explained-variance (EV) ratio, Var(E[G|Y])/Var(G).

They construct factors as stochastic integrals of the driving martingale, Y = ∫F(t)ᵀdM_t. Each column of F is a feedforward network that takes time alone as input. A deterministic F makes Y Gaussian, allowing exact draws of paths conditional on Y falling inside a bin through Gaussian conditioning of the discretised increments. Training fits F alongside a regression network h. It minimises E[(G − h(Y))²], whose minimum over h equals the unexplained variance.

The experiments price two simulated contracts. A G2++ range accrual pays 6% a year, accruing on days when the 10y CMS remains within 0.008 of its starting value and the 10y minus 2y spread remains within 0.004. The simulation uses 1260 daily steps. The other contract is a worst-of autocallable on three assets under a Wishart covariance model. It has an 8% coupon, quarterly calls after one year and 60% protection, simulated over 312 steps spanning three years.

Every stratified estimator draws 524,288 paths across 4096 equiprobable normal-quantile strata, with 128 paths per stratum. The grids use 4096 bins for one factor, 64×64 for two and 16×16×16 for three. Benchmarks include the authors' adaptations of the Glasserman, Heidelberger and Shahabuddin (GHS) drift and Hessian directions, calculated from smoothed payoff proxies. Model-based alternatives use level and slope directions for the CMS note and return principal components (return-PC) for the autocallable.

These experiments measure how precisely an estimator prices a model at fixed parameters. They contain no market data, calibration or P&L. We could not backtest the method: we have no price history for swaps, CMS notes or autocallables. Nor does a faster estimate of a model value define a position.

How much does learning add?

Measured against crude Monte Carlo, VR ranges from 1.903 to 5.448 for the CMS note and from 2.303 to 2.870 for the autocallable. The improvement over the strongest benchmark at the same dimension ranges between 5% and 16%.

For the CMS note, the learned factor produces VR of 1.903, 4.337 and 5.448 with one, two and three factors. GHS Hessian leads the benchmarks at every dimension, returning 1.809, 3.794 and 4.926. For the autocallable, the learned factor delivers 2.303, 2.870 and 2.847. Return-PC delivers 2.115, 2.482 and 2.537.

These gains come from single runs. We found neither repeated trials nor intervals for the VR column, and a consistency check suggests that estimation noise matters. With finite strata, population VR cannot exceed 1/(1 − EV): coarse bins explain less variance than the factor itself. The learned two-factor CMS result has EV of 0.762, implying a ceiling of 4.20. Its reported VR is 4.337. That discrepancy of 0.14 amounts to about a quarter of the 0.543 advantage over GHS Hessian, comparing 4.337 with 3.794.

Other rows breach their ceilings by smaller amounts. For the CMS note, level delivers VR of 1.803 against a ceiling of 1.80 (EV 0.444); slope delivers 1.393 against 1.38. The learned one-factor autocallable result is 2.303 against 2.30. Return-PC records 2.115 against 2.10 with one factor, then 2.482 against 2.44 with two.

Benchmark choice matters for the autocallable. The derivative-based directions perform poorly: a single GHS Hessian factor explains 0.233 of variance, compared with 0.524 for return-PC. A local Hessian evaluated at one point struggles to capture a worst-of payoff with call barriers. The authors also choose the smoothing proxies themselves, so some of the weakness may come from those proxies. Return-PC sets the relevant hurdle. The network clears it by about 0.04 to 0.08 of EV.

Sixteen bins per axis

With a fixed budget of 4096 strata, the third factor begins to cost too much grid resolution. Bins per axis fall from 64 to 16 when the extra coordinate is added. The autocallable's EV increases from 0.658 to 0.695, yet VR falls from 2.870 to 2.847.

The ceilings help locate the lost gain. They are estimates too, derived from the same noisy EV ratios. With two factors, the autocallable achieves 2.870 against a possible 2.92. With three, it achieves 2.847 against 3.28. The CMS note leaves more on the table: EV of 0.879 implies 8.26, while the coarse grid produces 5.448. The shortfall remains large even allowing for the estimation caveat. Wide strata can waste much of the benefit of a better conditioning variable. Anyone adopting this design should tune bins per axis, starting with the three-factor grid.

Stability requires keeping every direction

Can network approximation error spoil the conditional quantities? Theorems 3.4 and 3.7 establish continuity provided the approximating covariance matrices eventually share the limit's rank. The counterexamples explain the restriction. In one, factors (W, ε_n Z) converge to (W, 0). Rank falls from 2 to 1, while the conditional expectation error remains at 1. An approximation must therefore avoid shrinking an informative direction to zero weight.

Example 3.5 goes further: at a rank-deficient limit, covariances alone cannot determine the outcome. Two sequences have identical covariance matrices and the same singular limit, yet their conditional distributions end differently. One remains at least a fixed distance c0 away in the bounded-Lipschitz metric. Corollary 3.8 then establishes that networks attain the same supremum of explained variance as any non-degenerate deterministic-integrand factor.

Corollary 3.8 proves existence. As the authors put it, "our universal approximation result is nonconstructive." Their conclusion leaves convergence of training and the statistical properties of the learned factor and stratified estimator for future work. Remark 2.3 supplies a reason for caution: a strict local maximum is worth 2 when the global value is 8. Training can stall there.

We found no wall-clock times, training budgets, network sizes or restart counts in the numerical section. At variance reduction of 2.870, training pays only if it costs less than the roughly 1.9 crude estimators it replaces. The paper supplies no basis for testing that condition.

We would build the two-factor version for the autocallable, where its advantage over return-PC is largest (2.870 against 2.482) and VR remains below the 2.92 ceiling, then time it before committing further.