Pricing bounds and bang-bang analysis of the Polaris variable annuities

References

• Alonso-García, J., Wood, O. and Ziveyi, J., Pricing and hedging guaranteed minimum withdrawal benefits under a general Lévy framework using the COS method. Quant. Finance, 2018, 18(6), 1049–1075. doi: 10.1080/14697688.2017.1357832
   Web of Science ®Google Scholar
• American International Group, Polaris Choice IV Prospectus [online]. aig.com, 2016. Available online at: http://aig.onlineprospectus.net/AIG/ProductDocuments/index.html (accessed 23 October 2016).
   Google Scholar
• Azimzadeh, P. and Forsyth, P.A., The existence of optimal bang-bang controls for GMxB contracts. SIAM J. Financ. Math., 2015, 6, 117–139. doi: 10.1137/140953885
   Web of Science ®Google Scholar
• Azimzadeh, P., Forsyth, P.A. and Vetzal, K.R., Hedging costs for variable annuities under regime-switching. In Hidden Markov Models in Finance: Further Developments and Applications, Volume II, edited by R.S. Mamon and R.J. Elliott, pp. 133–166, 2014 (Springer: Boston, MA). Available online at: https://doi.org/10.1007/978-1-4899-7442-6_6.
   Google Scholar
• Barrera-Esteve, C., Bergeret, F., Dossal, C., Gobet, E., Meziou, A., Munos, R. and Reboul-Salze, D., Numerical methods for the pricing of swing options: A stochastic control approach. Methodol. Comput. Appl. Probab., 2006, 8, 517–540. doi: 10.1007/s11009-006-0427-8
   Web of Science ®Google Scholar
• Bauer, D., Gao, J., Moenig, T., Ulm, E.R. and Zhu, N., Policyholder exercise behavior in life insurance: The state of affairs. N. Am. Actuar. J., 2017, 21(4), 1–17. doi: 10.1080/10920277.2017.1314816
   Web of Science ®Google Scholar
• Bauer, D. and Ha, H., A least-squares Monte Carlo approach to the calculation of capital requirements. Department of Risk Management and Insurance, Georgia State University, 2013.
   Google Scholar
• Bauer, D., Kling, A. and Russ, J., A universal pricing framework for guaranteed minimum benefits in variable annuities. Astin Bull., 2008, 38, 621–651. doi: 10.1017/S0515036100015312
   Web of Science ®Google Scholar
• Belomestny, D., Kolodko, A. and Schoenmakers, J., Regression methods for stochastic control problems and their convergence analysis. SIAM J. Control Optim., 2010, 48, 3562–3588. doi: 10.1137/090752651
   Web of Science ®Google Scholar
• Boyd, S. and Vandenberghe, L., Convex Optimization, 2004 (Cambridge University Press: Cambridge).
   Google Scholar
• Brown, D.B., Smith, J.E. and Sun, P., Information relaxations and duality in stochastic dynamic programs. Oper. Res., 2010, 58, 785–801. doi: 10.1287/opre.1090.0796
   Web of Science ®Google Scholar
• Carmona, R. and Ludkovski, M., Valuation of energy storage: An optimal switching approach. Quant. Finance, 2010, 10, 359–374. doi: 10.1080/14697680902946514
   Web of Science ®Google Scholar
• Carriere, J.F., Valuation of the early-exercise price for options using simulations and nonparametric regression. Insurance Math. Econom., 1996, 19, 19–30. doi: 10.1016/S0167-6687(96)00004-2
   Web of Science ®Google Scholar
• Chen, Z., Vetzal, K. and Forsyth, P.A., The effect of modelling parameters on the value of GMWB guarantees. Insurance Math. Econom., 2008, 43, 165–173. doi: 10.1016/j.insmatheco.2008.04.003
   Web of Science ®Google Scholar
• Clément, E., Lamberton, D. and Protter, P., An analysis of a least squares regression method for American option pricing. Finance Stoch., 2002, 6, 449–471. doi: 10.1007/s007800200071
   Web of Science ®Google Scholar
• Cong, F. and Oosterlee, C.W., Multi-period mean–variance portfolio optimization based on Monte-Carlo simulation. J. Econ. Dyn. Control, 2016, 64, 23–38. doi: 10.1016/j.jedc.2016.01.001
   Web of Science ®Google Scholar
• Cui, Z., Feng, R. and MacKay, A., Variable annuities with VIX-linked fee structure under a Heston-type stochastic volatility model. N. Am. Actuar. J., 2017, 21, 458–483. doi: 10.1080/10920277.2017.1307765
   Web of Science ®Google Scholar
• Dai, M., Kwok, Y.K. and Zong, J., Guaranteed minimum withdrawal benefit in variable annuities. Math. Finance, 2008, 18, 595–611. doi: 10.1111/j.1467-9965.2008.00349.x
   Web of Science ®Google Scholar
• Del Moral, P., Rémillard, B. and Rubenthaler, S., Monte Carlo approximations of American options that preserve monotonicity and convexity. In Numerical Methods in Finance, edited by R.A. Carmona, P. Del Moral, P. Hu, and N. Oudjane, pp. 115–143, 2012 (Springer: Berlin).
