SSVI from zero

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Section 1

From SVI to SSVI

SVI fits one volatility smile at a single expiry. It does that well -- five parameters, a clean shape. But a volatility surface has many expiries stacked together. Fitting each slice independently creates a problem.

When you fit each slice with its own SVI parameters, nothing guarantees that total variance increases from one expiry to the next at every strike. If it doesn't, you have a calendar spread arbitrage -- a free-money trade where you sell the short-dated option and buy the long-dated one.

SSVI solves this by building the whole surface from a single parameterization. Instead of 5 parameters per slice, the entire surface is driven by:

The two key pieces

θ(t) = ATM total variance at time t

φ(θ) = ATM skew function (controls smile steepness)

θ(t) is observed directly from ATM option prices. φ(θ) encodes how the smile shape varies with maturity. Together with a single skew parameter ρ, they define the entire surface.

Analogy

Think of per-slice SVI like giving each floor of a building its own architect. Each floor may look great, but the staircases between floors might not line up. SSVI hires one architect for the whole building -- the individual floors are slightly less customized, but everything connects.

Section 2

The SSVI parameterization

One formula. Three controls. Move the sliders below and watch the smile reshape itself.

SSVI total variance

w(k, θ) = θ/2 · (1 + ρφ(θ)k + √((φ(θ)k + ρ)² + (1 − ρ²)))

k is log-moneyness (ln(K/F)). At k=0, you are at-the-money. Negative k is in-the-money calls (OTM puts).

Play with each parameter to build intuition for what it controls:

θ0.04

ρ-0.30

φ1.50

θ shifts the overall level -- higher ATM variance means the entire smile moves up. ρ tilts the smile -- negative ρ creates the put skew traders expect. φ controls how wide the wings are.

Section 3

Calendar spread arbitrage

The whole point of SSVI: calendar arbitrage freedom by construction. But only if φ is chosen correctly.

Calendar spread arbitrage means total variance at some strike decreases from a shorter expiry to a longer one. That cannot happen in a fair market -- it would give you free money.

Below, compare a bad choice of φ (constant, ignores maturity) with the good power-law form. Drag the slider on the left panel and watch the violation indicator.

Constant phi = 3.0No violations

phi3.0

Constant phi ignores maturity. The smile stays equally steep at every expiry, which can make total variance decrease as you move to longer maturities at some strikes.

Power-law phi(theta)No violations

The power-law phi decays as theta grows, flattening the smile at longer maturities. Total variance is monotonically increasing in time at every strike -- no calendar arbitrage.

The constant φ keeps the smile equally steep at every expiry. As θ grows, the steep smile pushes total variance at the wings higher than it should be for short maturities but not high enough for long maturities -- creating crossings.

The power-law φ decays as θ grows, naturally flattening the smile at longer maturities. This guarantees w(k, θ) is monotonically increasing in θ at every k.

Section 4

The power-law form

Two parameters control the entire surface. That is the payoff for all the constraints SSVI imposes.

Power-law phi

φ(θ) = η / (θ^γ (1 + θ)^1−γ)

η > 0 controls overall smile amplitude. γ ∈ [0, 1] controls how fast the smile flattens with maturity. Calendar freedom requires η(1 + |ρ|) < 2.

Move the sliders below. Watch the heatmap change -- the x-axis is log-moneyness, the y-axis is expiry, and color is implied volatility.

η0.50

γ0.50

ρ-0.30

SSVI implied volatility surface

Expiry (years)

1.00.530.05

-0.800.8

Log-moneyness (k)

IV

19%51%

η scales the smile amplitude everywhere. Crank it up and the wings widen across all expiries. γ changes how quickly the smile flattens with maturity. Low γ means long-dated smiles stay steep; high γ makes them flatten fast.

Three parameters (ρ, η, γ) plus the observed ATM variance curve θ(t). That is the entire surface. Compare that to 25+ parameters for five slices of per-slice SVI.

Where to go next:

SSVI reference -- the full formula, fitting tips, and comparison table

SVI parameterization -- the per-slice model that SSVI extends

How surfaces are built -- the production pipeline