Sufficient conditions for no-arbitrage for the f4 implied variance parametrization

Establish sufficient parameter conditions under which the parametric total implied variance function w(x) = p4 (tanh(p1 x) + p2) (p3 + x) + p5 is free of butterfly arbitrage and calendar arbitrage, analogous to the known sufficient conditions for the SVI (Stochastic Volatility Inspired) parametrization, thereby providing guarantees for absence of arbitrage when using this f4 slice parametrization in practice.

Background

The paper uses symbolic regression to discover alternative parametric representations of total implied variance as a function of log-moneyness. Among the discovered forms, the expression f4(x) = p4 (tanh(p1 x) + p2) (p3 + x) + p5 achieves lower complexity and lower loss than the SVI parametrization on a large SPX dataset.

For SVI, sufficient conditions ensuring absence of butterfly and calendar arbitrage are known in the literature. In contrast, for the newly discovered f4 parametrization, the authors point out that such an analysis has not yet been performed, leaving theoretical no-arbitrage guarantees unresolved for f4 despite its empirical performance.

References

As a caveat, we note that an analysis of sufficient conditions for the absence of butterfly- and calendar-arbitrage, as performed for SVI in , is left open for expression $f_4$.

— Discovering parametrizations of implied volatility with symbolic regression  (2603.21892 - Keller-Ressel et al., 23 Mar 2026) in Section 3, Subsection "A Closer Look at Expression f_4"