Non-Parametric & ML Models
No formula for the smile. These models learn the surface shape directly from market data using optimization, neural networks, or path-dependent rules.
At a Glance
What they share
All three approaches let the data determine the vol surface shape rather than imposing a formula. They differ in how they learn and what guarantees they provide.
How they relate to each other
SANOS is optimization-based: it solves a linear program to find the surface that best fits market prices while satisfying no-arbitrage constraints exactly. No neural networks, no training -- just a well-posed convex problem. Neural SDE takes the opposite approach: a neural network learns the volatility dynamics from data, which means it can capture patterns no closed-form model can express, but arbitrage freedom depends on the architecture and is not guaranteed by default. Path-Dependent Volatility sits in between. It uses the realized price path (via signature methods) to predict current vol, giving it a dynamic interpretation that SANOS lacks, but without the heavy training infrastructure of Neural SDEs.
Models in this section:
- SANOS — Non-parametric arbitrage-free surfaces
- Neural SDE / Deep Hedging — ML-learned vol dynamics
- Path-Dependent Volatility — Vol remembers the price path