With the advances in pharmaceutical industry, and the improvement of living standards, life expectancy has risen steadily since the 1960’s in the Europe and North America. Also significant underestimation of the longevity improvements and high uncertainty about future mortality has made longevity a high-profile risk for pension funds, insurers, and other companies. The longevity risk has brought a lot of pressure to the Social Security program. The U.S. government is suffering a deficit in the Social Security program. The Treasury will have to redeem the trust fund in 2020, and exhaust it in 2033. In this thesis, we propose that the U.S. government issue a mortality bond with payoff equal to the realized mortality rate, in order to provide a way to hedge the longevity risk faced by the government. A potential buyer analysis is carried out to investigate the potential market of the mortality bond. We then prove the existence of market equilibrium after introducing the mortality bond. The Lee-Carter framework (Lee and Carter, 1992) is adopted to model and forecast the mortality rate. We model the mortality time series in the Lee-Carter framework by the double exponential jump diffusion (DEJD) model in Kou (2002). And Laplace inversion algorithm (Cai, Kou and Liu, 2011) is used to calculate the distribution of the mortality time series. We derive suitable truncation and discretization constants in the inversion algorithm for the density function of the mortality time series, which is not given in Cai, Kou and Liu (2011), so that the error can be controlled within an acceptable range. Also, we derive an optimal Laplace transform parameter ξ, such that the truncation constant N, as well as the computational workload, is minimized. We propose to use the rational expectation asset pricing approach and the Wang transform to price the bond based on a new model specifying the correlation between the consumption growth rate and the mortality growth rate.