Option Pricing in ARCH-type Models: with Detailed Proofs

ARCH-models have become popular for modelling nancial time series. They seem, at rst, however, to be incompatible with the option pricing approach of Black, Scholes, Merton et al., because they are discrete-time models and posess too much variability. We show that completeness of the market holds for a broad class of ARCH-type models de ned in a suitable continuous-time fashion. As an example we focus on the GARCH(1,1)-M model and obtain, through our method, the same pricing formula as Duan (1995), who applied equilibrium-type arguments. This is an extended version of Kallsen and Taqqu (1995). It includes additional comments and detailed proofs. It also includes a chapter concerning \the equality of ltrations" which deals with the following issue. Trading strategies should be based on information ( ltration) that traders posess. In practice, however, one typically assumes that thay are predictable with respect to the ltration generated by a Brownian motion which serves as a background source of randomness. It is thus necessary to show that the two ltrations coincide. We do this here. Acknowledgements. The rst author was supported by a grant of the Deutscher Akademischer Austauschdienst for research at Boston University. He would like to thank the university for its hospitality. The second author was partially supported by the NSF grant DMS-9404093 at Boston University. We would like to thank Ofer Zeitouni for indicating a simpli cation to the original proof of Lemma 2.