SANOS (Non-Parametric Surfaces)
SANOS takes a different approach to building a vol surface. Instead of fitting a formula with a few knobs (like SVI's 5 parameters per slice), SANOS builds the surface directly from market data using a grid of local vol nodes and guaranteed no-arbitrage constraints. A smooth surface that can capture any shape the market produces -- including local features that parametric models structurally cannot fit.
Parametric vs. non-parametric
SVI and SABR decide the shape of the smile in advance (parabola-like, with 3-5 knobs). SANOS makes no shape assumption. It asks: "what is the smoothest surface that passes through the market data without creating arbitrage?" This produces better fits when the market deviates from parametric expectations. The implied volatility at each node is a free variable, not a formula output.
See It in Action
Toggle between views to see how SANOS compares to a parametric SVI fit, how the grid looks, and how arbitrage constraints are enforced.
SANOS Surface Construction
SVI (parametric) uses 5 parameters and may miss local features. SANOS (non-parametric) passes through market quotes while respecting arbitrage constraints.
How It Works
1. A grid of vol nodes instead of a formula
The surface is represented by a grid of local vol values -- one at each (strike, expiry) point. With 15 strikes and 5 expiries, you have 75 free variables instead of SVI's 25. More flexibility, but you need constraints to prevent the optimizer from producing nonsense.
2. No-arbitrage constraints baked in
The two fundamental no-arbitrage rules translate into simple constraints on the grid:
The key: with local vol nodes as the variables, all these constraints are linear. That means the optimizer can enforce them perfectly, every time.
3. Solved with linear programming
The constraints and objective are linear, so the whole thing is a linear program.
- No local minima -- the solver always finds the best answer, not just a nearby one
- No initialization sensitivity -- you do not need a good starting guess
- Fast -- modern LP solvers handle this in milliseconds
- Bid-ask native -- the LP naturally handles bid-ask spreads as ranges, not mid-prices
Why linear programming matters
SVI and SABR require nonlinear optimization: you need a good starting point and might land in a local minimum. SANOS sidesteps all of this. The LP always finds the globally best answer, fast and deterministically. Every strike and expiry node is jointly constrained against calendar arbitrage and butterfly violations in a single pass.
Bid-Ask Spread Handling
Most models fit mid-prices. But mid-prices are a fiction -- the market quotes a bid and an ask, and the "true" value is somewhere in that range. SANOS fits to bid-ask ranges directly: the model only needs to fall within the range at each point. Liquid quotes (tight spreads) tightly constrain the surface. Illiquid quotes (wide spreads) loosely constrain it. No artificial mid-price bias.
SANOS tradeoffs
Most flexible and cleanest approach to surface construction. No-arbitrage by design, native bid-ask handling, captures local features parametric models miss. The cost: new (2025), no dynamic interpretation (no skew dynamics prediction), more infrastructure.
SANOS vs. Parametric Models
Strengths and Limitations
Not a replacement for everything
SANOS solves the static fitting problem better than parametric models. But it does not address smile dynamics (use SABR), does not give you a compact representation for storage (SVI's 5 numbers per slice is hard to beat), and requires more infrastructure. Computing Greeks like delta and vega from a SANOS surface requires finite-difference bumping on the grid. Next generation of surface fitting, not a replacement for understanding the parametric models.
Relevance to Crypto
Crypto options markets have several features that favor non-parametric approaches:
- Sparse, irregular quotes: Not every strike has a quote at every expiry. SANOS handles irregular grids natively.
- Wide bid-ask spreads: Especially on smaller underlyings. SANOS uses spreads as constraints rather than discarding them.
- Structural events: Token unlocks, protocol upgrades, and airdrops create localized vol features that SVI cannot capture. These show up as ATM vol bumps at specific expiries.
- Rapid regime changes: The surface can change shape faster than a parametric model can be re-estimated. SANOS's LP fitting is fast enough to keep up. The term structure can shift dramatically intraday and SANOS adapts without manual intervention.
SANOS and local vol extraction
Because SANOS directly parameterizes local vol nodes, extracting a full Dupire local vol surface is trivial -- it is the fitted grid. This makes SANOS especially useful for pricing path-dependent exotics (barriers, cliquets) where local vol dynamics determine the payoff. Parametric models like Black-Scholes-based SVI require a separate extraction step that can introduce numerical artifacts.
Equation Explorer
Convert between implied vol, total variance, log-moneyness, and option prices.
Equation Explorer
💡 Tip: Try answering each question yourself before revealing the answer.
Building mathematical intuition
Learn SANOS from scratchInteractive lesson · no prerequisitesThis lesson explains SANOS as a node-grid surface rather than a fixed formula, then shows how the optimization and no-arbitrage constraints work together.
See also:
- SVI Parameterization -- The industry-standard parametric model
- SSVI (Surface SVI) -- Calendar-free parametric surfaces
- SABR Model -- Dynamic smile model
- Local Volatility -- Dupire's local vol surface
- Interpolation Methods -- All methods compared
- How Surfaces Are Built -- The full pipeline