Valuation of guaranteed minimum maturity benefits under generalised regime-switching models using the Fourier Cosine method

This paper presents a flexible valuation approach for variable annuity (VA) contracts embedded with guaranteed minimum maturity benefit (GMMB) riders written on an underlying fund that evolves according to a general regime-switching framework. Unlike the classical regime-switching models which only allow model parameters to change upon regime switches, our framework allows, more importantly, model structures to vary. With mild assumptions on the characteristic function of the log-stock price, our model settings enable the study of fundamental features of the market dynamics, such as stochastic volatility and jumps, on the underlying fund value of GMMB in a unified framework. This novel idea is illustrated by a three-regime model whose environments can be characterised by either the geometric Brownian motion process, the Heston (1993) stochastic volatility process, or double exponential process. Three versions of the GMMB riders are considered; a fixed or roll-up guarantee, a geometric average guarantee and a ratchet guarantee. With the Fourier Cosine (COS) method which utilises characteristic functions, explicit valuation expressions for various contracts are derived, and numerical illustrations are performed to analyse the efficiency of the approach in terms of computational speed and accuracy. The paper makes a unique contribution by presenting regime-dependent bounds and an algorithm for determining the optimal grid points required for the COS method to achieve a specific level of accuracy. Numerical experiments for the valuation framework reveal that i) as the likelihood of regime shifts increases, the price difference of VA contracts with different initial regimes diminishes, which is consistent with financial intuition; ii) the generalised regime-switching models are more suitable for VA pricing when the time-to-maturity spans over several decades, during which structural changes, although infrequent, are inevitable.