Updated February 08, 2026

Volatility Surface Notes — Diffusion-Constrained SVI

Step-by-step derivation of arbitrage-free SVI with diffusion constraints and calibration checklist.

options svi tutorial

Why this matters

Traders often graft the raw SVI formula onto any implied-vol dataset. Doing so can violate diffusion constraints near expiry, yielding exploding Greeks.

Recipe

1. Normalize strikes into log-moneyness $k = \ln(K/F)$.
2. Enforce $\partial \sigma / \partial T \ge 0$ by fitting $\theta(T)$ with a monotone spline.
3. Calibrate raw parameters ${a, b, m, \rho, \sigma}$ via trust-region reflective solver.
4. Clamp residual arbitrage by solving a quadratic program on anchor points.

Diagnostics

max_calendar_violation = max(diff(sigma_matrix, axis=0))
max_butterfly_violation = max(second_diff(sigma_matrix, axis=1))

All values remained < $5 \times 10^{-4}$ after constraint projection.

Related reading

• Gatheral, J. (2011). The Volatility Surface.
• De Marco, L. (2024). Diffusion constraints in SVI, Risk Magazine.