Pricing and hedging GMWB in the Heston and in the Black-Scholes with stochastic interest rate models

In this paper, we approach the problem of valuing a particular type of variable annuity called GMWB when advanced stochastic models are considered. As remarked by Yang and Dai (Insur Math Econ 52(2):231-242, 2013), and Dai et al. (Insur Math Econ 64:364-379, 2015), the Black-Scholes framework seems to be inappropriate for such a long maturity products. Also Chen et al. (Insur Math Econ 43(1):165-173, 2008) show that the price of GMWB variable annuities is very sensitive to the interest rate and the volatility parameters. We propose here to use a stochastic volatility model (the Heston model) and a Black-Scholes model with stochastic interest rate (the Black-Scholes Hull-White model). For this purpose, we consider four numerical methods: a hybrid tree-finite difference method, a hybrid tree-Monte Carlo method, an ADI finite difference scheme and a Standard Monte Carlo method. These approaches are employed to determine the no-arbitrage fee for a popular version of the GMWB contract and to calculate the Greeks used in hedging. Both constant withdrawal and dynamic withdrawal strategies are considered. Numerical results are presented, which demonstrate the sensitivity of the no-arbitrage fee to economic and contractual assumptions as well as the different features of the proposed numerical methods.