Implied volatility expansions allow calibration of sophisticated volatility models. They provide an
accurate t and parametrization of implied volatility surfaces that is consistent with empirical ob-
servations. Fine-grained higher order expansions o er a better t but pose the challenge of nding a
robust, stable and computationally tractable calibration procedure due to a large number of market
parameters and nonlinearities. We propose calibration schemes for second order expansions that take
advantage of the model's structure via exact parameter reductions and recoveries, reuse and scaling
between expansion orders where permitted by the model asymptotic regime and numerical iteration
over bounded signi cant parameters. We perform a numerical analysis over 12 years of real S&P 500
index options data for both multiscale stochastic and general local-stochastic volatility models. Our
methods are validated empirically by obtaining stable market parameters that meet the qualitative
and numerical constraints imposed by their functional forms and model asymptotic assumptions.