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Valuing Guaranteed Minimum Withdrawal Benefit (GMWB) has attractedsignificant attention from both the academic field and real world financialmarkets. As remarked by Yang and Dai, the Black and Scholes framework seems tobe inappropriate for such a long maturity products. Also Chen Vetzal andForsyth in showed that the price of these products is very sensitive tointerest rate and volatility parameters. We propose here to use a stochasticvolatility model (Heston model) and a Black Scholes model with stochasticinterest rate (Hull White model). For this purpose we present four numericalmethods for pricing GMWB variables annuities: a hybrid tree-finite differencemethod and a Hybrid Monte Carlo method, an ADI finite difference scheme, and aStandard Monte Carlo method. These methods are used to determine theno-arbitrage fee for the most popular versions of the GMWB contract, and tocalculate the Greeks used in hedging. Both constant withdrawal, optimalsurrender and optimal withdrawal strategies are considered. Numerical resultsare presented which demonstrate the sensitivity of the no-arbitrage fee toeconomic, contractual and longevity assumptions.

Pricing and Hedging GMWB in the Heston and in the Black-Scholes with Stochastic Interest Rate Models