  We apply a quadratic hedging scheme developed by Foellmer, Schweizer, and
Sondermann to European contingent products whose underlying asset is modeled
using a GARCH process and show that local risk-minimizing strategies with
respect to the physical measure do exist, even though an associated minimal
martingale measure is only available in the presence of bounded innovations.
More importantly, since those local risk-minimizing strategies are in general
convoluted and difficult to evaluate, we introduce Girsanov-like risk-neutral
measures for the log-prices that yield more tractable and useful results.
Regarding this subject, we focus on GARCH time series models with Gaussian
innovations and we provide specific sufficient conditions that have to do with
the finiteness of the kurtosis, under which those martingale measures are
appropriate in the context of quadratic hedging. When this equivalent
martingale measure is adapted to the price representation we are able to
recover out of it the classical pricing formulas of Duan and Heston-Nandi, as
well as hedging schemes that improve the performance of those proposed in the
literature.
