Pricing Asian Options in Affine Garch models

We derive recursive relationships for the m.g.f. of the geometric average of the underlying within some affine Garch models (namely Heston and Nandi 2000, Christoffersen, Heston and Jacobs 2006, Bellini and Mercuri, 2007, Mercuri, 2008) used for the semi-analytical valuation of geometric Asian options. Similar relationships are obtained for low order moments of the distribution of the arithmetic average of the underlying in the same models, that are used for approximate evaluation of arithmetic Asian options (Turnbull and Wakeman 1991). In both cases the accuracy of the semi-analytical procedure is assessed by means of a comparison with Montecarlo prices.