This paper considers valuation of life insurance contracts within a Markov model in the case where the interest rate and the transitions rates are dependent. We are working in an ane setup such that we are able to obtain results without needing to solve partial differential equations but only systems of ordinary differential equations as long as we consider models without cycles. Ane processes have for some time been used to model the interest rate and transitions rates in life insurance, see e.g. Dahl and Mller (2006) and Bis (2005). The results in this paper can be seen as a continuation of the work in Due et al. (2000). Moreover, we explore in which models one can obtain results without the needs of integration of mappings of the solutions of ordinary differential equations and in which models integration is indispensable.

The surrender option is embedded in many life insurance contracts and the attention to the modelling of surrender rates and valuation of life insurance contracts where one takes into account the surrender option, has increased highly the last years. This is, in particular, motivated by the forthcoming Solvency II legislation, where the lapse risk plays an important role. One way to model the surrender rate is to make it dependent of the interest rate, which is a fairly natural approach as shown in Kuo et al. (2003). Interest rate dependent surrender rates form our main example of our general results on dependent interest and transition rates.