   Google Scholar
• Egloff, D., Monte Carlo algorithms for optimal stopping and statistical learning. Ann. Appl. Probab., 2005, 15, 1396–1432. doi: 10.1214/105051605000000043
   Web of Science ®Google Scholar
• Egloff, D., Kohler, M. and Todorovic, N., A dynamic look-ahead Monte Carlo algorithm for pricing Bermudan options. Ann. Appl. Probab., 2007, 17, 1138–1171. doi: 10.1214/105051607000000249
   Web of Science ®Google Scholar
• Feng, R. and Shimizu, Y., Applications of central limit theorems for equity-linked insurance. Insurance Math. Econom., 2016, 69, 138–148. doi: 10.1016/j.insmatheco.2016.05.004
   Web of Science ®Google Scholar
• Forsyth, P. and Vetzal, K., An optimal stochastic control framework for determining the cost of hedging of variable annuities. J. Econ. Dyn. Control, 2014, 44, 29–53. doi: 10.1016/j.jedc.2014.04.005
   Web of Science ®Google Scholar
• Gao, J. and Ulm, E.R., Optimal consumption and allocation in variable annuities with guaranteed minimum death benefits. Insurance Math. Econom., 2012, 51, 586–598. doi: 10.1016/j.insmatheco.2012.08.003
   Web of Science ®Google Scholar
• Gao, J. and Ulm, E.R., Optimal allocation and consumption with guaranteed minimum death benefits, external income and term life insurance. Insurance Math. Econom., 2015, 61, 87–98. doi: 10.1016/j.insmatheco.2014.12.004
   Web of Science ®Google Scholar
• Glasserman, P. and Yu, B., Number of paths versus number of basis functions in American option pricing. Ann. Appl. Probab., 2004, 14, 2090–2119.
   Web of Science ®Google Scholar
• Hardy, M.R., Hedging and reserving for single-premium segregated fund contracts. N. Am. Actuar. J., 2000, 4, 63–74. doi: 10.1080/10920277.2000.10595903
   Google Scholar
• Horneff, V., Maurer, R., Mitchell, O.S. and Rogalla, R., Optimal life cycle portfolio choice with variable annuities offering liquidity and investment downside protection. Insurance Math. Econom., 2015, 63, 91–107. doi: 10.1016/j.insmatheco.2015.03.031
   Web of Science ®Google Scholar
• Huang, Y.T. and Kwok, Y.K., Regression-based Monte Carlo methods for stochastic control models: Variable annuities with lifelong guarantees. Quant. Finance, 2016, 16, 905–928. doi: 10.1080/14697688.2015.1088962
   Web of Science ®Google Scholar
• Huang, Y.T., Zeng, P. and Kwok, Y.K., Optimal initiation of guaranteed lifelong withdrawal benefit with dynamic withdrawals. SIAM J. Financ. Math., 2017, 8, 804–840. doi: 10.1137/16M1089575
   Google Scholar
• Insured Retirement Institute, Second-quarter 2017 annuity sales report [online]. irionline.org, 2017. Available online at: www.myirionline.org/newsroom/newsroom-detail-view/iri-issues-second-quarter-2017-annuity-sales-repor (accessed 23 September 2017),
   Google Scholar
• Kharroubi, I., Langrené, N. and Pham, H., A numerical algorithm for fully nonlinear HJB equations: An approach by control randomization. Monte Carlo Methods Appl., 2014, 20, 145–165. doi: 10.1515/mcma-2013-0024
   Google Scholar
• Knoller, C., Kraut, G. and Schoenmaekers, P., On the propensity to surrender a variable annuity contract: An empirical analysis of dynamic policyholder behavior. J. Risk Insurance, 2016, 83, 979–1006. doi: 10.1111/jori.12076
   Web of Science ®Google Scholar
• Kuznetsov, A., Kyprianou, A.E., Pardo, J.C. and van Schaik, K., et al., A Wiener–Hopf Monte Carlo simulation technique for Lévy processes. Ann. Appl. Probab., 2011, 21, 2171–2190. doi: 10.1214/10-AAP746
   Web of Science ®Google Scholar
• Kwok, Y.K., Mathematical Models of Financial Derivatives, 2008 (Springer: Berlin).
