Jump-Wings¶

Overview¶

The jump-wings parametrization (Gatheral 2004) re-expresses SVI in terms of financially interpretable quantities: ATM variance, ATM skew, wing slopes, and minimum variance. Same 5 degrees of freedom as raw SVI, but every parameter has a direct market meaning.

Model¶

Jump-wings parameters map to raw SVI \((a, b, \rho, m, \sigma)\) via a bijection:

\[b = \frac{p_t + c_t}{2}, \quad \rho = 1 - \frac{p_t}{b}, \quad \beta = \rho - \frac{2\psi_t\sqrt{T}}{b}\]

\[\alpha = \mathrm{sgn}(\beta)\sqrt{\frac{1}{\beta^2} - 1}, \quad m = \frac{(v_t - \tilde{v}_t)\,T}{b\left[-\rho + \mathrm{sgn}(\alpha)\sqrt{1 + \alpha^2} - \alpha\sqrt{1 - \rho^2}\right]}\]

\[\sigma = |\alpha \cdot m|, \quad a = \tilde{v}_t \cdot T - b\,\sigma\sqrt{1 - \rho^2}\]

Evaluation then proceeds through the raw SVI formula.

Parameters¶

Parameter	Meaning	Constraint
\(v_t\)	ATM variance \(\sigma_{\mathrm{ATM}}^2\)	\(v_t > 0\)
\(\psi_t\)	ATM skew	bounded
\(p_t\)	left (put) wing slope	\(p_t \geq 0\)
\(c_t\)	right (call) wing slope	\(c_t \geq 0\)
\(\tilde{v}_t\)	minimum implied variance	\(\tilde{v}_t > 0\)

Usage¶

from pysvi import get_model, calibrate_slice

model = get_model("jw")  # or "jumpwings"
T = float(df_slice["maturity"].iloc[0])
params = calibrate_slice(df_slice, model, T=T)
# params: {'v_t', 'psi_t', 'p_t', 'c_t', 'v_tilde_t', 'T', 'forward'}

Arbitrage behaviour¶

Soft parameter bounds by default (QUASI); NO_BUTTERFLY and NO_CALENDAR penalties available via the raw-SVI conversion — see Arbitrage freeness.

References¶

Gatheral, J. (2004). “A parsimonious arbitrage-free implied volatility parameterization with application to the valuation of volatility derivatives.”