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

Abstract

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 [15]. We show that the scaled direct control method has some advantages over the penalty method.

Inexact arithmetic considerations for direct control and penalty methods: American options under jump diffusion

“…We focus exclusively on methods which do not require knowledge of the structure of the exercise region. Our analysis extends some of the results in [20] for the case of a singular control problem. Our main results are:…”

“…Also, utilizing the properties of inexact arithmetic, it is possible to convert existing software (which uses a penalty method) to use the scaled direct control formulation in just a few lines of code. Finally, our analysis can also be applied to other HJB equations, such as singular control problems [20].…”

The valuation of GMWB variable annuities under alternative fund distributions and policyholder behaviours

“…Looking at Table 1, one can see that few papers go beyond the assumption of normality for the fund returns. Notable exceptions are [15] and [20], where the jump diffusion model of Merton is considered, and [9], where regime switching type processes are used.…”

Methods for Pricing American Options under Regime Switching

“…Using the methods in [24], we can estimate the largest values of C which can be used before round off prevents convergence. For both penalty and direct control formulations, the estimate for this maximum value of C (designated by C max ) which can be used in finite precision arithmetic is C max tolerance 2δ , (6.4) .…”