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SVI and SSVI implied-volatility parameterizations (Gatheral 2004; Gatheral & Jacquier 2014).
SVI (raw form) parameterizes one expiry’s total variance in
log-moneyness k = ln(K/F):
w(k) = a + b [ rho (k - m) + sqrt((k - m)^2 + sigma^2) ]five parameters per smile: level a, wing slope b, skew rho,
shift m, ATM curvature sigma. Wings are asymptotically linear
with slopes b(1 - rho) (put side) and b(1 + rho) (call side).
SSVI parameterizes the whole surface from the ATM total-variance
term structure theta_t and three global parameters (rho, eta, gamma) through the power-law curvature
phi(theta) = eta / (theta^gamma (1 + theta)^(1-gamma)):
w(k, t) = theta_t/2 [ 1 + rho phi k + sqrt((phi k + rho)^2 + 1 - rho^2) ]Both calibrate by Levenberg-Marquardt
(core::optimization) in transformed
parameter spaces, the same pattern as
heston::calibrate. Butterfly
arbitrage is checked through the Gatheral-Jacquier g(k) density
condition (SVI) and the power-law sufficient conditions (SSVI), and
fitted smiles sample into the pricing
VolSurface via
Ssvi::to_vol_surface.