Iterative methods for the solution of a singular control formulation of a GMWB pricing problem

摘要

Discretized singular control problems in finance result in highly nonlinear algebraic equations which must be solved at each timestep. We consider a singular stochastic control problem arising in pricing a guaranteed minimum withdrawal benefit (GMWB), where the underlying asset is assumed to follow a jump diffusion process. We use a scaled direct control formulation of the singular control problem and examine the conditions required to ensure that a fast fixed point policy iteration scheme converges. Our methods take advantage of the special structure of the GMWB problem in order to obtain a rapidly convergent iteration. The direct control method has a scaling parameter which must be set by the user. We give estimates for bounds on the scaling parameter so that convergence can be expected in the presence of round-off error. Example computations verify that these estimates are of the correct order. Finally, we compare the scaled direct control formulation to a formulation based on a block version of the penalty method (Huang and Forsyth in IMA J Numer Anal 32:320–351, 2012). We show that the scaled direct control method has some advantages over the penalty method.