A unified pricing of variable annuity guarantees under the optimal stochastic control framework

Pavel V. Shevchenko*, Xiaolin Luo

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)
27 Downloads (Pure)


In this paper, we review pricing of the 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. We also discuss pricing under the complete/incomplete financial market models, stochastic mortality and optimal/sub-optimal policyholder behavior, and in the presence of taxes. For numerical valuation of these contracts in the case of simple risky asset process, we develop a direct integration method based on the Gauss-Hermite quadratures 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 easier to implement and faster 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 present accurate numerical results for pricing of a Guaranteed Minimum Accumulation Benefit (GMAB) guarantee available on the market that can serve as a numerical benchmark for practitioners and researchers developing pricing of variable annuity guarantees to assess the accuracy of their numerical implementation.

Original languageEnglish
Article number22
Pages (from-to)1-31
Number of pages31
Issue number3
Publication statusPublished - Sep 2016
Externally publishedYes

Bibliographical note

Copyright 2016 by the authors; licensee MDPI, Basel, Switzerland. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.


  • variable annuity
  • guaranteed living and death benefits
  • guaranteed minimum accumulation benefit
  • optimal stochastic control
  • direct integration method


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