Our review of the paperwe backtested it

Conditioning the denoiser turns heavy-tailed diffusion noise into a nuisance

An AR(1) tempered-stable volatility clock with exact calibration, tested only on synthetic GARCH paths.

2026-08-01 · 5 min read

Read the paper on arxiv

Our backtest of this idea

Our automated quick test, not the paper's

Tempered-Stable AR(1) Volatility-Clock Risk Overlay with Optional ATM Straddle Volatility-Risk-Premium Overlay

Backtest period 2020-01-01 to 2025-10-08 · hypothetical, net of modelled costs

Why these figures are not the paper's (1)

Our own audit found this run does not follow the paper faithfully (9)

deviation left undescribed by the audit (invalidates: Conditioned diffusion forward-process claims; DDPM sampling claims)
deviation left undescribed by the audit (invalidates: Posterior-form sampling claims; conditioned v1 intervention-invariance experiment as an implemented result)
deviation left undescribed by the audit (invalidates: Per-step KL training claims; denoiser-whitening claims as implemented mechanics)
deviation left undescribed by the audit (invalidates: v2 control results; v2 filtering-quality results; empirical-Bayes memory-movement results)

5 further finding(s) are described in the note.

These are our findings about our own implementation, not criticisms of the paper. Read the figures below as a description of what we ran.

StartTotal -3.3%End

Sharpe: -0.33
Total Return: -3.3%
Max Drawdown: -5.8%
CAGR: -0.8%
Volatility: 2.3%
Trades: 103

The reusable idea in this paper is a theorem that kills the motivation for a whole line of heavy-tailed generators, plus a calibration map you can check by hand.

Heavy-tailed diffusion models replace Gaussian noise with a Gaussian variance mixture. DLPM (Shariatian et al., 2024) draws the mixing scale i.i.d. across coordinates; Student-t EDM (Pandey et al., 2025) shares one scale per sample. DSPM sits between them: the per-coordinate variance is a stationary AR(1) chain driven by tempered-stable increments, the discrete Barndorff-Nielsen-Shephard volatility process, running along the data axis. One memory parameter phi interpolates DDPM, DLPM and t-EDM as boundary cases. Kurtosis and squared-noise autocorrelation come out closed form, K = 3(1-alpha)/(theta(1+phi)) and rho_r2(h) = v*phi^h/(3v+2), with an analytic inverse and a falsifiable feasibility bound rho_r2(1) < K/(3(K+2)), always below 1/3. Daily equity indices (K around 8, rho_1 around 0.2) sit inside it. Recovery is unbiased: at targets (K, rho_1) = (8, 0.20), giving (phi, theta) = (0.750, 0.0643), 200 replications on 2,000-path panels return phi = 0.756 +/- 0.085 and theta = 0.066 +/- 0.007. The sampler matches the theoretical Laplace transform to within 2.1e-4 on u in [0.05, 4] with two million draws.

Why conditioning erases the mechanism

The result that should stop anyone building heavy-tailed samplers for finance is Proposition 5. The authors prove that when the denoiser is conditioned on the mixing variables, and the variance chain is drawn independently of the data in training, the mixing law has zero effect on the generated distribution in the exact-denoiser limit. Interventions on the chain do nothing. The whitening step in the loss cancels the variance factor coordinate by coordinate, so the network never sees a gradient that ties the mixing law to output structure.

The experiments land on this hard. Four conditioned models differing only in the law of A all reproduce the data's squared-return autocorrelation rho_r2(1) = 0.159 to within 0.017, whether their noise offers no clustering (plain DDPM with A fixed at 1), constant clustering, or the full calibrated chain. The between-model spread is no larger than the between-seed spread across three seeds. The network learns the clustering; the noise never hands it over. Then they design an eightfold volatility shock and feed it at sampling time: the conditioned envelope moves by a ratio of 0.88 +/- 0.03 against a naive sqrt(A) prediction of 2.83, and a calm A = 0.25 scenario gives 1.06 +/- 0.01 against a naive 0.50. Wrong sign, essentially flat. The trained network cancels the scenario almost exactly.

