Options on pension annuity

We introduce a European (exotic) call option on a pension annuity. The option gives its owner the right to buy, for a specified lump sum, an ordinary annuity (pension) that starts at a specified future date ("retirement age"). Thus, instead of contributing monthly to a pension fund, one could buy this option and insure the terms of their retirement. We price the options under stochastic interest rates in both discrete and continuous time regimes. In the discrete time case, we use an order 2 autoregressive process (AR(2)). In the continuous time case, we use simulations of the short interest rate according to various models, such as CIR, and simulations of GARCH process estimated from real data.