  We consider an asset whose risk-neutral dynamics are described by a general
class of local-stochastic volatility models and derive a family of asymptotic
expansions for European-style option prices and implied volatilities. Our
implied volatility expansions are explicit; they do not require any special
functions nor do they require numerical integration. To illustrate the accuracy
and versatility of our method, we implement it under five different model
dynamics: CEV local volatility, quadratic local volatility, Heston stochastic
volatility, $3/2$ stochastic volatility, and SABR local-stochastic volatility.
