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Risk-Neutral Valuation of GLWB Variable Annuities Interaction Between Providers and Customers
Abstract: I start this talk by describing a discrete model, proposed in [1], for pricing single premium variable annuities with GLWB riders under a stochastic mortality framework. Our set-up is very general and only requires the Markovian property for the mortality intensity and the asset price processes. The contract value is defined through an optimization problem which is solved by using dynamic programming. More in detail, it is obtained by assuming that the policyholder chooses dynamically a withdrawal strategy in order to maximize the expected present value of her future cash-flows under a risk-neutral probability measure selected by the insurer. We prove, by backward induction, the validity of the bang-bang condition for the set of withdrawal strategies of the problem. We assume constant interest rates, although our results still hold in the case of a Markovian interest rate process. After that, in [2] we consider an insurer that offers his policyholders different contracts structured as before. They differ for the contractual parameters or the risk/return profile of the reference fund in which the premium is invested, and are all fairly priced. To price them, we use the previous risk-neutral approach. However, the policyholder can have different preferences; in particular we assume that she acts in order to maximize the expected discounted utility, under the physical measure, of her cash-flows. This leads to a ranking of the various contracts from the policyholder’s perspective according to their (maximum) utility. Being the contracts fairly priced, the expected present value, under the insurer’s probability, of the cash flows induced by the actual policyholder’s strategy does not exceed the single premium, thus leaving room for a gain. Then a ranking of the contracts from the insurer’s perspective arises as well.