In this paper we analyse how the policyholder surrender behaviour is influenced by changes in various sources of risk impacting a variable annuity (VA) contract embedded with a guaranteed minimum maturity benefit rider that can be surrendered anytime prior to maturity. We model the underlying mutual fund dynamics by combining a Heston (1993) stochastic volatility model together with a Hull and White (1990) stochastic interest rate process. The model is able to capture the smile/skew often observed on equity option markets (Grzelak and Oosterlee, 2011) as well as the influence of the interest rates on the early surrender decisions as noted from our analysis. The annuity provider charges management fees which are proportional to the level of the mutual fund as a way of funding the VA contract. To determine the optimal surrender decisions, we present the problem as a 4-dimensional freeboundary partial differential equation (PDE) which is then solved efficiently by the method of lines (MOL) approach. The MOL algorithm facilitates simultaneous computation of the prices, fair management fees, optimal surrender boundaries and hedge ratios of the variable annuity contract as part of the solution at no additional computational cost. A comprehensive analysis on the impact of various risk factors in influencing the policyholder's surrender behaviour is carried out, highlighting the significance of both stochastic volatility and interest rate parameters in influencing the policyholder's surrender behaviour. With the aid of the hedge ratios obtained from the MOL, we construct an effective dynamic hedging strategy to mitigate the provider's risk and compare different hedging performances when the policyholders' surrender behaviour is either optimal or sub-optimal.
Continuous time mortality models, based on affine processes, provide many advantages over discrete-time models, especially for financial applications, where such processes are commonly used for interest rate and credit risks. This paper presents a multi-cohort mortality model for age-cohort mortality rates with common factors across cohorts as well as cohort specific factors. The mortality model is based on well developed and used techniques from interest rate theory and has many applications including the valuation of longevity-linked products. The model has many appealing features. It is a multi-cohort model that describes the whole mortality surface, it captures cohort effects, it allows for observed imperfect correlation between different cohorts, it is shown to fit historical data at pension-related ages very well, it has closed form expressions for survival curves and we show that it outperforms a number of other commonly used discrete-time mortality models in forecasting future survival curves.
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