Analysis of a penalty method for pricing a guaranteed minimum withdrawal benefit (GMWB)

摘要

The no-arbitrage pricing of guaranteed minimum withdrawal benefits contracts results in a singular stochastic control problem that can be formulated as a Hamilton-Jacobi-Bellman (HJB) variational inequality (VI). Recently, a penalty method has been suggested for solution of this HJB VI (Dai et al. (2008), Math. Finance, 18, 595-611). This method is very simple to implement. In this article we present a rigorous proof of the convergence of the penalty method to the viscosity solution of the HJB VI. Numerical tests of the penalty method are presented that show the experimental rates of convergence, and a discussion of the choice of the penalty parameter is also included.