Rough Heston on SPX: model, implementation, results

One page: what was built, how it was validated, and what the data actually said. Data: CBOE SPX delayed-quote chain, 2026-07-02 close (spot 7483.24, 11,858 filtered OTM quotes, 50 expiries, T = 4 days … 2.5 years).

Model and convention

Rough Heston (El Euch–Rosenbaum). Characteristic function of the de-drifted log return XT = log(ST/S_0) − (r−q)T (identical convention to the classical pricer, so the Fourier layer is shared):

ψ(u) = exp( κθ·I¹h(u,T) + v0·I^{1−α}h(u,T) ), α = H + ½, D^α h = −(u²+iu)/2 + (iuρξ−κ)h + (ξ²/2)h², h(0) = 0.

At α = 1 this is classical Heston exactly — the critical test gate.

Numerics

• Solver: product-trapezoidal (Diethelm–Ford–Freed weights) made fully implicit — each step is a scalar complex quadratic, solved in closed form. Chosen after the textbook explicit predictor blew up (NaN) at the large Fourier arguments short-dated pricing needs: explicit stability demands |iuρξ−κ|·Δ^α ≲ 1, which is untenable at u ~ 10³. Implicit is unconditionally stable in the linear part and needs no predictor.
• Observed convergence (vs N = 4096 reference): order 1.61 at H = 0.1 (theory: 1+α = 1.6) and 2.0 at H = 0.49 (smooth limit). See results/convergence.csv.
• Pricing: Gil-Pelaez on fixed Gauss-Legendre nodes; ALL strikes of an expiry share the 2n CF evaluations, so the O(N²) solve is paid per expiry, not per option. Full-chain (11,858 quotes) rough evaluation: ~1 s.
• AD: ForwardDiff gradients through solver + quadrature verified against finite differences to ~1e-9 on all six parameters including H (the 0^α NaN-partial landmine is handled explicitly).
• Noise floor, stated: solver error grows ~u^{3/2}, leaving an absolute price noise ~1e-3·(spot/100) — ≪ 1 bp of vol after vega weighting, visible only on microscopic far-wing prices.
• Validation stack: α→1 recovery to rtol 1e-6 (this gate caught two real bugs: the stability blow-up and a variable-shadowing bug in CF assembly); exact identities ψ(0)=ψ(−i)=1 for all H; classical CF cross-checked against an independent Monte Carlo simulation of the SDE (substituting for literature-value pinning — no trusted published table was available, and fabricating one would be worse; noted as open debt).

Results — the honest version

1. The market exhibits the rough signature. Model-free fit of |ATM skew| ∝ T^{H−1/2} across the chain (results/skew_powerlaw.csv):

H ≈ 0.11 sits squarely in the literature range (~0.05–0.15). The classical model's calibrated skew cannot hold this power law: it is too steep at the front (−2.0 vs market −1.4 at one week) and too flat at the back (−0.14 vs −0.18 at 2.5y).

2. Calibration nevertheless does not select roughness on this snapshot. Whole-surface, identical quote set / loss / optimizer for both models (results/rough_fit.csv, results/model_comparison.csv):

Rough Heston converged to the classical corner (κ, θ, ξ, ρ, v0 match the classical fit to 2–3%). The short-end-only duel (T ≤ 0.16, results/short_end_fit.csv) is a dead heat too: 157.6 vs 157.6 bps, H again at the bound, both models resorting to the extreme-κ contortion (κ ≈ 27, v0 ≈ 0).

3. Why both are true at once. The chain is a weekend snapshot: front-end implied vols are deflated by holiday calendar time, and the measured skews are non-monotone in maturity (the 4-day expiry is flatter than the 7-day) — distortions that are real for the market but unfittable by ANY time-homogeneous model, rough or classical. That misfit (~150 bps at the front) dominates the vega-weighted loss; the component roughness improves — the scaling of skew across maturities — is a small part of the total. Within that floor, the likelihood is flat in H and the optimizer drifts to the boundary. H is unidentified by this loss on this data, not rejected by it: the model-free skew diagnostic (H ≈ 0.11) and the calibration are answering different questions.

What would identify H: business-time day counts (or intraday data away from weekends/holidays), a skew-slope term in the loss, or joint fits across several snapshot dates. Listed as future work, deliberately not bolted on to manufacture a win.

The tooling for all three now exists (src/identification.jl, src/cboe.jl): prepare_chain(...; daycount = :business) measures T in NYSE trading days, calibrate_rough_heston_joint(quote_sets; skew_weight) adds a skew-slope term to the loss and fits (κ, θ, ξ, ρ, H) jointly across snapshots with v0 free per date. The results above are unchanged — they await a rerun on data.

Costs

O(N²) per CF evaluation (N = 96 calibration / 192 evaluation), amortized per expiry by the batch pricer. Rough calibration 68 s vs classical 9 s on the same 192-quote set (~7×); full-chain evaluation ~1 s. 194 tests green.