  A variable annuity contract with Guaranteed Minimum Withdrawal Benefit (GMWB)
promises to return the entire initial investment through cash withdrawals
during the policy life plus the remaining account balance at maturity,
regardless of the portfolio performance. Under the optimal withdrawal strategy
of a policyholder, the pricing of variable annuities with GMWB becomes an
optimal stochastic control problem. So far in the literature these contracts
have only been evaluated by solving partial differential equations (PDE) using
the finite difference method. The well-known Least-Squares or similar Monte
Carlo methods cannot be applied to pricing these contracts because the paths of
the underlying wealth process are affected by optimal cash withdrawals (control
variables) and thus cannot be simulated forward in time. In this paper we
present a very efficient new algorithm for pricing these contracts in the case
when transition density of the underlying asset between withdrawal dates or its
moments are known. This algorithm relies on computing the expected contract
value through a high order Gauss-Hermite quadrature applied on a cubic spline
interpolation. Numerical results from the new algorithm for a series of GMWB
contract are then presented, in comparison with results using the finite
difference method solving corresponding PDE. The comparison demonstrates that
the new algorithm produces results in very close agreement with those of the
finite difference method, but at the same time it is significantly faster;
virtually instant results on a standard desktop PC.
