A variable annuity contract with Guaranteed Minimum Withdrawal Benefit (GMWB) promises to return the entire initial investment through cash withdrawals during the contract plus the remaining account balance at maturity, regardless of the portfolio performance. Under the optimal(dynamic) withdrawal strategy of a policyholder, GMWB pricing becomes an optimal stochastic control problem that can be solved by backward recursion of Bellman equation. In this paper we develop a very efficient new algorithm for pricing these contracts in the case of stochastic interest rate not considered previously in the literature. Presently our method is applied to the Vasicek interest rate model, but it is generally applicable to any model when transition density or moments of the underlying asset and interest rate are known in closed form or can be evaluated efficiently. Using bond price as a numeraire the required expectations in the backward recursion are reduced to two-dimensional integrals calculated through a high order Gauss-Hermite quadrature applied on a two-dimensional cubic spline interpolation.
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