Joint Calibration Results
Heston 5-parameter · DE + L-BFGS-B
2026-03-24 · SPX = 6581.0 · r = 4.5%
κ (kappa)
4.6213
Mean reversion speed
Half-life = 37.8d
θ (theta)
0.07641
Long-run variance
√θ = 27.64% ann vol
σ (vol-of-vol)
0.8407
Vol of variance
Controls smile curvature
ρ (rho)
-0.99
Spot-vol correlation
AT BOUNDARY
v₀ (v-naught)
0.05613
Initial variance
√v₀ = 23.69% spot vol
SPX IV RMSE
3.833 vol pts
Moderate — acceptably close
VIX Futures RMSE
1.15 pts
Acceptable futures fit
VIX Options RMSE
37.14 pts
Model misspecification — Heston cannot joint-fit VIX options
Calibration Performance
42.6s
5,374 evaluations
DE (120 iter × 40) + L-BFGS-B
SPX Implied Volatility Smile — All Maturities
Calibrated Parameter Summary
| Parameter | Symbol | Value | Interpretation |
|---|---|---|---|
| Mean reversion | κ | 4.6213 | Half-life ≈ 38 trading days |
| Long-run vol | √θ | 27.64% | Steady-state implied vol target |
| Vol-of-vol | σ | 0.8407 | High: drives strong skew / smile |
| Correlation | ρ | -0.99 ⚠ BOUNDARY | Extreme leverage effect — needs jumps |
| Spot vol | √v₀ | 23.69% | Current instantaneous vol |
| Risk-free rate | r | 4.5% | USD 6M SOFR (2026-03) |
| Dividend yield | q | 1.3% | SPX continuous div yield |
| Feller margin | 2κθ − σ² | -0.000447 | FAIL — variance can touch zero |
SPX level: 6581.0
As-of date: 2026-03-24
Fit time: 42.6s
DE evaluations: 5,374
RMSE Decomposition — Loss Components