As human life expectancy continues to increase, longevity risk has become a major concern for pension plan sponsors and annuity providers. To hedge the risk, longevity-linked securities have been developed. Since these securities often have payoffs linked to mortality rates of multiple populations, it is important to investigate the relationship between them. In this thesis, we use England and Wales (EW) and Canadian mortality data for illustration. We consider the long-term equilibrium between the mortality indexes of the two populations through cointegration analysis. Our test shows that structural change occurred in the equilibrium. To capture changes in both equilibrium and autoregression structure, we adopt the Threshold Vector Error Correction Model (TVECM). We find that the TVECM model provides adequate fit to our data. This model is further applied to pricing an illustrative longevity bond. Our numerical results indicate that the changes in the long-term equilibrium have a significant impact on longevity bond prices.