The Forward Smile in Local-Stochastic Volatility Models

We introduce an asymptotic expansion for forward start options in a multi-factor local-stochastic volatility model. We derive explicit approximation formulas for the so-called forward implied volatility which can be useful to price complex path-dependent options, as cliquets. The expansion involves only polynomials and can be computed without the need for numerical procedures or special functions. Recent results on the exploding behaviour of the forward smile in the Heston model are confirmed and generalized to a wider class of local-stochastic volatility models. We illustrate the effectiveness of the technique through some numerical tests. Mathematica codes are freely available on the authors' website.