The model choice is based on our experience with practitioners. We know that at least in three major German life insurance companies one-factor Cox-Ingersoll-Ross models together with geometric brownian motions have been considered in the context of Asset Liability Management. CIR-1 is used to model the debt securities market and interest rates whereas the geometric brownian motions (in reminiscence of the Black-Scholes model) are used to model ``stocks''. The mentioned insurance companies are interested in multi-factor models although not yet using them. The combination and correlation of the models as proposed in this paper seems to be new.
For insurance companies (known to be conservative), an important aspect of such models is the acceptance by the scientific public. This enlightens the decision for standard models like geometric brownian motions or the CIR-1 model.
As mentioned in Fischer, May and Walther (2003), for instance the Vasicek-2 (Gaussian) model behaves in some way better than CIR-2 (concerning parameter estimation or the values of the likelihood function; see also Babbs and Nowman (1998)). Nonetheless, insurance companies seem to prefer CIR, as under the respective parameter constraints CIR assures positive interest rates. Facing a possible deflation in the Eurozone (especially in Germany), one might want to reconsider this philosophy.
From the academic point of view it is clear that alternative models like the Vasicek model should also be examined with respect to the optimization problem. However, for several reasons which will become clear later we recommend to stay inside the class of so-called affine term structure models.
Another more theoretical problem is whether the probably for optimization purposes used derivatives (23), (24) and (25) really exist. Depending on the considered model, this might not be trivial, see also Appendix A for some comments on the differentiability problem. We have not proven the existence of the derivatives for the proposed model (in this case, the Vasicek model may be easier to handle, too). However, for our purposes, this unsolved theoretical problem (which relies on the used model) is no drawback as our optimization routines are subject to ``back-testing'' by the SI methods. Nonetheless, the GS methods work very well in the search of local extrema.
The problems coming in line with differentiation of quantile expressions could be avoided by using risk measures which have more suitable differentiability properties as e.g. the risk measures depending on one-sided moments which are proposed in Fischer (2003).
The authors admit that the proposed model has not been examined for absence of arbitrage. This is postponed to further research. Actually, the model is used like an econometric framework. In this sense, the philosophy of our approach is pragmatic.