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A variable annuity contract with Guaranteed Minimum Withdrawal Benefit (GMWB)promises to return the entire initial investment through cash withdrawalsduring the contract plus the remaining account balance at maturity, regardlessof the portfolio performance. Under the optimal(dynamic) withdrawal strategy ofa policyholder, GMWB pricing becomes an optimal stochastic control problem thatcan be solved by backward recursion of Bellman equation. In this paper wedevelop a very efficient new algorithm for pricing these contracts in the caseof stochastic interest rate not considered previously in the literature.Presently our method is applied to the Vasicek interest rate model, but it isgenerally applicable to any model when transition density or moments of theunderlying asset and interest rate are known in closed form or can be evaluatedefficiently. Using bond price as a numeraire the required expectations in thebackward recursion are reduced to two-dimensional integrals calculated througha high order Gauss-Hermite quadrature applied on a two-dimensional cubic splineinterpolation. Numerical results from the new algorithm for a series of GMWBcontracts for both static and optimal cases are presented. As a validation,results of the algorithm are compared with the closed form solutions for simplevanilla options, and with Monte Carlo and finite difference results for thestatic GMWB. The comparison demonstrates that the new algorithm issignificantly faster than finite difference or Monte Carlo for all thetwo-dimensional problems tested so far. For dynamic GMWB pricing, we found thatfor positive correlation between the underlying asset and interest rate, theGMWB price under the stochastic interest rate is significantly higher comparedto the case of deterministic interest rate, while for negative correlation thedifference is less but still significant.

Valuation of Variable Annuities with Guaranteed Minimum Withdrawal Benefit under Stochastic Interest Rate