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FOURIER SPACE TIME-STEPPING ALGORITHM FOR VALUING GUARANTEED MINIMUM WITHDRAWAL BENEFITS IN VARIABLE ANNUITIES UNDER REGIME-SWITCHING AND STOCHASTIC MORTALITY

Published online by Cambridge University Press:  07 August 2017

Katja Ignatieva ,
Andrew Song  and
Jonathan Ziveyi
Katja Ignatieva
Affiliation:
School of Risk and Actuarial Studies, Business School, University of New South Wales, Sydney, NSW 2052, Australia E-Mail: k.ignatieva@unsw.edu.au
Andrew Song
Affiliation:
School of Risk and Actuarial Studies, Business School, University of New South Wales, Sydney, NSW 2052, Australia E-Mail: k.ignatieva@unsw.edu.au
Jonathan Ziveyi*
Affiliation:
School of Risk and Actuarial Studies, Business School, University of New South Wales, Sydney, NSW 2052, Australia E-Mail: k.ignatieva@unsw.edu.au
*
E-Mail: j.ziveyi@unsw.edu.au
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Abstract

This paper introduces the Fourier Space Time-Stepping algorithm to the valuation of variable annuity (VA) contracts embedded with guaranteed minimum withdrawal benefit (GMWB) riders when the underlying fund dynamics evolve under the influence of a regime-switching model. Mortality risk is introduced to the valuation framework by incorporating a two-factor affine stochastic mortality model proposed in Blackburn and Sherris (2013). The paper considers both, static and dynamic policyholder withdrawal behaviour associated with GMWB riders and assesses how model parameters influence the fees levied on providing such guarantees. Our numerical experiments reveal that the GMWB fees are very sensitive to regime-switching parameters; a percentage increase in the force of interest results in significant decrease in guarantee fees. The guarantee fees increase substantially with increasing volatility levels. Numerical experiments also highlight an increasing importance of mortality as maturity of the VA contract increases. Mortality has less impact on shorter maturity contracts regardless of the policyholder's withdrawal behaviour. As much as mortality influences pricing results for long maturities, the associated guarantee fees are decreasing functions of maturities for the VA contracts. Robustness checks of the Fourier Space Time-Stepping algorithm are performed by making numerical comparisons with several existing valuation approaches.

Keywords

C63G22G13G12Guaranteed minimum withdrawal benefitsvariable annuitymortality riskregime-switchingFourier space time-stepping
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 48 , Issue 1 , January 2018 , pp. 139 - 169
Copyright
Copyright © Astin Bulletin 2017 

