Heat kernel expansion for a family of stochastic volatility models : delta-geometry

Abstract

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.