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Parametric Smile Models

These models describe the shape of the smile using a formula with a few knobs. Give them market data, they find the best-fitting parameters, and you get a smooth smile evaluable at any strike.

The workhorse approach for crypto and equity options.

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Parametric = shape description, not explanation

The model decides the general shape (parabola-like for SVI, polynomial for Quintic). The parameters control the specifics: how steep, how curved, where the minimum is.

At a Glance​

Model
Parameters
Key feature
Used in crypto?
<a href="/docs/reference/svi">SVI</a>
5
Industry standard. Bounded wings. Simple constraints.
Yes (primary)
<a href="/docs/reference/orc-wing">ORC Wing</a>
5
SVI reparameterized with trader-friendly knobs (ATM vol, wing slopes).
Some desks
<a href="/docs/reference/ssvi">SSVI</a>
3 + curve
SVI extended to the full surface. Calendar-free by construction.
Growing
<a href="/docs/reference/quintic-polynomial">Quintic Polynomial</a>
5-6
No shape assumption. Can fit any smile. Recent (2023).
Experimental

What they share​

All four models do the same job: take a set of market-observed option prices for a single expiry and produce a smooth implied-vol curve you can evaluate at any strike. They differ in what shape they assume and what guarantees they offer.

Model
Shape assumption
Arbitrage constraints
Surface version?
Fitting speed
SVI
Hyperbola (linear wings)
Manual checks needed
No (per-slice)
Fast
ORC Wing
Same as SVI (reparameterized)
Same as SVI
No (per-slice)
Fast
SSVI
Hyperbola + power-law term structure
Calendar-free by construction
Yes
Fast
Quintic
None (5th-degree polynomial)
Enforced via coefficient constraints
No (per-slice)
Very fast

How they relate to each other​

SVI is the starting point. ORC Wing is the same model with different knobs -- a reparameterization that replaces abstract SVI parameters with ATM vol, slope, and curvature that a trader can reason about. SSVI takes SVI one level up: it parameterizes how SVI parameters change across expiries, so the full surface is consistent by construction. Quintic Polynomial is the alternative track entirely. It drops the hyperbolic assumption and fits an arbitrary 5th-degree polynomial, which means it can match smile shapes that SVI cannot. The trade-off is that it has no built-in surface extension and needs separate arbitrage checks.

How to choose​

  • Standard crypto/equity surface fitting? → SVI. It's the default for a reason.
  • Need calendar arbitrage freedom? → SSVI. Guarantees consistency across expiries.
  • Want to edit the smile by hand? → ORC Wing. Same math as SVI but with trader-friendly knobs.
  • Smile has unusual features SVI can't capture? → Quintic Polynomial. Maximum flexibility.

Models in this section:

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