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The fair valuation problem of guaranteed annuity options:
The stochastic mortality environment case
Laura Ballotta ,Steven Haberman
• 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
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.
• Assumption:1. The mortality risk is independent of the financial
risk.2. Single premium S0(ignore any expense)
3. The market is frictionless and competitive market with continuous trading
• Model1. Heath-Jarrow-Morton for interest rate2.Bullotta and Haberman for mortality intensity
riskmortality andrisk financial
:risk of sources by two affected iscontract GAO that theshow (3)~Eqs.(1)
where
(3) ˆ
is xageder then policyhold aby 0 at time enteredcontract GAO theof Tt0 at time valueThe
y. ageder policyhold a of lifetime remaining the
ngrepresenti r.v. a is andpayment annuity theof times theare T where
(2) ˆ
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age, retirement normal theis N 0, at time xaged iser policyhold theIf
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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”
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