  Valuing Guaranteed Minimum Withdrawal Benefit (GMWB) has attracted
significant attention from both the academic field and real world financial
markets. As remarked by Yang and Dai, the Black and Scholes framework seems to
be inappropriate for such a long maturity products. Also Chen Vetzal and
Forsyth in showed that the price of these products is very sensitive to
interest rate and volatility parameters. We propose here to use a stochastic
volatility model (Heston model) and a Black Scholes model with stochastic
interest rate (Hull White model). For this purpose we present four numerical
methods for pricing GMWB variables annuities: a hybrid tree-finite difference
method and a Hybrid Monte Carlo method, an ADI finite difference scheme, and a
Standard Monte Carlo method. These methods are used to determine the
no-arbitrage fee for the most popular versions of the GMWB contract, and to
calculate the Greeks used in hedging. Both constant withdrawal, optimal
surrender and optimal withdrawal strategies are considered. Numerical results
are presented which demonstrate the sensitivity of the no-arbitrage fee to
economic, contractual and longevity assumptions.
