Calibration — Vol System

The Heston correlation parameter has hit its lower bound during calibration. This indicates the model cannot fully reproduce the steep 2026 SPX left skew with a pure diffusion structure — jump risk or a non-affine model (e.g. rough Heston) is required. The extreme negative ρ is the model's best approximation of the leverage effect, revealing model misspecification. VIX options leg is disabled (w₃ = 0.0) — RMSE was 37.14 vol pts, a structural failure of the CIR density, not a calibration failure. Bates SVJ calibration was attempted — jump params degenerate on daily data (see Barndorff-Nielsen & Shephard 2004). Heston remains the active model.

2κθ = 2 × 4.9997 × 0.04314 = 0.43133 | σ² = 0.6568² = 0.43133 | Margin = -1.10e-07
The constraint is active, not broken. The calibrator applies a soft exterior penalty 50·max(0, σ²−2κθ)² whose gradient vanishes at the boundary, so when the SPX smile wants more vol-of-vol than Feller permits the optimiser settles a whisker outside the feasible set — here σ agrees with √(2κθ) to six significant figures. Reporting that as a failure would be a rounding artefact. What it does mean is economically real: the market smile is asking for more vol-of-vol than a Feller-satisfying Heston can supply, so σ is pinned by the constraint rather than by the data.

Pinned at a boundary: rho, sigma. A pinned parameter is set by its constraint, not by the market data, so this 5-parameter model is effectively fitting with 3. ρ pins at its −0.95 floor on nearly every joint fit: the SPX skew wants more spot–vol correlation than Heston is allowed to express. That is the misspecification this project documents, stated as a number.
Fit from 9 SPX options and 4 VIX tenors, all taken on the closed SPX session of 2026-09-11. The VIX-futures leg needs at least 3 tenors to identify a curve; fits below that threshold are rejected, because a single tenor is one data point that Heston matches exactly and the joint fit silently collapses into an SPX-only calibration.

κ (kappa)

4.9997

Mean reversion speed

Half-life = 34.9d

θ (theta)

0.04314

Long-run variance

√θ = 20.77% ann vol

σ (vol-of-vol)

0.6568

Vol of variance

Controls smile curvature

ρ (rho)

-0.95

Spot-vol correlation

AT BOUNDARY

v₀ (v-naught)

0.01088

Initial variance

√v₀ = 10.43% spot vol

SPX IV RMSE

0.215 vol pts

Good fit

VIX Futures RMSE

0.266 pts

Good futures fit

VIX Options RMSE

DISABLED

Heston CIR density underestimates vol-of-vol path uncertainty. RMSE was 37.14 pts — structural failure. w₃ = 0.0 (was 0.2).

Calibration Performance

7.3s

8,406 evaluations

DE (120 iter × 40) + L-BFGS-B

SPX Implied Volatility Smile — All Maturities

Calibrated Parameter Summary

Parameter	Symbol	Value	Interpretation
Mean reversion	κ	4.9997	Half-life ≈ 35 trading days
Long-run vol	√θ	20.77%	Steady-state implied vol target
Vol-of-vol	σ	0.6568	High: drives strong skew / smile
Correlation	ρ	-0.95 ⚠ BOUNDARY	Extreme leverage effect — needs jumps
Spot vol	√v₀	10.43%	Current instantaneous vol
Risk-free rate	r	3.91%	USD 6M SOFR (2026-03)
Dividend yield	q	0.0%	SPX continuous div yield
Feller margin	2κθ − σ²	-1.10e-07	BINDING — constraint active; σ set by Feller, not by the data
Effective d.o.f.	5 − \|pinned\|	3 / 5	Pinned: rho, sigma — set by constraints, not data
Calibration inputs	n_SPX / n_VIX	9 / 4	SPX options / VIX tenors — ≥3 tenors required to identify the VIX leg

SPX level: 7657.0

As-of date: 2026-09-11

Fit time: 7.3s

DE evaluations: 8,406

RMSE Decomposition — Loss Components

HMM Regime Classifier — Limitations

HMM trained 2010–2019. Out-of-distribution on 2025 tariff regime — misclassifies 2025-04-09 as R1. Use XGBoost for production classification. HMM is available as a research alternative via HMMRegimeClassifier in regime_classifier.py.

C13 — Two-Factor Quintic OU Model (Research)

σ(t) = g₀(t) · p(Z_t) · Z_t = θX_t + (1−θ)Y_t · p(z) = α₀ + α₁z + α₃z³ + α₅z⁵ · VIX priced via 2D Gauss-Hermite ≈ 500 pts

Parameter	Symbol	Value	Description
Fast OU speed	λx	168.48	Fast factor half-life ≈ 1.04d
Slow OU speed	λy	10.543	Slow factor half-life ≈ 16.6d
Factor weight	θ	2.8539	Fast-factor loading in Z_t
Poly constant	α₀	0.4061	Sets vol floor
Poly linear	α₁	-0.9404	Linear skew driver
Poly cubic	α₃	0.8877	Smile curvature
Poly quintic	α₅	-0.4895	Wing correction
Leverage (corr)	ε	-0.1823	Spot-vol correlation (leverage)

Metric	Heston	Quintic OU	Δ
SPX IV RMSE (vp)	0.215	2.343	+2.128
VIX Fut RMSE (pts)	0.266	0.001	-0.265
VIX Opt RMSE (vp)	DISABLED	17.53	—

Fit time: 2751.8s

Evaluations: 2,813

MC paths: 6,000/expiry

VIX quad: 18² = 324 pts

Key advance over Heston: VIX options are analytically tractable via 2D Gauss-Hermite quadrature — no structural failure of the pricing kernel. The non-affine polynomial structure captures steep SPX left skew without rho saturating. VIX futures fit to 0.001 pts — by construction, not model skill: ξ₀ is bootstrapped from the same VIX term structure being fit (and no tradable futures prices exist in the DB; the "futures" leg is an index proxy). Quote the SPX-leg RMSE (2.34 vp) as the real fit quality. Model: models/quintic_ou.py · Calibrator: calibration/quintic_calibrator.py

Honest limitation: VIX-option RMSE of ~17.5 vp beats Heston's 37.1 vp (a ~53% reduction) but does not yet reach the <10 vp research target. The residual is in the VIX-smile wings — the degree-5 polynomial under-fits far-OTM VIX calls. A higher-degree map or a second stochastic driver is the next step. Reported as measured, not as a hit target.