Valuation of GMWB under stochastic volatility

In this paper, we consider the pricing of the variable annuities with Guaranteed Minimum Withdrawal Benefit (GMWB) options, which are deferred, fund-linked annuity contracts, usually with a single premium payment up front. We use the Heston model for the financial market, which assumes the risky asset dynamic follows a stochastic volatility model. The structure of the GMWB policy enables us to adopt the pricing problem of a barrier option. We derive a GMWB pricing partial differential equation (PDE) and the insurance fee is computed by solving an optimization problem. The computed insurance fee is found to be underpriced in the market with a stochastic volatility model. A sensitivity analysis is performed to see the impacts of various parameters on the value of the policy, and the sensitivity of the pricing function with respect to the market risk.