  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. Numerical results from the new algorithm for a series of GMWB
contracts for both static and optimal cases are presented. As a validation,
results of the algorithm are compared with the closed form solutions for simple
vanilla options, and with Monte Carlo and finite difference results for the
static GMWB. The comparison demonstrates that the new algorithm is
significantly faster than finite difference or Monte Carlo for all the
two-dimensional problems tested so far. For dynamic GMWB pricing, we found that
for positive correlation between the underlying asset and interest rate, the
GMWB price under the stochastic interest rate is significantly higher compared
to the case of deterministic interest rate, while for negative correlation the
difference is less but still significant.
