Pricing Maturity Guarantee with Dynamic Living Benefit

1. Pricing Maturity Guarantee with Dynamic Living Benefit • 숭실대학교 • 정보통계 보험수리학과 • 고방원 • bko@ssu.ac.kr

2. I-1.Dynamic Fund Protection • 풋옵션을 업그레이드한 보증유형 (A Strengthened Version of Put Option) • 계약기간 동안 펀드 계좌의 금액이 보증수준 K이하로 떨어지지 않도록 보증 • 펀드 계좌의 금액이 K이하가 되면 보증 판매자는 적당한 금액을 즉시 펀드에 추가하도록 설계 • 흔히 Reset Guarantee라 불림

3. I-2.Dynamic Fund Protection • F(t) : 시간 t에서의 unprotected펀드계좌의 잔고 • 시간 t에서의 DFP펀드계좌의 잔고 F(0) Protection Level K F(t) Time t

4. I-3.Dynamic Fund Protection • H. Gerber & E. S. W. Shiu(1998, 1999) • - Dynamic fund protection 도입 • - Perpetual protection • - Ruin theory approach • H. Gerber &G. Pafumi (2000) • - A closed form expression for finite time protection • - Geometric Brownian motion • J. Imai & P. P. Boyle (2001), H-K. Fung & L. K. Li (2003) • - CEV (Constant Elasticity of Variance) process • - Discretely monitored protection • - Numerical approach • H. Gerber & E. S. W. Shiu (2003) • - Dynamic fund protection with stochastic barrier • - Optimal exercise strategy

5. I-4.Dynamic Fund Protection • H. Gerber &G. Pafumi (2000)’s Assumption • Under Black Sholes Framework, assume • , W(t): Standard B.M. & F(0) ≥ K • All dividends are reinvested. • No transaction costs, no arbitrage opportunity etc. • The main idea of pricing DFP is the relationship between F(t) and such that • If drops to K, just enough money will be added so that does not fall below K.

6. I-5.Dynamic Fund Protection • More precisely, • Why? • Consider as the number of fund units. • Note that n(0) = 1 & n(t) is nondecreasing. • The equal sign is chosen to minimize the guarantee cost. • See Gerber & Shiu (2003).

7. I-6.Dynamic Fund Protection • An interpretation of the process • Consider when • After simple algebra, • By Graversen and Shiryaev (2000), we recognize as • a reflecting Brownian motion with drift = μ, volatility = σ,started at

8. I-7.Dynamic Fund Protection • A useful result about a reflecting B. M. with drift from Graversen and Shiryaev (2000) • For any • where satisfies the stochastic differential equation • Sometimes, |μt + W(t)| is called a reflecting B. M. with drift.

9. I-8.Dynamic Fund Protection 0 0 t t

10. I-9.Dynamic Fund Protection • For a reflecting B. M. with drift, an explicit expression of the transition density is available. • See Cox & Miller (1965) for the derivation. • Let denote the probability that a reflecting B.M. started at will be observed in the interval between x and • x + dx after time T.

11. I-10.Dynamic Fund Protection • Pricing formula for DFP – Gerber & Pafumi (2000) • By the fundamental theorem of asset pricing, • And, • After some tedious calculation, one may obtain the following formula:

12. I-11.Dynamic Fund Protection • Pricing formula for DFP – Gerber & Pafumi (2000)

13. I-12.Dynamic Fund Protection • Esscher Transform • Discussion paper by Y-C. Huang and E. S. W. Shiu (2000, NAAJ) derives the pricing formula by using the reflection principleand the method of EsscherTransforms.

14. I-13.Dynamic Fund Protection • Numerical Illustration – Table 3 from Gerber & Pafumi (2000) • When F(0) = 100, T = 1, σ = 0.2, r = 0.04 • Interesting Fact • One may verify that

15. II-1.Maturity Guarantee with DLB • Maturity Guarantee with Dynamic Living Benefit의 제안 • - 펀드의 잔고가 미리 정한 일정 수준 (B)을 넘어가면 그 초과액을 고객에게 배당금과 같은 형태로 바로 지급하고 만약 만기일에 펀드잔고가 보장수준 (K) 이하로 떨어지면 부족한 부분을 보장 • Maturity Guarantee with Dynamic Living Benefit의 제안 배경 • 변액연금에서GLB (Guaranteed Living Benefit) 상품인 GMWB, GMIB, GMAB의 선택비율이 높음 • Dynamic Fund Protection의 쌍대 (Dual) 문제로 명시적 가격 결정공식 유도가 가능 • B와 K를 동시에 조정,Dynamic Fund Protection보다 Cheap

16. II-2.Maturity Guarantee with DLB DLB payment level B F(0) Protection Level K Deficit covered by protection issuer

17. II-3.Maturity Guarantee with DLB • F(t) : 시간 t에서의 펀드계좌의 잔고 • 시간 t에서의 DLB를 지급하는펀드계좌의 잔고 • F(t)와 의 관계식 • 이 성립함

18. II-4.Maturity Guarantee with DLB • Under the same framework with Gerber and Pafumi (2000), • 0 < K≤ F(0) = 1 ≤ B • Denote k = lnK, b = lnB (k≤ 0 ≤b) • VL(B, T): time-0 value of the aggregate DLB payments • VP(K, B, T): time-0 value of the maturity guarantee with payoff • 6. Investor pays 1 + VP(K, B, T) at the beginning of the contract.

