In this paper, we study a family of stochastic volatility processes ; this family features a mean reversion term for the volatility and a double CEV-like exponent that generalizes SABR and Heston's models. We derive approximated closed form formulas for the digital prices, the local and implied volatilities. Our formulas are efficient for small maturities. Our method is based on differential geometry, especially small time diffusions on riemanian spaces. This geometrical point of view can be extended to other processes, and is very accurate to produce variate smiles for small maturities and small moneyness.

We derive a direct link between local and implied volatilities in the form of a quasilinear degenerate parabolic partial differential equation. Using this equation we establish closed-form asymptotic formulae for the implied volatility near expiry as well as for deep in-and out-of-the-money options. This in turn leads us to propose a new formulation near expiry of the calibration problem for the local volatility model, which we show to be well posed.In the Black-Scholes-Merton model [4,24], it is assumed that the price of a non-dividend paying stock S t follows the lognormal stochastic differential equationwhere t is time, µ and σ are constants and W t is a standard Brownian motion. The parameter σ is called the volatility of the stock S t . It is well-known that the price C(S t , t; K, T ) of a European call option written on S t with strike K and maturity T satisfies the linear parabolic partial differential equationwhere r is the risk-free short-term interest rate. Such options are commonly traded on markets, however σ is not directly observable. Therefore it is common practice to start from the observed prices and invert the closed-form solution to (2) in order to find that constant σ -called implied volatility-for which the solution to (2) agrees with the market price at today's value of the stock. It is widely observed that calls having different strikes and otherwise identical have different implied volatilities. This phenomenon, usually referred to as the smile effect, clearly violates the Black-Scholes-Merton model, since in this framework a constant σ is supposed to determine the dynamics of the underlying stock S t through (1) regardless of options, strikes and maturities.To overcome this difficulty, the model has to be extended. One widely used approach is to consider that the volatility also follows a stochastic diffusion process. Another case is when the volatility σ is not a constant any more but rather a (deterministic) function of the underlying asset and the time. Actually this can be seen as a particular case of the previous approach. These types of models are called local volatility models. For them, the dynamics of the underlying asset is governed by the stochastic differential equationThere are two problems which are relevant in practice. Firstly one needs to compute accurately the implied volatilities of option prices-the pricing problem. If one follows a traditional approach (e.g. solve the PDE corresponding to each model and then invert Black-Scholes formula), this is known to be