A risk desk should value an auditable volatility repair more than a small score advantage, and this construction makes every adjustment visible. Its optimal control is algebraic: sigma* = ((sigma_ref)^-1 + Z/sqrt(dt))^-1. The network learns only Z, the projected adjoint. Each move away from calibrated volatility can therefore be inspected step by step.

We cannot trade ES, NQ or YM in the form we would run this. Our implementable version uses SPY, QQQ and DIA. Futures session conventions, leverage and liquidity differ enough that the paper's reported figures do not transfer to that version. When we built it, the closed form from El Boustany, Basseras, Mekkaoui, Alouadi, Hafsi and Pham's construction held up. It is the part we would keep.

What changes, and what remains frozen

The method begins with a fixed reference model. It is a price-only, path-dependent volatility specification in the Guyon-Lekeufack style, driven by short and long memory signals for trend and activity. A ridge regression on path features supplies the drift. Volatility parameters are estimated by Gaussian negative log-likelihood on 80% of the training paths, then chosen on the remaining 20%. After selection they stay fixed, with marginal volatilities clipped for numerical stability. For the synthetic tasks, the paper gives the clip as [1e-3, 0.6].

The control leaves b_ref unchanged and alters sigma alone. Its objective has three pieces: squared MMD between generated and target path laws, squared MMD between pushed-forward volatility features, and a running specific-entropy penalty relative to the reference volatility matrix. MMD is the kernel distance the paper uses between distributions.

A discrete-time, path-dependent McKean-Vlasov stochastic maximum principle supplies the optimality conditions. Automatic differentiation produces the adjoint A in the implementation. A ridge first regresses it on reference path features. The result is projected onto a one-layer 96-unit GRU, and the procedure repeats.

For the real-data study, the authors aggregate Databento depth-one records for ES, NQ and YM into one-second prices. The training set contains 496 sessions through May 2025. Validation covers 144 sessions from June to December 2025, followed by 123 held-out test sessions from January to June 2026. Each instrument is fitted separately with four seeds and 8,192 paths per split. Every experiment uses d=1.

The tree earns the strongest claim

The cleanest evidence comes from a GARCH scenario tree. With N=6 and three branches at each date, it has 729 complete paths and an optimum that can be calculated exactly. The sampled procedure hides the exact conditional signal and runs with P=128 and K=300. It recovers 96.4% of the objective improvement delivered by the exact solution. There is a known optimum here, so the estimator has a genuine benchmark.

Results higher up the synthetic ladder are less decisive than the abstract implies. On Heston, realized-volatility W1 drops from 0.089 to 0.014. Absolute-return ACF error falls from 0.068 to 0.016, early-future error from 0.214 to 0.018, and max-drawdown W1 from 0.059 to 0.022. Path distance deteriorates from 0.060 to 0.062.

SBTS beats the correction on each volatility metric, posting 0.005, 0.004 and 0.012, while its path distance is worse at 0.089. LS4 records the best SWD at 0.039. For a single Heston process, SBTS wins every volatility comparison and LS4 wins on path distance.

The structure pays off on the eight-regime Heston mixture. All diagnostics improve relative to the pooled reference, which the authors explicitly describe as deliberately misspecified. SWD moves from 0.347 to 0.237, RV W1 from 0.479 to 0.095, regime TVD from 0.629 to 0.331, and MDD W1 from 0.309 to 0.009. SBTS learns the complete generator rather than repairing one. Even so, it finishes behind the naive reference on SWD at 0.407, on RV W1 at 0.493 against 0.479, and on drawdown W1 at 0.478.

The delayed-volatility task improves as the horizon lengthens, contrary to what I expected. The correction removes 31.6% of reference distributional error at N=128, 39.7% at N=256 and 62.2% at N=512. Dependence error also improves at every horizon, by 4.8%, 6.9% and 3.0%.

Direct minimisation of the same objective with the same network removes 15.7%, 14.1% and 52.6% of distributional error. Temporal dependence becomes worse than the reference at each horizon: -6.7%, -3.4% and -4.5%, compared with +3.0% for the correction at N=512. Direct training does achieve a lower RV W1 at N=512, 0.046 versus 0.055, on 2 of 4 completed seeds. The authors warn that these statistics use completed runs and contain no correction for that selection.

