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.
- Variable annuity
- optimal stochastic control
- guaranteed minimum withdrawal benefit
- Gauss–Hermite quadrature
- cubic spline