The TSLL Model: A Simple, Parametric Fit for the Implied Volatilities of VIX Options Inspired by the Shifted Lognormal Model

The standard shifted lognormal model, defined by just two parameters, provides a remarkably good fit to the market implied volatilities of VIX options. Inspired by an analytic approximation derived by Lee and Wang, we propose a simple, intuitive extension that provides better empirical fits while retaining analytical tractability. In essence, by introducing a third parameter that controls the tilt of the surface beyond the shifted lognormal baseline we can better control the behavior of the fit for large strikes. We call this extended model TSLL: tilted and shifted lognormal-like. Finally, we suggest an alternative parameterization in terms of the ATM volatility, volatility floor and tilt parameter that is better suited to help set cutoffs and to rule out arbitrage violations.