19. II-5.Maturity Guarantee with DLB • Similarly in DFP, • Thus, the process is a reflecting B. M. started at bwith drift (– μ), volatility σ, and reflecting barrier at 0. • The pricing formulas for VL(B, T) and VP(K, B, T) can be found by using the transition density.

20. II-6.Maturity Guarantee with DLB • VL 공식 • By the fundamental theorem of asset pricing, • 여기서, Q는 Equivalent Martingale Measure, 은 drift가 반대부호

21. II-7.Maturity Guarantee with DLB • VP 공식

22. II-8.Maturity Guarantee with DLB • In the derivation of the pricing formulas, we have used two extensions from Gerber and Pafumi (2000, NAAJ): • Similarly with DFP, • Because VP ≥ BSP, the sum of the last terms should always be positive.

23. II-9.Maturity Guarantee with DLB • The pricing formulas can be derived by using the method of Esscher Transforms. • The pricing formulas can be easily extended to the case with exponentially varying barriers.

24. II-10.Maturity Guarantee with DLB • Numerical Illustration – 1 (r = 5%, σ = 20%) VL(B, T) 1.0 0.8 : B = 1.0 :B = 1.5 : B = 2.0 : B = 2.5 0.6 0.4 0.2 0 20 40 60 80 Maturity (Years)

25. II-11.Maturity Guarantee with DLB • Numerical Illustration – 2 (r = 5%, σ = 20%) VP(K, B, T) 0.10 K = 1.0 0.08 K = 0.9 0.06 K = 0.8 0.04 K = 0.7 0.02 K = 0.6 0 Maturity (Years) 20 40 60 80

26. II-12.Maturity Guarantee with DLB • Numerical Illustration – 3 (r = 5%, σ = 20%) • Table.VP(K, B, T)와 DFP의 가격비교

27. II-13.Maturity Guarantee with DLB • Numerical Illustration – 4 (r = 5%, σ = 20%) • VP(K, B, T) 와 European Put Price의 가격비 B = 1.0 3 B = 1.1 B = 1.2 2 : K = 1.0 :K = 0.9 : K = 0.8 1 Maturity (Years) 0 4 8 12 16

28. II-14.Maturity Guarantee with DLB • Asymptotic Result • By the asymptotic formula in Abramowitz and Stegun (1972), it can be shown that for 0 < K≤ 1,

29. II-15.Maturity Guarantee with DLB • Numerical Illustration – 5 (r = 5%, σ = 20%) 0.4 VL(B, T = 5) 0.3 0.2 K = 1.0 Break-even if B = 2.01 K = 0.9 0.1 K = 0.8 VP(K, B, T = 5) 0 1 2 3 4 B

30. II-16.Maturity Guarantee with DLB • Future Research • For reflected processes more general than Brownian Motion, see Linetsky (2005). • What if reflection is replaced by refraction? See, for example, Gerber & Shiu (2006). Withdrawal Level Protection Level

31. 참고문헌 • Abramowitz, M. and Stegun, I. (1972) Handbook of Mathematical Functions. Dover Publications: New York • Cox, D. R. and Miller, H. (1965) The Theory of Stochastic Processes. Chapman & Hall • Fung, H-K. and Li, L. K. (2003) Pricing Discrete Dynamic Fund Protections. North American Actuarial Journal7(4): 23-31. • Graversen, S. E. and Shiryaev, A. N. (2000) An Extension of P. Lévy’s Distributional Properties to the Case of a Brownian Motion with Drift. Bernoulli 6(4): 615-620. • Gerber, H. U. and Pafumi, G. (2000) Pricing Dynamic Investment Fund Protection. North American Actuarial Journal 4(2): 28-37. Discussion Paper by Huang, Y-C. & Shiu, E. S. W.

32. Gerber, H. U. and Shiu, E. S. W. (1998) Pricing Perpetual Options for Jump • Processes. North American Actuarial Journal 2(3): 101-107. • Gerber, H. U. and Shiu, E. S. W. (1999) From Ruin Theory to Pricing Reset Guarantees and Perpetual Put Options. Insurance: Mathematics and Economics 24(1): 3-14. • Gerber, H. U. and Shiu, E. S. W. (2003) Pricing Perpetual Fund Protection with Withdrawal Option. North American Actuarial Journal7(2): 60-92. • Gerber, H. U. and Shiu, E. S. W. (2006) On Optimal Dividends: From Reflection to Refraction. Journal of Computational and Applied Mathematics 186: 4-22. • Imai, J. and Boyle, P. P. (2001) Dynamic Fund Protection. North American Actuarial Journal5(3): 31-51. • Linetsky, V. (2005) On the Transition Densities for Reflected Diffusions. Advances in Applied Probability 37: 435-460.