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Pricing Maturity Guarantee with Dynamic Living Benefit

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Pricing Maturity Guarantee with Dynamic Living Benefit

- 숭실대학교
- 정보통계 보험수리학과
- 고방원
- [email protected]

- 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

- F(t) : 시간 t에서의 unprotected펀드계좌의 잔고
- 시간 t에서의 DFP펀드계좌의 잔고

F(0)

Protection Level K

F(t)

Time t

- 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

- 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.

- 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).

- 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

- 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.

0

0

t

t

- 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.

- 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:

- Pricing formula for DFP – Gerber & Pafumi (2000)

- 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.

- 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

- 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

DLB payment level

B

F(0)

Protection Level K

Deficit covered by protection issuer

- F(t) : 시간 t에서의 펀드계좌의 잔고
- 시간 t에서의 DLB를 지급하는펀드계좌의 잔고
- F(t)와 의 관계식
- 이 성립함

- 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.

- 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.

- VL 공식
- By the fundamental theorem of asset pricing,
- 여기서, Q는 Equivalent Martingale Measure, 은 drift가 반대부호

- VP 공식

- 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.

- 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.

- 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)

- 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

- Numerical Illustration – 3 (r = 5%, σ = 20%)
- Table.VP(K, B, T)와 DFP의 가격비교

- 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

- Asymptotic Result
- By the asymptotic formula in Abramowitz and Stegun (1972), it can be shown that for 0 < K≤ 1,

- 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

- 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

- 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.

- 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.