Research On Longevity Bond Pricing Under Dependent Mortality Intensity Model

Longevity risk is the risk that the realized future mortality improved trend exceeds current assessments,which leads to the extension of repayment periods and the increase of payment amount.Involving systemic risk,this risk could not be diversified.It is becoming a major problem faced by many life insurers and annuity providers.The ef-fective way of managing this risk is securitization,issuing derivatives based on longevity risk,also known as longevity bonds.In recent years,financial institutions have succes-sively tried to issue various types of longevity bonds.From the experience of financial practice,the reasonable pricing of longevity bonds is the key to the success of issuance.In this paper,we study the pricing of longevity bonds with the principle of equivalent utility and the optimal investment strategy of investors.Pricing is based on reasonable measurement of mortality risk.Different from the existing literature,we introduce the Gaussian random field and stochastic string to model mortality intensity,so as to investigate both the stochastic evolvements of mor-tality from two dimensions of time and age simultaneously,and the relationship between mortality intensities of different age cohorts.This model is more consistent with the actual mortality data,which suggests that the decline trend of mortality is inconsistent across age cohorts.We identify the indifference price of zero-coupon longevity bond based on this model.Then we use the principle of equivalent utility to determine the price of longevity bond in incomplete market.Specifically,our work includes the following three aspects:(1)A Gaussian random field model was developed to investigate the dependen-cies across different age cohorts.The covariance of morality was gained under certain conditions.For a fixed age x,this model degenerated to the classical term structure model.Then,we brought up the?~2random field to overcome the disadvantage of the Gaussian random field,that the mortality is not definitely non-negative.Finally,the stochastic partial differential equation(SPDE)for the longevity bond price was obtained by risk-neutral pricing,this equation is similar to the classical Black-Scholes equation.(2)We study the pricing of longevity bonds by using the principle of equivalent utility in the incomplete market based on a stochastic string model.Stochastic string is a concept introduced from physics to model mortality to provide more covariance structures and shapes of the term structure.Brownian motion model,Gaussian random field model and etc.can be regarded as special cases of stochastic string model.We focus on the OU sheet model and modified chi~2random field model to gain the quadratic variation for mortalities between different age cohorts.In the case of fixed age,these two stochastic string models degenerate to affine term structure models.The dynamic evolutions of the two affine models were obtained and compared.In the incomplete financial market,we assume a constant interest rate,the price of longevity bonds and the optimal investment strategy of investors are obtained by using the equivalent utility principle.We demonstrate the difference between the indifference prices of investors and bond issuers.Finally,the SPDEs that the price of two kinds of longevity bonds satisfied under exponential utility and CRRA type power utility function are given.(3)We establish the term structure of interest rate to price the long term longevity bonds.Under this hypothesis,the price processes for risky assets,such as stocks and zero-coupon longevity bonds,are no longer independent.After the price of zero-coupon longevity bond is determined based on OU sheet model,we obtain the price of longevity bonds and the optimal investment strategy of investors by using the equivalent utility principle.Finally,the SPDEs that the price of two kinds of longevity bonds satisfied under exponential utility and CRRA type power utility are given.Compared with the results in(2),it can be seen that the setting of random interest rate has a great influence on the optimal investment strategy and the price of longevity bonds.