Pricing Maturity Guarantee with Dynamic Living Benefit
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Pricing Maturity Guarantee with Dynamic Living Benefit. 숭실대학교 정보통계 보험수리학과 고방원 [email protected] I-1. Dynamic Fund Protection. 풋옵션을 업그레이드한 보증유형 (A Strengthened Version of Put Option) 계약기간 동안 펀드 계좌의 금액이 보증수준 K 이하로 떨어지지 않도록 보증 펀드 계좌의 금액이 K 이하가 되면 보증 판매자는 적당한 금액을 즉시 펀드에 추가하도록 설계

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Pricing maturity guarantee with dynamic living benefit

Pricing Maturity Guarantee with Dynamic Living Benefit

  • 숭실대학교

  • 정보통계 보험수리학과

  • 고방원

  • [email protected]


I 1 dynamic fund protection

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


I 2 dynamic fund protection

I-2.Dynamic Fund Protection

  • F(t) : 시간 t에서의 unprotected펀드계좌의 잔고

  • 시간 t에서의 DFP펀드계좌의 잔고

F(0)

Protection Level K

F(t)

Time t


I 3 dynamic fund protection

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


I 4 dynamic fund protection

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.


I 5 dynamic fund protection

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).


I 6 dynamic fund protection

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


I 7 dynamic fund protection

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.


I 8 dynamic fund protection

I-8.Dynamic Fund Protection

0

0

t

t


I 9 dynamic fund protection

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.


I 10 dynamic fund protection

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:


I 11 dynamic fund protection

I-11.Dynamic Fund Protection

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


I 12 dynamic fund protection

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.


I 13 dynamic fund protection

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


Ii 1 maturity guarantee with dlb

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


Ii 2 maturity guarantee with dlb

II-2.Maturity Guarantee with DLB

DLB payment level

B

F(0)

Protection Level K

Deficit covered by protection issuer


Ii 3 maturity guarantee with dlb

II-3.Maturity Guarantee with DLB

  • F(t) : 시간 t에서의 펀드계좌의 잔고

  • 시간 t에서의 DLB를 지급하는펀드계좌의 잔고

  • F(t)와 의 관계식

  • 이 성립함


Ii 4 maturity guarantee with dlb

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.


Ii 5 maturity guarantee with dlb

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.


Ii 6 maturity guarantee with dlb

II-6.Maturity Guarantee with DLB

  • VL 공식

  • By the fundamental theorem of asset pricing,

  • 여기서, Q는 Equivalent Martingale Measure, 은 drift가 반대부호


Ii 7 maturity guarantee with dlb

II-7.Maturity Guarantee with DLB

  • VP 공식


Ii 8 maturity guarantee with dlb

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.


Ii 9 maturity guarantee with dlb

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.


Ii 10 maturity guarantee with dlb

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)


Ii 11 maturity guarantee with dlb

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


Ii 12 maturity guarantee with dlb

II-12.Maturity Guarantee with DLB

  • Numerical Illustration – 3 (r = 5%, σ = 20%)

  • Table.VP(K, B, T)와 DFP의 가격비교


Ii 13 maturity guarantee with dlb

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


Ii 14 maturity guarantee with dlb

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,


Ii 15 maturity guarantee with dlb

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


Ii 16 maturity guarantee with dlb

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


Pricing maturity guarantee with dynamic living benefit

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.


Pricing maturity guarantee with dynamic living benefit

  • 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.


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