2014/10/30 by Xiaolin Luo, Luo, Xiaolin, Pavel V. Shevchenko +1
Economics, Econometrics and Finance · Social Sciences · #FOS: Economics and business #Insurance, Mortality, Demography, Risk Management #Monetary Policy and Economic Impact #Pricing of Securities (q-fin.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1410.8609
openalex publication_date 2014/10/30 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
A variable annuity contract with Guaranteed Minimum Withdrawal Benefit (GMWB)\npromises to return the entire initial investment through cash withdrawals\nduring the policy life plus the remaining account balance at maturity,\nregardless of the portfolio performance. Under the optimal withdrawal strategy\nof a policyholder, the pricing of variable annuities with GMWB becomes an\noptimal stochastic control problem. So far in the literature these contracts\nhave only been evaluated by solving partial differential equations (PDE) using\nthe finite difference method. The well-known Least-Squares or similar Monte\nCarlo methods cannot be applied to pricing these contracts because the paths of\nthe underlying wealth process are affected by optimal cash withdrawals (control\nvariables) and thus cannot be simulated forward in time. In this paper we\npresent a very efficient new algorithm for pricing these contracts in the case\nwhen transition density of the underlying asset between withdrawal dates or its\nmoments are known. This algorithm relies on computing the expected contract\nvalue through a high order Gauss-Hermite quadrature applied on a cubic spline\ninterpolation. Numerical results from the new algorithm for a series of GMWB\ncontract are then presented, in comparison with results using the finite\ndifference method solving corresponding PDE. The comparison demonstrates that\nthe new algorithm produces results in very close agreement with those of the\nfinite difference method, but at the same time it is significantly faster;\nvirtually instant results on a standard desktop PC.\n