1. The fair valuation problem of guaranteed annuity options: The stochastic mortality environment case Laura Ballotta,Steven Haberman

2. 1. introduction 2. A valuation approach for guaranteed annuity options 3. A stochastic approach to mortality risk: the basic model and its extensions 4. A model for the financial risk and the GAO valuation formula 5. Numerical calculations and sensitivity analysis

3. Guaranteed annuity option Guaranteed annuity option(GAO) is a contract giving the holder the right to receive at retirement the greater of (a) a cash payment equal to the current value of the investment in the equity fund, S, (b)the expected present value of the life annuity obtained by converting this investment at the guaranteed rate.

4. Assumption: 1.The mortality risk is independent of the financial risk. 2.Single premium S 0 (ignore any expense) 3.The market is frictionless and competitive market with continuous trading Model 1. Heath-Jarrow-Morton for interest rate 2.Bullotta and Haberman for mortality intensity

7. Ornstein-Uhlenbeck model has the desirable property of mean reversion

11. 對於評價與死亡力相關的商 品，不能在 ”risk neutral measure under financial risk”, 而是應該找出一個 ”risk neutral measure under mortality risk”