Fourier Space Time-Stepping Algorithm for Valuing Guaranteed Minimum Withdrawal Benefits in Variable Annuities Under Regime-Switching and Stochastic Mortality

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.

The Investigation of a Forward-Rate Mortality Framework

“…Presently, we assume that the market price of longevity risk is constant, and, for convenience, zero; this has previously been done in, e.g., in Ziveyi et al (2013) and Ignatieva et al (2018) in relation to the pricing of life insurance products. The result of this work would not be impacted if arbitrary λ ∈ R were selected to represent the market price of longevity risk, since it is true that, in practice, investors will certainly demand a risk premium.…”

ε-monotone Fourier methods for optimal stochastic control in finance

“…These optimal stochastic problems arise in many important financial applications. This includes problems such as asset allocation (Li and Ng, 2000;Huang, 2010;Forsyth and Vetzal, 2017;Cong and Oosterlee, 2016), pricing of variable annuities (Bauer et al, 2008;Dai et al, 2008;Ignatieva et al, 2018;Alonso-Garcia et al, 2018;Huang et al, 2017), and hedging in discrete time (Remillard and Rubenthaler, 2013;Angelini and Herzel, 2014) to name just a few.…”

“…These methods are based on Fourier Space Timestepping (FST) (Jackson et al, 2008), the CONV technique (Lord et al, 2008) or the COS algorithm (Fang and Oosterlee, 2008). Fourier methods have been applied to pricing of exotic variance products and volatility derivatives (Zheng and Kwok, 2014), guaranteed minimum withdrawal benefits (Ignatieva et al, 2018;Alonso-Garcia et al, 2018;Huang et al, 2017) and equity-indexed annuities (Deng et al, 2017) to name just a few.…”

“…;Ignatieva et al (2018);Huang et al (2017) for applications in finance.Algorithm 1 FST/CONV Fourier method. x • y is the Hadamard product of vectors x,y.Require: G = {G(ω j , ∆τ )}, j = −N/2, .…”

Option pricing under regime-switching models: Novel approaches removing path-dependence

“…Hardy (2003) pioneered the use of regime-switching models to value long-term options embedded in variable annuities. A non-exhaustive list of other papers which use this family of models to either price or hedge equity options attached to equity-linked insurance contracts or variable annuities is hereby provided: Lin et al (2009), Jin et al (2011), Ng and Li (2011), Ngai and Sherris (2011), Wang and Yin (2012) Azimzadeh et al (2014), Fan et al (2015), Siu et al (2015), Ignatieva et al (2016), Wang et al (2017), Trottier et al (2018a), Trottier et al (2018b) and Ignatieva et al (2018).…”