Price Bounds for Guaranteed Annuity Options in a Model-Independent Setting

In this paper, we consider the valuation of Guaranteed Annuity Options (GAOs) in a very generalised modelling framework where both interest rate and mortality risk are stochastic and correlated. It is extremely hard to price these type of options in the correlated environment and as a result there is an absence of a closed form solution in the literature. We utilise doubly stochastic stopping times to incorporate the randomness about the time of death and employ a suitable change of measure to make the valuation of survival benefit possible, there by adapting the pay-off of the GAO in terms of the pay-off of a basket call option. We derive general price bounds for GAOs by employing the theory of comonotonicity, the Rogers-Shi ([72]) approach and the general closed form basket option pricing bounds as discussed in [16]. The theory derived is then applied to affine models to generate some very interesting formulae for the bounds under the affine set up. Numerical examples are furnished and benchmarked against Monte Carlo simulations to estimate the price of a GAO for the well known Vasicek model.