'Effective' Parameters for Stochastic Volatility Models

This paper tackles the issue of approximated formula for stochastic model with time dependent model parameters, using an averaging principle. The idea lies in finding a similar model but with constant parameters that is the closest to our initial process, along the same lines as results proven by Gyouml;ngy (1986) for general stochastic processes. We extend previous results found by Piterbarg (2005) for the particular case of SABR model (Hagan (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