﻿ 含有多种最低利益保证的变额年金定价
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 同济大学学报(自然科学版)  2019, Vol. 47 Issue (9): 1375-1382.  DOI: 10.11908/j.issn.0253-374x.2019.09.020 0

### 引用本文

DONG Bing, XU Wei. Pricing Variable Annuities Embedding Various Guaranteed Minimum Benefits[J]. Journal of Tongji University (Natural Science), 2019, 47(9): 1375-1382. DOI: 10.11908/j.issn.0253-374x.2019.09.020

### 文章历史

Pricing Variable Annuities Embedding Various Guaranteed Minimum Benefits
DONG Bing , XU Wei
School of Mathematical Sciences, Tongji University, Shanghai 200092, China
Abstract: With the acceleration of population aging in China, the variable annuities have drawn a lot of attentions due to its anti-inflation, pension and investment functions. This paper studies the valuation of variable annuities embedding various guaranteed minimum benefits. We propose a willow tree method for pricing variable annuities under the jump-diffusion model in three guarantee types, including return of premium, Roll-up and Ratchet types. Our tree structure can be easily extended to other stochastic models. The willow tree construction and pricing procedure are independent of each other. Thus, for different stochastic models, it does not need extra work and has better applicability. Finally, numerical experiments demonstrate the accuracy and high efficiency of the proposed method compared with Monte Carlo method.
Key words: variable annuity valuation    guaranteed minimum benefits    jump-diffusion model    willow tree method

1 变额年金合约介绍

 $\begin{array}{*{20}{l}} {{V_0}(\xi ) = \sum\limits_{n = 1}^N {{{\tilde P}_{{x_0},{t_{n - 1}}}} \cdot {{\tilde Q}_{{x_0} + {t_{n - 1}},\varDelta t}} \cdot } }\\ \begin{array}{l} {\mathit{\boldsymbol{E}}_Q}\left[ {{{\rm{e}}^{ - rT}}\left( {{W_T}\left( {{t_n};\xi } \right) + {D_T}\left( {{t_n};\xi } \right)} \right)} \right] + \\ {{\tilde P}_{{x_0},T}}{\mathit{\boldsymbol{E}}_Q}\left[ {{{\rm{e}}^{ - rT}}\left( {{L_T}(T + 1;\xi ) + {W_T}(T + 1;\xi )} \right)} \right] \end{array} \end{array}$ (1)

2 柳树法定价变额年金

2.1 风险资产价格柳树的构建

 图 1 含有4个时刻和5个空间节点的柳树示意 Fig.1 Graphical depiction of the willow tree lattice with 4 time nodes and 5 space nodes

 $\begin{array}{*{20}{c}} {\frac{{{\rm{d}}S(t)}}{{S(t)}} = (r - \lambda \bar k){\rm{d}}t + \sigma {\rm{d}}B(t) + }\\ {\left[ {Y(t) - 1} \right]{\rm{d}}N(t)} \end{array}$ (2)

2.1.1 资产价格估计

 $\left\{ \begin{array}{l} \begin{array}{*{20}{l}} {\mu = \left( {r - {\sigma ^2}/2 - \lambda \bar k + \lambda {\alpha _J}} \right)t}\\ {v = \left( {{\sigma ^2} + \lambda \alpha _J^2 + \lambda \sigma _J^2} \right)t} \end{array}\\ {\kappa _3} = \frac{1}{{\sqrt t }}\left( {\frac{{\lambda \left( {\alpha _J^3 + 3{\alpha _J}\sigma _J^2} \right)}}{{{{\left( {{\sigma ^2} + \lambda \alpha _J^2 + \lambda \sigma _J^2} \right)}^{3/2}}}}} \right)\\ {\kappa _4} = 3 + \frac{1}{t}\left( {\frac{{\lambda \left( {\alpha _J^4 + 6\alpha _J^2\sigma _J^2 + 3\sigma _J^4} \right)}}{{{{\left( {{\sigma ^2} + \lambda \alpha _J^2 + \lambda \sigma _J^2} \right)}^2}}}} \right) \end{array} \right.$

