In this paper, we apply the stratified approximation method by Lognormal and Gamma distribution, which was introduced for pricing Asian options, to price variable annuities with period and single premium. By using the moment matching method to find a proper Lognormal or Gamma distribution to approximate the conditional distribution of the average of the underlying assets given its terminal value, the highly oscillating double integrals for computing the expectation of some function of the average on the underlying asset is simplified to a single integral, which greatly reduces the computation time for pricing the variable annuities with period premium. Compare with the existing research on the under priced problem of the variable annuities with period premium, we obtain more accurate result by the stratified approximation method than the numerical method by solving partial differential equations, and we found that the under priced problem of variable annuities with period premium is even more severe than it was stated in the existing literature. We further investigate the price gap of variable annuities with period premium and variable annuities with single premium in different parameter settings to find the key impact parameter on the price gap between variable annuities with period premium and single premium. The robustness of the stratified approximation method is tested by numerical examples.
