A desk could use this as a front-month SPX scenario generator. Wiese, Buehler and Horvath derive the risk-neutral drift for every node of a short-dated surface, then simulate across 5,001 possible instruments. Options trade only inside a moneyness eligibility band. Simulated returns are consistent with zero drift, with spot and eight volatility factors driving the system. The group's companion DYSANOS paper reached risk neutrality numerically, in practice for 49 instruments per simulated date. The scale difference is roughly a hundredfold, 5,001 against 49, though DYSANOS imposed Q numerically on a model built under P.
Prices first, then drift
Discrete local variances encode the surface. The fixed grid has 25 daily expiries, 200 interior cash strikes and two far boundary nodes. A tridiagonal matrix built from squared local vols takes the discrete density from one expiry to the next. With non-negative local variances, that matrix is an M-matrix. The authors can therefore write that "mere positivity of the local volatilities is sufficient to ensure the option market is free of static arbitrage at all times." Local vol is a coordinate system for prices here. Spot is the product of this discrete process and an independent Black-Scholes factor at 5% vol; the latter smooths the call curve between strikes.
The paper's advance is the dynamics. Each squared local vol follows a lognormal SDE, with loadings ξ on d Brownian factors. Making every density coordinate a martingale determines its drift: at each node, minus the loadings dotted with that density's relative volatility. Those relative volatilities are calculated expiry by expiry, beginning at the front. The front density and spot share one factor. They follow a Gaussian information bridge, which collapses onto a grid strike at expiry.
Theorem 2.1 establishes a unique strong solution when the vol-of-vol is locally Lipschitz and bounded by a square-integrable deterministic function. There is a grid artifact. Spot settles on a strike at each expiry and is absorbed if it reaches a boundary node. The drift restriction has precedents in Wissel's working paper (extending Schweizer and Wissel) and in Carmona and Nadtochiy. The authors cite both, describing their model as a joint extension of the Wissel and Carmona-Nadtochiy papers with an implementation attached.
How much SPX fits?
The data comprise OptionMetrics SPX European quotes from 891 dates, 3 January 2022 to 29 August 2025. There are 2,192,458 bid-ask intervals, limited to at most 25 business days to expiry and standardized log-moneyness in [-2.5, 1.5]. A footnote attaches that restriction to the abstract's more-than-95% figure. The fit concerns the front month.
Freeing all 5,000 local-variance coordinates on each date puts 97.32% of quotes inside bid-ask; midpoint error is 0.237 half-spreads. Daily quote counts range from 1,590 to 5,096. Parameters outnumber quotes on thinner days, which limits what that in-sample coverage shows. Lower-rank fits tell us more. A shared affine state using 200 codes per date gets 95.02%; rank 100 gets 89.68%, and rank 400 gets 96.01%. The simulations begin from the direct state.
What the martingale tests see
The continuous-time drift eliminates infinitesimal drift exactly. Simulation introduces discretisation bias because densities are nonlinear functions of local vols. The authors acknowledge this. They propose an exact discrete-branch correction in Remark 2.4 but leave it untested, so the simulation tests reflect discretisation bias alongside the theory.
The restricted test covers spot and 44 options: 570 day-instrument coordinates over 100,000 paths. No coordinate exceeds five standard errors. For the conditional tests, an affine policy plus a two-layer, width-32 network is fitted on half the paths and scored on the rest. No statistic exceeds 1.755. Model D, rank 16 with eight steps a day, records a Tsqr of 0.039. Yet Model D has 22 coordinates beyond 2.576, against roughly 5.7 nominally; Model B has 10. Dependence between coordinates and the simultaneous screen mean those counts are diagnostics rather than family-wise p-values, as the paper notes. It marks them orange, signalling a moderate departure from the Gaussian benchmark without explaining its cause. We did not find a discussion of why the richest model has the most 1% exceedances.
The full grid is the result to remember.
Model C, at rank 8 and eight steps a day, allows 5,001 instruments and trades options only within a moneyness band. It screens 62,569 coordinates. There are 635 above 2.576 versus 625.69 expected, 188 above 3 versus 168.9, and none above 5. Conditional strategies earn 3.64e-8 spot units a day, with a standard error of 4.32e-8 and a T of 0.843. Model C alone was tested at this scale; the authors leave the other ranks for future work. Each test is, in their words, "a conditional test relative to the state and strategy class in (30), rather than a proof over all measurable strategies."
Under P
The physical-drift result is thinner. Eight PCA factors retain 72.88% of local-vol innovation variance; 16 retain 84.40%. Reaching 90% requires 26.
The simulations use one constant market price of risk. It accounts for 42.15% at rank 8 and 47.24% at rank 16 of empirical P-minus-Q drift energy. The abstract says the model fits market dynamics well, but the measure change actually simulated captures less than half that target. A state-dependent λ reaches 62.20% and 71.95%, the upper bound for the retained diffusion span. At rank 16 with constant λ, distance to the unconditional historical drift is 13.73, with an interval excluding zero. The authors concede that neither retained factor system reproduces empirical physical drift exactly.
Under P, entropy statistics range from 89.205 to 646.645. Across more than 100,000 paths, the fitted premium is plainly nonzero and state-predictable. We saw no check of its economic plausibility. The authors call their four matched paths "not a physical model validation."
We could not rerun these results. Our end-of-day options data lacks bid and ask, which the calibration objective requires. We also cannot confirm coverage of the SPX European chain; substituting SPY would introduce American exercise.
For a desk, GROMM offers front-month SPX surface scenarios free of static arbitrage by construction. It passes dynamic-arbitrage tests within the paper's strategy class across 5,000 options (200 strikes by 25 expiries). Its P dynamics are not yet a forecast. A rank-16 full-grid run with conditional statistics still inside ±1.96 would make it something I would build on.