Price expansion formulas for model combining local and stochastic volatility

This paper consists in introducing an option price expansion for model combining local and stochastic volatility with tight error estimates. The local volatility part is considered as general but has to satisfy some growth and boundedness assumptions. For the stochastic part, we choose a square root process, which is usually used for modelling the behaviour of the variance process. In the particular case of Call options, we also provide expansions of the Black-Scholes implied volatility which allow to obtain very simple and rapid formulas in comparison to the Monte Carlo approach while maintaining a very competitive accuracy.