Variable Annuities with a Threshold Fee: Valuation, Numerical Implementation and Comparative Static Analysis

In this paper we deal with a variable annuity which provides guarantees at death and maturity financed through the application of a state-dependent fee structure of the threshold type. Our first aim is to test the use of least squares Monte Carlo methods (LSMC) for the numerical implementation of the valuation model. In fact, special care is needed when applying LSMC, due to the shape of the surrender region. To this end we introduce a quite general framework, under which we derive a theoretical result that allows us to stem the numerical errors arising in the regression step of the valuation algorithm. The second aim of the paper is to analyse numerically the interaction between the various contract components, in particular fee rates/thresholds and surrender penalties, under alternative policyholder behaviours. This analysis turns out to be very useful, in particular when addressing the problem of the contract design.