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References

Bacinello, A., Millossovich, P., Olivieri, A. and Pitacco, E. (2011) Variable annuities: A unifying valuation approach. Insurance: Mathematics and Economics, 49, 285–297.Google Scholar
Bacinello, A.R., Millossovich, P. and Montealegre, A. (2016) The valuation of GMWB variable annuities under alternative fund distributions and policyholder behaviours. Scandinavian Actuarial Journal, 5, 446–465.Google Scholar
Bauer, D., Kling, A. and Russ, J. (2008) A universal pricing framework for guaranteed minimum benefits in variable annuities. ASTIN Bulletin-Actuarial Studies in Non Life Insurance, 38 (2), 621.Google Scholar
Bernard, C., Hardy, M. and Mackay, A. (2014) State-dependent fees for variable annuity guarantees. ASTIN Bulletin - The Journal of the International Actuarial Association, 44 (3), 559–585.Google Scholar
Biffis, E. (2005) Affine processes for dynamic mortality and actuarial valuations. Insurance: Mathematics and Economics, 37 (3), 443–468.Google Scholar
Blackburn, C. and Sherris, M. (2013) Consistent dynamic affine mortality models for longevity risk applications. Insurance: Mathematics and Economics, 53 (1), 64–73.Google Scholar
Buffington, J. and Elliott, R. (2002) American options with regime switching. International Journal of Theoretical and Applied Finance, 5 (5), 497–514.Google Scholar
Carr, P. and Madan, B. (1999) Option valuation using the fast Fourier transform. The Journal of Computational Finance, 2, 61–73.Google Scholar
Chen, Z. and Forsyth, P. (2008) A numerical scheme for the impulse control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit (GMWB). Numerische Mathematik, 109 (4), 535–569.CrossRefGoogle Scholar
Chen, Z., Vetzal, K. and Forsyth, P. (2008) The effect of modelling parameters on the value of GMWB guarantees. Insurance: Mathematics and Economics, 43 (1), 165–173.Google Scholar
Coleman, T., Li, Y. and Patron, M. (2006) Hedging guarantees in variable annuities under both equity and interest rate risks. Insurance: Mathematics and Economics, 38 (2), 215–228.Google Scholar
Da Fonseca, J. and Ziveyi, J. (2017) Valuing variable annuity guarantees on multiple assets. Scandinavian Actuarial Journal, 2017 (3), 209–230.Google Scholar
Dai, M., Kwok, Y. and Zong, J. (2008) Guaranteed minimum withdrawal benefit in variable annuities. Mathematical Finance, 18 (4), 595–611.Google Scholar
Donnelly, R., Jaimungal, S. and Robisov, D.H. (2014) Valuing guaranteed withdrawal benefits with stochastic interest rates and volatility. Quantitative Finance, 14 (2), 369–382.Google Scholar
Du, D. and Martin, C. (2014) Variable annuities-recent trends and the use of captives. Federal Reserve Bank of Boston, available at: https://www.bostonfed.org/bankinfo/publications/variable-annuities.pdf.Google Scholar
Elliott, R., Aggoun, L. and Moore, J. (1994) Hidden Markov Models: Estimation and Control. New York: Springer-Verlag.Google Scholar
Elliott, R., Chan, L. and Siu, T. (2005) Option pricing and Esscher transform under regime switching. Annals of Finance, 1 (4), 423–432.CrossRefGoogle Scholar
Forsyth, P. and Vetzal, K. (2014) An optimal stochastic control framework for determining the cost of hedging of variable annuities. Journal of Economic Dynamics and Control, 44, 29–53.Google Scholar
Gerber, H. and Shiu, E. (1994) Option pricing by Esscher transforms. Transactions of the Society of Actuaries, 46, 99–191.Google Scholar
Girsanov, I. (1960) On transforming a certain class of stochastic processes by absolutely continuous substitution of measures. Theory of Probability and its Applications, 5 (3), 285–301.Google Scholar
Guo, X. (2001) An explicit solution to an optimal stopping problem with regime switching. Journal of Applied Probability, 38 (2), 464–481.Google Scholar
Holland, D. and Simonelli, A. (2015) Variable Annuity Sales Increase Sharply, VA Net Assets Approaching $2 Trillion Mark; Fixed Indexed Annuity Sales Record Second Strongest Quarter Ever. Insured Retiment Institute Issues Second-Quarter 2015 Annuity Sales Report. Available at http://irionline.org/newsroom/newsroom-detail-view/iri-issues-second-quarter-2015-annuity-sales-report.Google Scholar
Horneff, V., Maurer, R., Mitchell, O.S. and Rogalla, R. (2015) Optimal life cycle portfolio choice with variable annuities offering liquidity and investment downside protection. Insurance: Mathematics and Economics, 63, 91–107.Google Scholar
Huang, Y. and Forsyth, P.A. Analysis of a penalty method for pricing a Guaranteed MinimumWithdrawal Benefit (GMWB). IMA Journal of Numerical Analysis, 32 (1), 320–351.Google Scholar
Hurd, T. and Zhou, Z. (2010) A Fourier transform method for spread option pricing. SIAM Journal on Financial Mathematics, 1 (1), 142–157.Google Scholar
Ignatieva, K., Song, A. and Ziveyi, J. (2016) Pricing and hedging of guaranteed minimum benefits under regime-switching and stochastic mortality. Insurance: Mathematics and Economics, 70, 286–300.Google Scholar
Jackson, K., Jaimungal, S. and Surkov, V. (2008) Fourier space time-stepping for option pricing with Lévy models, Working Paper.Google Scholar
Lamberton, D. and Lapeyre, B. (2011) Introduction to Stochastic Calculus Applied to Finance, 2nd edition. CRC Press.Google Scholar
Lippa, J. (2013) A Fourier Space Time-stepping Approach Applied to Problems in Finance. Master's thesis, University of Waterloo.Google Scholar
Liu, R., Zhang, Q. and Yin, G. (2006) Option pricing in a regime-switching model using the fast Fourier transform. Journal of Applied Mathematics and Stochastic Analysis, 2006, 1–22.Google Scholar
Luo, X. and Shevchenko, P. (2015) Fast numerical method for pricing of variable annuities with guaranteed minimum withdrawal benefit under optimal withdrawal strategy. International Journal of Financial Engineering, 2 (3).Google Scholar
Milevsky, A. and Salisbury, S. (2006) Financial valuation of guaranteed minimum withdrawal benefits. Insurance: Mathematics and Economics, 38 (1), 21–38.Google Scholar
Moenig, T. and Bauer, D. (2016) Revisiting the risk-neutral approach to optimal policyholder behavior: A study of withdrawal guarantees in variable annuities. Review of Finance, 20 (2), 759–794.Google Scholar
Peng, J., Leung, K., and Kwok, Y. (2012) Pricing guaranteed minimum withdrawal benefits under stochastic interest rates. Quantitative Finance, 12 (6), 933–941.Google Scholar
Shah, P. and Bertsimas, D. (2008) An analysis of the guaranteed withdrawal benefits for life option. Working Paper. Available at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1312727.Google Scholar
Shen, Y., Fan, K. and Siu, T. (2014) Option valuation under a double regime-switching model. The Journal of Futures Markets, 34 (5), 451–478.Google Scholar
Sherris, M., Xu, Y. and Ziveyi, J. (2015) The application of affine processes in multi-cohort mortality model. Working Paper 2015/13. Available at http://www.cepar.edu.au.Google Scholar
Surkov, V. and Davison, M. (2009) Efficient construction of robust hedging strategies under jump models. Canadian Applied Mathematics Quarterly, 17 (4), 755–776.Google Scholar
Wong, H. and Guan, P. (2011) An FFT network for Levy option pricing. Journal of Banking and Finance, 35 (4), 988–999.Google Scholar