GARCH options via local risk minimization

We apply the quadratic hedging scheme developed by Föllmer, Schweizer, and Sondermann to European contingent products whose underlying asset is modeled using a GARCH process. The main contributions of this work consist of showing 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 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 and multinomial innovations and we provide specific conditions under which those martingale measures are appropriate in the context of quadratic hedging. In the Gaussian case, those conditions have to do with the finiteness of the kurtosis and, for multinomial innovations, an inequality between the trend terms of the prices and of the volatility equations needs to be satisfied.