Fast numerical method for pricing of variable annuities with guaranteed minimum withdrawal benefit under optimal withdrawal strategy

Abstract

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.

A Unified Pricing of Variable Annuity Guarantees under the Optimal Stochastic Control Framework

“…is not known in closed form but one can find its moments, then the integration can also be done with similar efficiency and accuracy by the method of matching moments as described in [7,26]. The method also works very well in the two-dimensional case, see, e.g., [12] where it was applied for GMWB pricing in the case of stochastic interest rate.…”

“…Also, there is no death benefit; it is assumed that the beneficiary will maintain the contract in the case of a policyholder death. This contract has only basic features facilitating comparison of results from different academic studies, such as [4,5,7,14].…”

“…More efficient quadrature methods (requiring less points to approximate the integral) exist. In particular, in the case of a geometric Brownian motion process for the risky asset, it is very efficient to use the Gauss-Hermite quadrature as developed in [26] and applied for GMWB pricing in [7]. Section 6.3 provides detailed description of the method for pricing VA riders in general.…”

“…The authors of [1] develop a unified approach with numerical estimation via MC and direct integration methods. The direct integration method was developed further in [7,8] using the Gauss-Hermite quadrature and cubic interpolations. The authors of [9] consider many VA riders under the stochastic interest rate and stochastic volatility if the policyholder withdraws at the pre-defined contractual rate or completely surrenders the contract.…”

“…This method can be applied when transition density of the underlying asset between the contract cashflow event dates or its moments are known in closed form. We have used this for pricing specific financial derivatives and some simple versions of the VA guarantees in [7,26]. Here, we adapt and extend the method to handle pricing VA riders in general.…”