Risk Minimizing Strategies for Life Insurance Contracts with Surrender Option

Abstract

We study the hedging strategies for insurance contracts with surrender option. In contrast to the usual description of a surrender time as an optimal stopping time (see Bacinello [1, 2], Grosen and Jorgensen [9], Tak Kuen Siu [15] or Shen and Xu [16] for examples), we assume here the surrender times are not stopping times with respect to the filtration generated by the financial prices. This assumption leads to an incompleteness of the insurance market which does not appear in the usual surrender models. It is then worthwhile to study the hedging strategies an insurer should follow to minimize its exposure to this surrender risk. In this paper, we choose to study the risk minimizing hedging strategies which have already been applied with success in insurance by Moller [11, 12], Dahl and Moller [7] or Riesner[13]. We extend these results in two ways. Firstly, the random times of payment (here the surrender times) are not independent of the financial market and accordingly not independent of each other. Secondly, the financial market is not the Black and Scholes model and is not necessarily complete. We only assume the prices of the tradable financial assets follow local martingales. In this framework, we derive the form of the risk-minimizing strategies for a single policy and for a portfolio of policies. We generalize Moller and Riesner's conclusions by showing that even tough the dates of payment are not independent, we can still combine risk-minimizing strategies and diversification to reduce the relative risk of a portfolio down to to a limit related to the degree of incompleteness of the financial market.