Explicit implied volatilities for multifactor local-stochastic volatility models

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. We also establish rigorous error estimates for these quantities. 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 four different model dynamics: constant elasticity of variance local volatility, Heston stochastic volatility, three-halves stochastic volatility, and SABR local-stochastic volatility.