  Valuing Guaranteed Lifelong Withdrawal Benefit (GLWB) has attracted
significant attention from both the academic field and real world financial
markets. As remarked by Forsyth and Vetzal the Black and Scholes framework
seems to be inappropriate for such long maturity products. They propose to use
a regime switching model. Alternatively, 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 GLWB 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 GLWB contract, and to
calculate the Greeks used in hedging. Both constant withdrawal and optimal
withdrawal (including lapsation) strategies are considered. Numerical results
are presented which demonstrate the sensitivity of the no-arbitrage fee to
economic, contractual and longevity assumptions.