 $X_n^i = \varepsilon {g^{ - 1}}\left( {\frac{{{z_i} - \gamma }}{\delta }} \right) + \nu$

2.1.2 转移概率计算

 $\begin{array}{*{20}{c}} {p_{ij}^n = P\left( {A < X_{n + 1}^j < B|X_n^i} \right) = }\\ {\int_A^B {\sum\limits_{l = 0}^\infty {\frac{{{{\rm{e}}^{ - \lambda \varDelta t}}{{(\lambda \varDelta t)}^l}}}{{l!}}} } \frac{1}{{\sqrt {2{\rm{ \mathsf{ π} }}} {\sigma _l}}}\exp \left( { - \frac{{{{\left( {x - {\mu _l}} \right)}^2}}}{{2\sigma _l^2}}} \right){\rm{d}}x} \end{array}$

2.2 变额年金的定价方法

 $A_n^ - = A_{n - 1}^ + \frac{{{S_n}}}{{{S_{n - 1}}}}{{\rm{e}}^{ - \alpha \varDelta t}}$
 $A_n^ + = \max \left\{ {A_n^ - - {\xi _n},0} \right\}$

 $A_n^ - = \max \left\{ {{A_0}{S_n}{{\rm{e}}^{ - n\varDelta t}} - {A_0}\sum\limits_{k = 1}^{n - 1} {{\xi _k}} {{\left( {y{{\rm{e}}^{ - \alpha \varDelta t}}} \right)}^{n - k}},0} \right\}$
 $\begin{array}{l} A_n^ + = \max \left\{ {A_n^ - - {\xi _n},0} \right\} = \\ \;\;\;\;\;\;\;\;\max \left\{ {{A_0}{S_n}{{\rm{e}}^{ - n\alpha \varDelta t}} - {A_0}\sum\limits_{k = 1}^n {{\xi _k}{{\left( {y{{\rm{e}}^{ - \alpha \varDelta t}}} \right)}^{n - k}}} ,0} \right\} \end{array}$

t=t1, 平均收益yS1, A1=A0yeαΔt, A1+=max{A1ξ1, 0}=max{A0yeαΔtξ1, 0}.

 $A_{n - 1}^ - = \max \left\{ {{A_0}{S_{n - 1}}{{\rm{e}}^{ - (n - 1)\alpha \varDelta t}} - {A_0}\sum\limits_{k = 1}^{n - 2} {{\xi _k}} {{\left( {y{{\rm{e}}^{ - \alpha \varDelta t}}} \right)}^{n - k - 1}},0} \right\}$
 $\begin{array}{l} A_{n - 1}^ + = \max \left\{ {A_{n - 1}^ - - {\xi _{n - 1}},0} \right\} = \\ \;\;\;\;\;\;\;\;\;\max \left\{ {{A_0}{S_n}{{\rm{e}}^{ - n\alpha \varDelta t}} - {A_0}\sum\limits_{k = 1}^n {{\xi _k}} {{\left( {y{{\rm{e}}^{ - \alpha \varDelta t}}} \right)}^{n - k}},0} \right\} \end{array}$

 $A_n^ - = \max \left\{ {{A_0}{S_n}{{\rm{e}}^{ - n\alpha \varDelta t}} - {A_0}\sum\limits_{k = 1}^{n - 1} {{\xi _k}} {{\left( {y{{\rm{e}}^{ - \alpha \varDelta t}}} \right)}^{n - k}},0} \right\}$
 $\begin{array}{l} A_n^ + = \max \left\{ {A_n^ - - {\xi _n},0} \right\} = \\ \;\;\;\;\;\;\;\;\max \left\{ {{A_0}{S_n}{{\rm{e}}^{ - n\alpha \varDelta t}} - {A_0}\sum\limits_{k = 1}^n {{\xi _k}} {{\left( {y{{\rm{e}}^{ - \alpha \varDelta t}}} \right)}^{n - k}},0} \right\} \end{array}$

