Reply to the discussion of “ Non - Gaussian OU based models and some of their uses in financial economics , ”

Alternative models A number of discussants have pointed clearly to alternative models which share features, such as second order properties, with our OU based volatility models. We mentioned in our paper some diffusion based alternatives and these are highlighted in the comments by Valentine Genon-Catalot and Catherine Larédo; Eric Renault; Nour Meddahi. These diffusion alternatives are generally non-linear processes with Gaussian increments, with the non-linearity forcing the process to be positive. Our approach is to advocate linear processes with non-Gaussian increments for volatility. Although diffusions have many advantages, only in the CIR case (to our knowledge) is it possible to easily analytically study the cumulant functional of x∗(t), σ2∗(t)|σ2(0). This is the vital issue in option pricing theory. We think our models open up a new class of analytic option pricing models. This is studied, following our initial work, by Nicolato and Venardos (2000) and Tompkins and Hubalek (2000). Eric Renault points out the work of Andersen on discrete time autoregressive volatility models. It is clear we should have referenced this important and related work. Of course moving to continuous time does change the model structure very considerably as time aggregation means discrete time increments to integrated volatility do not have an autoregressive structure (although instantaneous volatility does). This point is made forcefully in the work by Meddahi and Renault quoted above. Professor Renault worries that our OU based model does not allow the conditional variance of volatility to be proportional to the conditional mean. This fear is shared by Nour Meddahi. However, Figure 6 shows this is actually the case when one conditions on returns, rather than on the unobserved instantaneous volatility. Peter Brockwell and Richard Davis make an interesting contribution, introducing ARMA type Lévy based continuous time volatility models. They give conditions on the volatility process so that it is positive. We look forward to thinking about this process in detail. In a sense their comment has answered one of the queries of Maurice Priestley. The other point that Professor Priestley makes is that we should compare the fit of our model to alternative non-linear diffusion based models. This is surely right, although statistical fit is only one criteria for use. Another, equally important one, is that of tractability.