Effective parameters for stochastic volatility models

This paper tackles the issue of approximated formula for stochastic models with time dependent model parameters, using an averaging principle. The objective is to find a similar model but with constant parameters that is the closest to our initial process, along the same lines as the proof by Gyöngy (1986) for general stochastic processes. We extend previous results found by Piterbarg (2005) for the particular case of the SABR model [Hagan et al. (2002)]. The resulting formula can be evaluated very quickly, solving the implied Riccati equations. We compare the approximation with exact solution of the corresponding partial differential equation using an ADI method. Numerical results show that the approximation works well for short term maturities.