 $A_n^{i - } = \max \left\{ {{A_0}S_n^i{{\rm{e}}^{ - n\alpha \varDelta t}} - {A_0}\sum\limits_{k = 1}^{n - 1} {{\xi _k}} {{\left( {y{{\rm{e}}^{ - \alpha \varDelta t}}} \right)}^{n - k}},0} \right\}$ (3)
 $\begin{array}{l} A_n^{i + } = \max \left\{ {A_n^{i - } - {\xi _n},0} \right\} = \\ \;\;\;\;\;\;\;\;\;\max \left\{ {{A_0}S_n^i{{\rm{e}}^{ - n\alpha \varDelta t}} - {A_0}\sum\limits_{k = 1}^{n - 1} {{\xi _k}} {{\left( {y{{\rm{e}}^{ - \alpha \varDelta t}}} \right)}^{n - k}},0} \right\} \end{array}$ (4)

(1) 在初始时刻t=0, 死亡收益和累积提取收益为零, 即D0+=0, W0+=0.如果合约含有相应的最低利益保证, 则G0D/W/A/I+=A0, G0E+=XwG0W.反之, 对应的最低利益保证为零.

(2) 如果投保者在(tn－1, tn](1≤nN)内死亡, 需要计算DT(tn; ξ)和WT(tn; ξ)的值.在tn时刻, 投资账户价值Ani可以根据式(3)、(4)计算.如果投保者在(tn－1, tn]内死亡, 当风险资产价格为Sni时, 会得到收益为maxGn, iD, Ani.所以, 对应的死亡收益DTi(tn; ξ)为

 $D_T^i\left( {{t_n};\xi } \right) = {{\rm{e}}^{r(N - n)\varDelta t}}\max \left\{ {G_{n,i}^{D - },A_n^{i - }} \right\},i = 1,2, \cdots ,m$

 ${W_T}\left( {{t_n};\xi } \right) = W_{n - 1}^ + {e^{r(N - n + 1)\varDelta t}}$

(3) 若投保者在(tn－1, tn]时间内存活, tn时刻投保者从账户中提取ξn, 累积提取收益Wn

 $W_n^ - = W_{n - 1}^ + {{\rm{e}}^{r\varDelta t}}$
 $W_n^ + = \left\{ {\begin{array}{*{20}{l}} {W_n^ - + {\xi _n},}&{{\xi _n} \le {E_n}}\\ {W_n^ - + {E_n} + (1 - \eta )\left( {{\xi _n} - {E_n}} \right),}&{{\xi _n} > {E_n}} \end{array}} \right.$

(4) 在合约到期日T, 如果投保者仍然存活, 累积提取收益为

 ${W_T}(T + 1;\xi ) = W_N^ +$

 $L_T^i(T + 1;\xi ) = L_T^{A + }\;或\;L_T^{I + },\quad i = 1,2, \cdots ,m$

 $G_n^{{\rm{D}} - } = G_{n - 1}^{{\rm{D}} + } = \frac{{A_n^ - }}{{{{\rm{e}}^{ - n\alpha \varDelta t}}{S_n}}}$
 $G_n^{{\rm{D}} + } = G_n^{{\rm{D}} - }\frac{{A_n^ + }}{{A_n^ - }} = \frac{{A_n^ + }}{{{{\rm{e}}^{ - n\alpha \varDelta t}}{S_n}}}$

t0时刻, G0D+=A0.

tn－1时刻, 若$G_{n-1}^{\text{D}-}=\frac{A_{n-1}^{-}}{{{\text{e}}^{-\left( n-1 \right)\alpha \varDelta t}}{{S}_{n-1}}}$成立, 则在tn时刻, 有

