Option Pricing in ARCH-type Models: with Detailed Proofs

ARCH-models have become popular for modelling financial time series. The seem, at first, 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 defined 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 filtrations which deals with the following issue. Trading strategies should be based on information (filtration) that traders posess. In practice, however, one typically assumes that they are predictable with respect to the filtration generated by a Brownian motion which serves as a background source of randomness. It is thus necessarey to show that the two filtrations coincide. We do this here.     «

ARCH-models have become popular for modelling financial time series. The seem, at first, 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 defined 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 (...     »