  In this paper, we review pricing of variable annuity living and death
guarantees offered to retail investors in many countries. Investors purchase
these products to take advantage of market growth and protect savings. We
present pricing of these products via an optimal stochastic control framework,
and review the existing numerical methods. For numerical valuation of these
contracts, we develop a direct integration method based on Gauss-Hermite
quadrature with a one-dimensional cubic spline for calculation of the expected
contract value, and a bi-cubic spline interpolation for applying the jump
conditions across the contract cashflow event times. This method is very
efficient when compared to the partial differential equation methods if the
transition density (or its moments) of the risky asset underlying the contract
is known in closed form between the event times. We also present accurate
numerical results for pricing of a Guaranteed Minimum Accumulation Benefit
(GMAB) guarantee available on the market that can serve as a benchmark for
practitioners and researchers developing pricing of variable annuity
guarantees.