Nine wins against a reference the authors fitted

The conditional test uses path shadowing. After conditioning on the first 65 log-prices, it finds the 256 nearest histories among 8,192 generated paths and scores the next 32 returns with CRPS. Deep MKV improves every score against its matched reference across all three indices. That produces the 9/9 result.

For realized-volatility CRPS, ES improves from 0.897 to 0.634 (x1000), NQ from 1.092 to 0.746, and YM from 0.704 to 0.574. Cumulative-return changes are much smaller. ES moves from 1.045 to 1.003, NQ from 1.383 to 1.334, and YM from 1.030 to 1.013.

The remainder of the ES column weakens the headline. Session bootstrap reaches 0.620 on realized-volatility CRPS and SBTS reaches 0.623, both ahead of 0.634. On ES cumulative return, block bootstrap scores 1.042. It is almost level with the reference at 1.045 and trails the correction at 1.003.

Retrieval uses the last 32 returns, a subsampled path shape, rolling volatility over 5, 10 and 20 steps, and absolute and squared return autocorrelations at lags 1, 2, 5 and 10. Those features resemble the quantities the generator was trained to match. Some of the conditional score therefore rewards its own training objective. The paper reports no significance test for the gaps, while the held-out sample spans one six-month regime in three US large-cap equity index futures.

Is 1.99 worth the move from 1.85?

The downstream decision sets exposure from the 90th percentile of maximum drawdown across the 256 retrieved continuations, with exposure capped at 5. The loss budget is 1%, and the required violation ceiling is 10%. Deep-MKV-TS produces average exposure of 1.99 with 8.13% violations. The fitted reference reaches 1.85 with 7.11%. Both remain below the ceiling.

Across four seeds, exposure sds are 0.06 for the corrected model and 0.09 for the reference. Violation-rate sds are 0.66% and 1.02%.

The reference already sat comfortably within its budget, which means part of the added capacity comes from moving along the frontier. Matching violation rates gives the more useful comparison. At 8.13%, block bootstrap reaches 1.84 and session bootstrap 1.95, versus 1.99 for the correction. The gain is 0.15 of exposure over block bootstrap and 0.04 over resampled whole training days. It rests on 123 sessions, with a seed sd of 0.06.

The authors acknowledge the closeness directly: "The historical baselines attain similar violation rates and, in the case of the session bootstrap, a similar average exposure." Their stated interpretation is that the experiment "improves the decision usefulness of the reference scenarios, rather than as evidence of universal dominance over alternative scenario generation procedures". For held-out futures forecasts, the abstract describes performance as "comparable to flexible generative and historical baselines". Its next sentence makes the stronger claim: "The resulting scenarios also support greater exposure than the reference under a fixed drawdown-risk target."

The introduction defines the narrower objective: "Our objective is therefore to measure how much a structured path-dependent correction can improve a given reference model; flexible generators serve only to contextualize the quality reached after correction." Under that framing, the bootstraps provide context. The target evidence is the rise from 1.85 to 1.99 in exposure and the 9/9 CRPS improvements over the frozen reference. The authors fitted and froze that reference themselves, so their improvement is measured against a baseline of their own construction. Transaction costs appear nowhere. The paper openly presents this as risk capacity rather than trading profit.

There is no Sharpe, no P&L.

Open implementation choices

We made two smaller choices as well. The records are one-second observations, while the locked settings specify N=128 and give Δt as one session. We chose the rule that reduces a session of prints to 128 steps. We also carried the synthetic-task volatility limits [1e-3, 0.6] into the futures runs.

For futures, Table 6 gives zero as the drift rule. Tables 4 and 5 describe the same futures baseline as "its matched fitted-drift reference" and "Fitted reference". The text never reconciles those descriptions, leaving the real-data reference drift unclear. The component frozen and defended on real data is the path-dependent volatility parameterisation.

We previously examined a diffusion generator whose denoiser nearly cancelled the calibrated volatility clock attached to it (our note). This construction addresses that failure mode. The closed-form sigma* exposes every move away from reference volatility, and Figure 3 plots the adjustment against the reference in the delayed-volatility task.

A second held-out regime would change my view of its practical value. One six-month window, four seeds and 0.04 of exposure over resampling real training days on the frozen ES drawdown-risk decision do not justify choosing a GRU-projected adjoint over a bootstrap on a risk desk. The oracle experiment shows that the machinery works. The futures table has yet to show that anyone needs it.