Natural SVI¶

Overview¶

The natural SVI parametrization (Gatheral & Jacquier 2014). Same 5 degrees of freedom as raw SVI, connected by an explicit bijection, but the parameters map more directly to ATM level, skew, and curvature — often better behaved in calibration, and useful as an initialisation coordinate system for raw SVI.

Model¶

\[w(k) = \Delta + \frac{\omega}{2}\left[1 + \zeta\rho\,(k - \mu) + \sqrt{\left(\zeta(k - \mu) + \rho\right)^2 + 1 - \rho^2}\right]\]

The bijection to raw SVI \((a, b, \rho, m, \sigma)\):

\[a = \Delta + \frac{\omega(1-\rho^2)}{2}, \qquad b = \frac{\omega\zeta}{2}, \qquad m = \mu - \frac{\rho}{\zeta}, \qquad \sigma = \frac{\sqrt{1-\rho^2}}{\zeta}\]

and its inverse:

\[\zeta = \frac{\sqrt{1-\rho^2}}{\sigma}, \qquad \omega = \frac{2b\sigma}{\sqrt{1-\rho^2}}, \qquad \mu = m + \frac{\rho\sigma}{\sqrt{1-\rho^2}}, \qquad \Delta = a - b\sigma\sqrt{1-\rho^2}\]

Both directions are exposed as natural_to_raw and raw_to_natural.

Parameters¶

Parameter	Meaning	Constraint
\(\Delta\)	vertical variance shift	unconstrained
\(\mu\)	log-moneyness translation	unconstrained
\(\rho\)	skew (correlation)	\(\|\rho\| < 1\)
\(\omega\)	overall variance scale	\(\omega > 0\)
\(\zeta\)	curvature / smile-width scale	\(\zeta > 0\)

Usage¶

from pysvi import get_model, calibrate_slice

model = get_model("natural")  # or "nsvi"
params = calibrate_slice(df_slice, model)
# params: {'delta', 'mu', 'rho', 'omega', 'zeta', 'forward'}

No extra keyword arguments are required. Converting a fit between conventions:

from pysvi import natural_to_raw, raw_to_natural

raw = natural_to_raw(params["delta"], params["mu"], params["rho"],
                     params["omega"], params["zeta"])

Arbitrage behaviour¶

Identical to raw SVI (the curves are the same family): soft parameter bounds by default (QUASI); NO_BUTTERFLY and NO_CALENDAR penalties supported via the raw-SVI conversion — see Arbitrage freeness. Wing slopes for the Lee-bound diagnostics are the raw-SVI asymptotes \(b(1 \mp \rho)\).

References¶

Gatheral, J., Jacquier, A. (2014). “Arbitrage-free SVI volatility surfaces.” Quantitative Finance 14(1), section 3.