Abstract: Along with the raise of living standard and improvement of medical technology, average life expectancy in China is rising substantially. The unexpected improvement of average life expectancy renders the insurers to face the challenge of excessive annuity payout, and even brings about to bankrupt risk to them. This type of risk is called longevity risk. The studies on how to manage longevity risk is very popular all over the world, since China is not the only country suffering from longevity and aging. Many scholars point out that we can employ the special financial instruments, which are called longevity derivatives, into hedging the longevity risk. Compared with the mechanism design, previous and present studies are more interested on how to price the longevity derivatives appropriately. In this article, the typical derivative, the longevity bond is priced which is issued by European Investment Bank and BNP Paribas in year 2004. The future mortality of male and female aged people is simulated with Cairns-Blake-Dowd two-factor model. Then EIB/BNP bond is priced by using the risk-neutral measure theory. Since there are infinitely risk-neutral measures in an incomplete market, the relative entropy methods are introduced, to select the optimal risk-neutral measure. Three different types of relative entropy, namely Kullback-Leibler relative entropy, Tsallis relative entropy and Rényi,are applied in order to ensure the robustness. The pricing result also incorporating the market prices of risk as the useful information, in order to make the longevity bond rationally priced. The difference between pricing male mortality linked longevity bonds and that of female morality linked longevity bonds are compared, finding that high risk premia is required when hedging the longevity risk arising from female aged people, which means that improvement of female mortality reveals more uncertainty than that of male mortality, especially when the maturity is long. The case when incorporating unique market price of risk and the case when incorporating total 6 market prices of risk are also compared. It can be concluded that the pricing result varies greatly according to the number of market prices provided by the market. The simulation outcomes when using Kullback-Leibler relative entropy, Tsallis relative entropy and Rényi relative entropy are highly consistent. Thus it is also concluded that the when pricing longevity bonds with relative entropy approach,the outcomes rely much on the completeness of market. This indicates that relative entropy method is very flexible, and the more complete the financial market is, the more reasonable the pricing result will reveal. If the longevity risk is to be managed effectively by means of financial derivatives, the most important issue is to perfect the financial market. And since the relative entropy approach shows great superiority in pricing longevity derivatives, this method can be further developed, and wide application can be made in the field of longevity risk management.

Key words: longevity bond, risk-neutral measure, Kullback-Leibler relative entropy, Tsallis entropy, Rényi relative entropy