Forward implied volatility expansion in time-dependent local volatility models

We introduce an analytical approximation to efficiently price forward start options on equity in time-dependent local volatility models as the forward start date, the maturity or the volatility coefficient are small. We use a conditional expectation argument to represent the price as an expectation of a Black-Scholes formula computed with a stochastic implied volatility depending on the value of the equity at the forward date. Then we perform a volatility expansion to derive an analytical approximation of the forward implied volatility with a precise error estimate. We also illustrate the accuracy of the formula with some numerical experiments. Some results and tools of this work were presented at the conference SMAI 2013 in the mini-symposium ”Méthodes asymptotiques en finance”