SVI Parameterization

info

This page covers the SVI model in depth. For context on how it fits into the vol surface pipeline, see How Surfaces Are Built. For a comparison with other methods, see Interpolation Methods.

SVI (Stochastic Volatility Inspired) is the industry standard for fitting volatility smiles in crypto and equity options. It uses 5 parameters to describe the shape of the smile at a single expiry. The name comes from the fact that its functional form can be derived from a simplified stochastic volatility model.

Explore the Parameters

Adjust each parameter below to see how it changes the smile. Use the presets to jump between common configurations.

SVI Parameter Explorer

Typical equity/crypto smile. Put wing elevated.

a (level)0.040

Shifts the entire smile up or down

b (slope)0.250

How steep the wings are

ρ (skew)-0.40

Negative = put skew, positive = call skew

m (shift)0.00

Where the smile minimum sits

σ (curvature)0.200

Small = sharp V, large = smooth U

ATM IV

104.6%

Put wing slope

0.350

Call wing slope

0.150

What each parameter does

• a (level): Shifts the entire smile up or down. Higher $a$ = higher overall IV. Think of it as the "baseline" variance.
• b (slope): Controls how steep the wings are. Higher $b$ = steeper wings = more expensive OTM options.
• $\rho$ (skew): Tilts the smile. Negative $\rho$ = put skew (normal). Positive $\rho$ = call skew (rare). Zero = symmetric.
• m (shift): Moves the minimum of the smile left or right. Usually near zero (minimum at ATM).
• $\sigma$ (curvature): Controls the roundness of the smile bottom. Small $\sigma$ = sharp V-shape. Large $\sigma$ = smooth U-shape.

Wing behavior

As you move far from ATM, the smile approaches straight lines. The slopes are:

• Put wing (left): slope = $b(1 - \rho)$
• Call wing (right): slope = $b(1 + \rho)$

With typical put skew ($\rho < 0$), the put wing is steeper than the call wing. This bounded, linear wing behavior is one of SVI's key advantages: it never extrapolates to absurd values.

Fitting to Market Data

Given a set of observed IV values at different strikes (for a single expiry), SVI finds the 5 parameters that best reproduce them.

The process:

1. Collect all IV observations at the target expiry (typically 5-15 data points).
2. Convert each to total variance: observed IV squared times time to expiry.
3. Run a weighted least-squares optimizer to find the parameters that minimize the gap between the SVI curve and the observations.
4. Weight by confidence: ATM options get more weight (more liquid), tight bid-ask spreads get more weight (more reliable), deep OTM gets less weight.
5. Enforce arbitrage constraints during optimization (see below).

Speed: A single expiry fits in under 10 milliseconds. The entire surface (all expiries) rebuilds in real time as quotes update.

Arbitrage Constraints

SVI can be constrained to prevent arbitrage. These constraints are simple inequalities on the parameters.

Butterfly constraint (no negative local variance):

The smile must be convex enough that no butterfly spread is free money. This requires:

$b(1 + |\rho|) \leq \frac{4}{T}$

Non-negative variance at minimum:

The smile minimum must be above zero:

$a + b\sigma\sqrt{1 - \rho^2} \geq 0$

Calendar constraint (across expiries):

Total variance must increase with maturity at every strike. This is enforced by checking $w_1(k) \leq w_2(k)$ for all $k$ between consecutive expiry slices.

Variants

SVI has two important variants, each with its own page:

ORC Wing (Jump-Wing) reparameterizes SVI using trader-friendly quantities: ATM variance, ATM skew, put wing slope, call wing slope, and minimum variance. Same smile, different knobs. Useful for editing the smile by hand.

SSVI (Surface SVI) extends SVI to the full surface, guaranteeing calendar arbitrage freedom by construction. Instead of fitting each expiry independently, SSVI links them through a shared skew parameter and a smile-steepness function. Fewer degrees of freedom, but no cross-expiry fixups needed.

Equation Explorer

SVI works in total variance and log-moneyness. Use this to convert between IV, total variance, and strike representations.

Equation Explorer

k = ln(K / F)log-moneyness = ln(strike / forward)

Strike (K)

$

The option strike price

Forward (F)

$

Forward price (≈ spot for short-dated)

Log-Moneyness (k)

-0.0513

Moneyness (K/F)

0.9500

Type

-5.0% OTM Put

Log-moneyness is the x-axis used by SVI and most parametric models. k = 0 is ATM. Negative k = OTM put side. Positive k = OTM call side.

Why SVI?

• 5 parameters is enough to fit nearly any observed smile. Fewer parameters than splines, more flexible than 3-parameter models.
• Bounded wings prevent runaway extrapolation. The worst SVI can do is extrapolate to a finite slope.
• Simple constraints guarantee no butterfly arbitrage. Just two inequalities.
• Fast fitting means the surface can update in real time as the market moves.
• Industry standard for crypto (Deribit, Hypercall) and most equity vol desks.

Building mathematical intuition

Learn SVI from scratchInteractive lesson · 5 sections · no prerequisites

The interactive lesson above covers SVI from first principles: what SVI parameterizes and why, each of the 5 parameters in isolation, the variance-to-vol conversion, no-arbitrage constraints, and hands-on calibration to market data.

Open source implementations

Repo	Why inspect it
SVI-Vol-Surface	Python SVI calibration with visualization
QuantLib	SVI parameterization in C++

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SVI variants: ORC Wing (Jump-Wing) | SSVI (Surface SVI)

See also: How Surfaces Are Built | Interpolation Methods | SABR Model | Skew