Efficient valuation of a variable annuity contract with a surrender option

We study the partial differential equation (PDE) approach for efficient and accurate valuation of a variable annuity (VA) contract with a surrender option. Specifically, using the Laplace-Carson Transform (LCT), we derive an analytic pricing formula for a VA contract with a surrender option which is formulated as a PDE with an optimal surrender boundary. To demonstrate the efficiency and accuracy of our approach, we show that our pricing formula efficiently provides the exact value of a VA contract. Moreover, we compare the performance of three numerical methods for Laplace inversion to find the most efficient method. Among these methods, we found that the Gaver-Stehfest method is the most efficient pricing for a VA contract with a surrender option.