Các ước tính được tiệm bình thường (đa biến bình thường nếu một vector), với nghĩa là bằng các tham số (s) là ước tính, và phương sai Tài sản của các ước lượng Khả năng xảy ra tối đa BẢNG 9,4 biện pháp rủi ro cho lợi ích GMDB VA-type, hợp đồng 30 năm, tỷ lệ phần trăm của giá trị quỹ ban đầu. | 140 DYNAMIC HEDGING FOR SEPARATE ACCOUNT GUARANTEES If in addition the management charge is set to zero H 0 t3 reduces to the form Ẵ0 Ps ti 1 P 10 2 - S0 We can split equation into the benefit due at each maturity or renewal or rollover date which allows us to apply survival probabilities. Furthermore we can generalize to include the death benefit under the GMAB contract. On death between t1 and t2 say the insurer is liable for the first rollover benefit at t1 as part of the survival benefit the insurer is also liable for the guarantee liability at the date of death when the amount due is the guarantee which has been reset at t1 less the fund value at t. We define Pk t for t tk to be the option price at time t 0 for the survival benefit due at tk given that the policy is still in force at that time and Pk t for tk t tk 1 to be the option price at t 0 for the death benefit due if the life dies at time t after k rollovers. Then H 0 t3 P1 t1 P2 t2 P3 t3 and P1 t Ps t P2 t S0 1 - m. Ps t1 P t - t1 t t1 P3 t S0 1 - m t2 Ps t1 1 - m 1 So 1 - m 1 Ps t1 P t2 - t1 P t - t2 t t2 The only terms in H 0 t that involve the stock price s0 are Ps and the terms in s0 1 - m . The first is a straightforward put option and the derivative with respect to s0 was derived in Chapter 7 so deriving the split between stocks and bonds for the hedge portfolio for the GMAB is not difficult giving the stock part of the hedge at time t 0 as so d H 0 t3 So -d1 t1 s0 1 - m f1 Ơ so X Í1 P t2 - t1 1 P t3 - t2 1 - m t2 h P t3 - t2 - so 1 - m f1 Allowing for exits the cost of the GMAB survival benefit hedge for a policyholder age x assuming final maturity at age t3 is P1 t1 t1 px P2 t2 t2px P3 t3 t3Px For the additional death benefit the hedge price at time t 0 is r P1 w wpx ddwdw it2 P2 w wpx l wdw it3 P3 w wpx xwdw 0 t1 t2 Black-Scholes Formulae for Segregated Fund Guarantees 141 TABLE Example hedge price percentage of fund for GMAB death and survival benefit. Guarantee of Fund
