The Risk Management of Power Options Embedded in Life–insurance Products

We analyse contracts which pay out a guaranteed minimum rate of return and a fraction of a positive excess rate, which is specified on the basis of a benchmark portfolio. These contracts are closely related to life–insurance products and can be considered as alternatives to a direct investment in the underlying benchmark portfolio. In particular, the effects of modelling a roll–over savings plan along the lines of Miltersen and Persson (1998) are analysed where the participation in the excess return is given in an exponential form. We discuss some important features of this contract specification, which heretofore have not been considered in the literature. For one, the option embedded into the savings plan is in fact a power option with a payoff which is neither convex nor concave in the value of the underlying benchmark. The power of the embedded option is equal to the participation rate α in the excess return above the guarantee, and the strike depends on α as well as the minimum guaranteed rate g. Thus the specification of the “fair” contract parameters is closely related to well known features of power options. The long maturity of life–insurance products makes it necessary to lift the Black/Scholes assumptions and consider an uncertain volatility scenario. This gives rise to an interesting pricing and hedging problem. In particular, we show how to determine the contract parameters conservatively and implement robust risk management strategies. This highlights the necessity of a careful choice of guarantees which are granted to the insurance customer.