Blind and conditioned, side by side

Flip the one design bit and hide the mixing chain from the denoiser, and the mechanism transmits precisely as calibrated: blind DSPM samples show rho_r2(1) = 0.151 against the calibrated 0.161 and rho_r2(5) = 0.019 against the theoretical 0.027, the geometric decay fingerprint intact, while the blind i.i.d. variant's clustering collapses to 0.008 because an i.i.d. mechanism has nothing to transmit. The cost of blindness is a marginal that explodes: kurtosis near 70 (blind DSPM 71.7, blind iid-TS 68.9) because unwhitened heavy noise cannot be cleanly removed. This is a deliberately crude ablation, not a faithful DLPM re-implementation, and the authors say so.

Control at the cost of the tails

The escape route is to break the independence of data and chain. The v2 model learns a variational volatility encoder trained with the stochastic-volatility likelihood whose log-determinant the simplified denoising loss drops. With it, the same eightfold shock scales the envelope by 3.00 +/- 0.14 (naive 2.83), the calm scenario by 0.52 +/- 0.01 (naive 0.50), and the encoder tracks the true GARCH latent volatility at per-path correlation 0.567 +/- 0.001 (0.762 pooled, inflated by cross-path level differences). Learned memory moves from the calibrated 0.638 toward the GARCH persistence a+b = 0.95, landing at 0.803. The bill is stated plainly: v2's unconditional excess kurtosis falls to 2.86 +/- 0.12, below the target 6.31, below v1's 5.23, and below even the Gaussian baseline's 4.59, with rho_r2(1) = 0.114 below all four conditioned models. v1 gives fidelity without control, v2 gives control without fidelity, and v2 as implemented runs on a log-normal clock rather than the tempered-stable chain of the construction. Two adjacent members of the same class, not one model.

Everything above is on synthetic GARCH(1,1)-t paths, d = 64, 28,000 of them (20,000 train, 8,000 test), roughly half a million parameters, minutes on a laptop. No real index data, and the authors flag the pricing test of path-dependent structures as future work. The paper reports no trading performance for itself, which is correct for what it is.

What we built, and why it barely tests the paper

We implemented the one component a trader could act on: the calibrated volatility clock as a long-only risk overlay on the top 150 US stocks and ETFs by capitalization, daily, 2020-01-01 to 2025-10-08, scaling exposure down as a symbol's forecast-vol percentile rises past 0.60 and to zero past 0.95, with an optional ATM straddle sleeve on real EOD option chains when a 20-day realized-vol forecast diverges from implied by more than five to eight vol points. We charged five basis points a side on equities and sixty-five cents a contract on options.

Two honest points about this run. First, it is not the paper. The generative sampler, the conditioning, the invariance result and the v2 filter are all absent from what we traded; we kept only the tempered-stable AR(1) calibration as a forecasting device. Second, and this is the part that undercuts our own test, with the paper's calibrated phi = 0.638 the 20-day forecast is dominated by rolling unconditional variance, so the volatility clock collapses to an ordinary EWMA vol filter over that horizon. Any edge or lack of it in our overlay is a statement about volatility-managed beta, not about DSPM. We could not turn the paper's contribution into a distinct trading signal in one pass, and the reason is structural, not a coding accident.

The result worth carrying to a desk is the negative one. If you were tempted to buy the story that heavy-tailed diffusion noise injects the stylized facts of returns into a generator, Proposition 5 says the noise law is invisible the moment you condition properly, and the numbers back it (0.017 spread across mixing laws, 0.88 shock ratio). The transmittable object is the exact, feasibility-bounded calibration between (phi, theta) and (K, rho_1). Before I trusted any generated path from a model like this, I would want its implied volatility forecast graded against realized on out-of-sample equity data, which this paper does not attempt.