Quintic Polynomial Model

SVI is the industry standard for fitting a vol smile -- 5 parameters, one slice at a time. But SVI bakes in a specific shape assumption: the smile is always a translated, scaled hyperbola. When the market does something SVI cannot produce, the fit degrades. The Quintic Polynomial model (Gauthier & Possamai, 2023) drops the shape assumption entirely. It fits total implied variance as a polynomial in log-moneyness -- a 4th or 5th degree polynomial with 5 or 6 coefficients. It can fit any smile shape the market produces, including ones SVI structurally misses.

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SVI without the shape constraint

Same parameter count as SVI. Same one-slice-at-a-time fitting. But where SVI forces a hyperbolic shape, the polynomial lets the data decide. The tradeoff: you lose SVI's built-in wing behavior and need explicit constraints for arbitrage freedom. Skew and curvature are independent knobs.

See It in Action

Drag the sliders to explore how each coefficient shapes the smile. Try the "Double bump" preset for a shape SVI cannot produce.

Quintic Polynomial Smile Explorer

Parabolic shape typical of SVI. Symmetric wings with moderate skew.

ATM level0.045

Sets the overall vol level

Skew-0.015

Tilts the smile left (put skew) or right

Curvature0.080

How wide the smile opens

Asymmetry-0.010

Makes one wing steeper than the other

Wing steepness0.020

Controls how fast wings rise. High values = steep tails.

Try "Double bump" and toggle "Show SVI reference" to see a shape the polynomial can produce that SVI structurally cannot.

How It Works

1. Total variance as a polynomial

For a given expiry $T$, total implied variance $w(k) = \sigma^2(k) \cdot T$ is modeled as a polynomial in log-moneyness $k = \log(K/F)$:

$$w(k) = c_0 + c_1 k + c_2 k^2 + c_3 k^3 + c_4 k^4$$

Each coefficient has a direct trader interpretation:

Coefficient

Trader name

What it controls

c0

ATM level

Overall vol level. Higher c0 = higher ATM implied vol.

c1

Skew

Tilts the smile. Negative = put skew (left wing higher).

c2

Curvature

How wide the smile opens. Controls butterfly richness.

c3

Asymmetry

Makes one wing steeper than the other. Odd-power effect.

c4

Wing steepness

Controls how fast wings rise at extreme strikes.

2. Arbitrage constraints are simple bounds

For the polynomial to be arbitrage-free (positive variance, convex call prices), the constraints reduce to inequalities on the coefficients. No need for complex numerical checks -- just bound the coefficients during fitting.

3. Fitting is fast

Fitting a polynomial to market data is a least-squares problem, solvable in microseconds. The fit is weighted toward ATM strikes where liquidity is highest. Add the coefficient bounds as linear constraints and you have a small QP (quadratic program) -- faster and more robust than SVI's nonlinear optimization.

ℹ️

Higher-degree polynomials oscillate in the wings

Degree 6 or 7 polynomials oscillate in the wings (Runge's phenomenon). Degree 4-5 has enough flexibility to capture real smile shapes without creating artifacts beyond the last liquid strike. For deep OTM wing behavior, you need explicit extrapolation rules.

Quintic vs. SVI

Feature

SVI

Quintic Polynomial

Parameters per slice

5

5 (quartic) or 6 (quintic)

Shape assumption

Hyperbolic (baked in)

None

Fit quality

Good for typical smiles

Can fit any shape

Wing extrapolation

Linear (bounded)

Polynomial (diverges)

Arbitrage constraints

Complex nonlinear

Simple coefficient bounds

Fitting method

Nonlinear optimization

Least-squares / QP

Industry adoption

Decades of use

New (2023)

SSVI-like surface version

Yes (SSVI)

Research stage

Relevance to Crypto

Crypto smiles are often asymmetric in ways that SVI struggles with -- sharp put skew from liquidation cascades, unusual call-side bumps from airdrop optionality, or "kinked" smiles around popular strike prices with concentrated open interest. The polynomial model fits these shapes without forcing a hyperbolic structure. Delta and vega computed from the polynomial smile are smooth by construction. The main limitation: crypto options have sparse strikes, and polynomials can misbehave between data points if not constrained carefully.

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SVI's simplicity without its shape bias

Fits smiles that SVI structurally cannot produce. The cost: you lose SVI's well-behaved wing extrapolation and must handle arbitrage constraints explicitly. Multi-expiry surfaces need separate term structure constraints. Best for markets where the smile is unusual or SVI's fit residuals are too large.

Equation Explorer

Convert between implied vol, total variance, log-moneyness, and option prices.

Equation Explorer

w = σ² × Ttotal variance = IV² × time

Implied Vol (σ)

%

The implied volatility

Time to Expiry

days

Calendar days to expiration

Total Variance (w)

0.022225

Annualized Variance (σ²)

0.2704

Round-trip IV

52.00%

Total variance is what SVI and other models fit. It scales with time, so a 50% vol for 30 days has less total variance than 50% vol for 90 days.

Test your understanding before moving on.

Q: Why can the Quintic Polynomial fit smile shapes that SVI cannot?

Q: What is the main disadvantage of using a polynomial for wing extrapolation?

Q: You are fitting a 3-day expiry on a crypto asset with only 6 liquid strikes. Would you prefer SVI or the polynomial?

💡 Tip: Try answering each question yourself before revealing the answer.

Building mathematical intuition

Learn quintic from scratchInteractive lesson · no prerequisites

This lesson explains why a polynomial fit buys you extra smile flexibility, how the total-variance polynomial works, and why stronger arbitrage checks matter as soon as the shape is allowed to move more freely.

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See also:

• SVI Parameterization -- The industry-standard parametric model this extends
• SSVI (Surface SVI) -- Calendar-consistent surface extension of SVI
• SANOS (Non-Parametric Surfaces) -- Full non-parametric approach with LP fitting
• Neural SDE / Deep Hedging -- Data-driven approach that learns dynamics end-to-end
• Interpolation Methods -- All methods compared