Fast Numerical Method for Pricing of Variable Annuities with Guaranteed Minimum Withdrawal Benefit Under Optimal Withdrawal Strategy

International Journal of Financial Engineering 2(3), [26 pages], 2015, DOI: 10.1142/S2424786315500243

24 Pages Posted: 1 Nov 2014 Last revised: 6 Dec 2015

See all articles by Xiaolin Luo

Xiaolin Luo

Government of the Commonwealth of Australia - CSIRO (Commonwealth Scientific and Industrial Research Organisation)

Pavel V. Shevchenko

Macquarie University - Department of Actuarial Studies and Business Analytics

Date Written: October 24, 2014

Abstract

A variable annuity contract with Guaranteed Minimum Withdrawal Benefit (GMWB) promises to return the entire initial investment through cash withdrawals during the policy life plus the remaining account balance at maturity, regardless of the portfolio performance. Under the optimal withdrawal strategy of a policyholder, the pricing of variable annuities with GMWB becomes an optimal stochastic control problem. So far in the literature these contracts have only been evaluated by solving partial differential equations (PDE) using the finite difference method. The well-known Least-Squares or similar Monte Carlo methods cannot be applied to pricing these contracts because the paths of the underlying wealth process are affected by optimal cash withdrawals (control variables) and thus cannot be simulated forward in time. In this paper we present a very efficient new algorithm for pricing these contracts in the case when transition density of the underlying asset between withdrawal dates or its moments are known. This algorithm relies on computing the expected contract value through a high order Gauss-Hermite quadrature applied on a cubic spline interpolation. Numerical results from the new algorithm for a series of GMWB contract are then presented, in comparison with results using the finite difference method solving corresponding PDE. The comparison demonstrates that the new algorithm produces results in very close agreement with those of the finite difference method, but at the same time it is significantly faster; virtually instant results on a standard desktop PC.

Keywords: Variable Annuity, Optimal Stochastic Control, Guaranteed Minimum Withdrawal Benefit, Gauss-Hermite Quadrature, Cubic Spline

Suggested Citation

Luo, Xiaolin and Shevchenko, Pavel V., Fast Numerical Method for Pricing of Variable Annuities with Guaranteed Minimum Withdrawal Benefit Under Optimal Withdrawal Strategy (October 24, 2014). International Journal of Financial Engineering 2(3), [26 pages], 2015, DOI: 10.1142/S2424786315500243, Available at SSRN: https://ssrn.com/abstract=2517094 or http://dx.doi.org/10.2139/ssrn.2517094

Xiaolin Luo

Government of the Commonwealth of Australia - CSIRO (Commonwealth Scientific and Industrial Research Organisation) ( email )

Riverside Corporate Park
Julius Avenue
Sydney, NSW 2113
Australia

HOME PAGE: http://www.cmis.csiro.au

Pavel V. Shevchenko (Contact Author)

Macquarie University - Department of Actuarial Studies and Business Analytics ( email )

Australia

HOME PAGE: http://www.mq.edu.au/research/centre-for-risk-analytics/pavel-shevchenko

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