   Google Scholar
• Li, K.C., Asymptotic optimality for Cp, CL, cross-validation and generalized cross-validation: Discrete index set. Ann. Statist., 1987, 15958–975. doi: 10.1214/aos/1176350486
   Web of Science ®Google Scholar
• Longstaff, F.A. and Schwartz, E.S., Valuing American options by simulation: A simple least-squares approach. Rev. Financ. stud., 2001, 14, 113–147. doi: 10.1093/rfs/14.1.113
   Web of Science ®Google Scholar
• Luo, X. and Shevchenko, P.V., Valuation of variable annuities with guaranteed minimum withdrawal and death benefits via stochastic control optimization. Insurance Math. Econom., 2015, 62, 5–15. doi: 10.1016/j.insmatheco.2015.02.003
   Web of Science ®Google Scholar
• Meyer, M.C., Inference using shape-restricted regression splines. Ann. Appl. Stat., 2008, 2(3), 1013–1033. doi: 10.1214/08-AOAS167
   Web of Science ®Google Scholar
• Milevsky, M.A. and Salisbury, T.S., Financial valuation of guaranteed minimum withdrawal benefits. Insurance Math. Econom., 2006, 38, 21–38. doi: 10.1016/j.insmatheco.2005.06.012
   Web of Science ®Google Scholar
• Moenig, T. and Bauer, D., Revisiting the risk-neutral approach to optimal policyholder behavior: A study of withdrawal guarantees in variable annuities. Rev. Financ., 2015, rfv018.
   Web of Science ®Google Scholar
• Newey, W.K., Convergence rates and asymptotic normality for series estimators. J. Econom., 1997, 79, 147–168. doi: 10.1016/S0304-4076(97)00011-0
   Web of Science ®Google Scholar
• Pasdika, U. and Wolff, J., Coping with longevity: The new German annuity valuation table DAV 2004 R. The Living to 100 and Beyond Symposium, Orlando FL, 2005.
   Google Scholar
• Shen, Z. and Weng, C., A backward simulation method for stochastic optimal control problems. arXiv preprint arXiv:1901.06715, 2019.
   Google Scholar
• Shevchenko, P.V. and Luo, X., A unified pricing of variable annuity guarantees under the optimal stochastic control framework. Risks, 2016, 4, 1–31. doi: 10.3390/risks4030022
   Web of Science ®Google Scholar
• Steinorth, P. and Mitchell, O.S., Valuing variable annuities with guaranteed minimum lifetime withdrawal benefits. Insurance Math. Econom., 2015, 64, 246–258. doi: 10.1016/j.insmatheco.2015.04.001
   Web of Science ®Google Scholar
• Stentoft, L., Convergence of the least squares Monte Carlo approach to American option valuation. Manage. Sci., 2004, 50, 1193–1203. doi: 10.1287/mnsc.1030.0155
   Web of Science ®Google Scholar
• Tsitsiklis, J.N. and Van Roy, B., Regression methods for pricing complex American-style options. IEEE Trans. Neural Netw., 2001, 12, 694–703. doi: 10.1109/72.935083
   PubMed Web of Science ®Google Scholar
• Wang, J. and Ghosh, S.K., Shape restricted nonparametric regression based on multivariate Bernstein polynomials. Technical report, North Carolina State University. Dept. of Statistics, 2012a.
   Google Scholar
• Wang, J. and Ghosh, S.K., Shape restricted nonparametric regression with Bernstein polynomials. Comput. Stat. Data Anal., 2012b, 56, 2729–2741. doi: 10.1016/j.csda.2012.02.018
   Web of Science ®Google Scholar
• Zanger, D.Z., Convergence of a least-squares Monte Carlo algorithm for bounded approximating sets. Appl. Math. Finance, 2009, 16, 123–150. doi: 10.1080/13504860802516881
   Google Scholar
• Zanger, D.Z., Quantitative error estimates for a least-squares Monte Carlo algorithm for American option pricing. Finance Stoch., 2013, 17, 503–534. doi: 10.1007/s00780-013-0204-9
   Web of Science ®Google Scholar