Replace five SVI parameters per slice with two surface parameters plus an ATM-variance curve. Gatheral-Jacquier (2014): SSVI is automatically calendar-arbitrage-free under a mild condition on the curvature function $\phi$.
SVI fits each maturity independently with 5 parameters. With $M$ maturities you have $5M$ parameters, and there's nothing forcing consistency across maturities — you can fit two slices arbitrage-free individually and still have a calendar arb between them. SSVI (Gatheral & Jacquier 2014) is the surface-level extension: parameterize the *whole* surface by an ATM-total-variance curve $\theta(T)$ plus two surface-wide parameters $\eta, \gamma$ that determine the smile shape at every maturity. The total parameter count drops to $2 + M$ (one ATM-variance per maturity for $\theta(T)$, plus $\eta, \gamma$ for the smile shape), and the surface is calendar-arb-free *by construction* under a simple condition on $\phi$.
β Intro Β· expand
Try first (productive failure)
Before the worked example: spend 60 seconds taking your best shot at this.
A guess is fine β being briefly wrong about a problem makes the explanation
land harder when you read it. This appears once per tutorial; skip
if you already know the trick.
60s
β Try first Β· expand
Worked example
An SSVI fit has $\eta = 1.0$, $\gamma = 0.5$, $\rho = -0.4$, and the ATM total variance at $T = 1$ year is $\theta_T = 0.04$. (a) Compute the curvature function value $\phi(\theta_T)$. (b) Compute the total variance $w(k=0.5, T)$ at log-moneyness $k = 0.5$. (c) Convert to implied vol.
β Worked example Β· expand
Practice 1 of 3Type a fraction, decimal, or expression β mathjs parses it.
β Practice Β· expand
Reflection
SSVI factorises the surface into 'ATM term-structure' (the $\theta(T)$ curve) and 'smile shape' (the two parameters $\eta, \gamma$ and the correlation $\rho$). Why is this factorisation natural for vol surfaces, and where does it break down in practice?