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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.

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Data decides the shape

Parametric models (SVI, SABR) choose a shape in advance. These models let the data decide. The trade-off: more flexible, harder to implement, slower to calibrate, less battle-tested.

At a Glance​

Model
Year
Key idea
Maturity
2025
Non-parametric surfaces via linear programming. Arb-free by construction.
New
2019+
Neural networks learn vol dynamics from data. Deep Hedging.
Research
2023
Vol depends on where price has been, not just where it is now.
New

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.

Model
Calibration method
Speed
Arbitrage-free?
Dynamic interpretation?
SANOS
Linear programming
Moderate
Yes (by construction)
No
Neural SDE
Neural network training
Slow (training), fast (inference)
Depends on architecture
Yes
Path-Dependent Vol
Signature-based regression
Moderate
Not guaranteed
Yes

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:

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