 $G_n^{{\rm{D}} - } = G_{n - 1}^{{\rm{D}} + } = \frac{{A_{n - 1}^ + }}{{{{\rm{e}}^{ - (n - 1)\alpha \varDelta t}}{S_{n - 1}}}} = \frac{{A_n^ - \frac{{{S_{n - 1}}}}{{{S_n}}}{{\rm{e}}^{\alpha \varDelta t}}}}{{{{\rm{e}}^{ - (n - 1)\alpha \varDelta t}}{S_{n - 1}}}} = \frac{{A_n^ - }}{{{{\rm{e}}^{ - n\alpha \varDelta t}}{S_n}}}$
 $G_n^{{\rm{D}} + } = G_n^{{\rm{D}} - }\frac{{A_n^ + }}{{A_n^ - }} = \frac{{A_n^ + }}{{{{\rm{e}}^{ - n\alpha \varDelta t}}{S_n}}}.$

 $\left\{ {\begin{array}{*{20}{l}} {G_{n,i}^{{\rm{D}}/{\rm{A}}/{\rm{I}} - } = \frac{{A_n^{i - }}}{{{{\rm{e}}^{ - n\alpha \varDelta t}}S_n^i}}}\\ {G_{n,i}^{{\rm{D}}/{\rm{A}}/{\rm{I}} + } = \frac{{A_n^{i + }}}{{{{\rm{e}}^{ - n\alpha \varDelta t}}S_n^i}}} \end{array}} \right.$ (5)

 $G_{n,i}^{{\rm{D}}/{\rm{A}}/{\rm{I}} - } = \frac{{A_n^{i - }{{\left( {1 + {i_r}} \right)}^{\tilde n}}}}{{{{\rm{e}}^{ - n\alpha \varDelta t}}S_n^i}},\;\;\;\;G_{n,i}^{{\rm{D}}/{\rm{A}}/{\rm{I + }}} = \frac{{A_n^{i + }{{\left( {1 + {i_r}} \right)}^{\tilde n}}}}{{{{\rm{e}}^{ - n\alpha \varDelta t}}S_n^i}}.$

 $\tilde n = \left\{ {\begin{array}{*{20}{l}} {n,\;\;\;\;\;\;\;\;如果\;{t_n}\;前没有取款}\\ {{n_1} - 1,\;\;\;如果\;{t_{{n_1}}}\;时刻第1次取款} \end{array}} \right.$

GnD/A/I－=Gn－1D/A/I+, GnD/A/I+= max{ GnD/A/I+·$\frac{A_{n}^{+}}{A_{n}^{-}}, A_{n}^{+}$}.

n=1时, 有

 $G_{1,i}^{{\rm{D}} - } = G_0^{{\rm{D}} + } = {A_0}$

n>1时, 最低死亡利益保证估计为

 $G_{n,i}^{{\rm{D}} - } = \sum\limits_{j = 1}^m {p_{ji}^{n - 1}} G_{n - 1,j}^{{\rm{D}} + }$

 $G_{n,i}^{{\rm{D}} + } = \max \left\{ {G_{n,i}^{{\rm{D}} - } \cdot \frac{{A_n^{i + }}}{{A_n^{i - }}},A_n^{i - }} \right\}$

 $G_n^{{\rm{W}}/{\rm{E}} - } = G_{n - 1}^{{\rm{W}}/{\rm{E}} + }$
 $G_n^{{\rm{W}} + } = G_n^{{\rm{W}} - } - {\xi _n},\;\;\;\;G_n^{E + } = G_n^{E - } \cdot \frac{{G_n^{{\rm{W}} + }}}{{G_n^{{\rm{W}} - }}}$

 $\begin{array}{l} {V_0} = \sum\limits_{n = 1}^N {{{\tilde P}_{{x_0},{t_{n - 1}}}}} \cdot \\ \;\;\;\;\;{{\tilde Q}_{{x_0} + {t_{n - 1}},\varDelta t}}\sum\limits_{i = 1}^m {q_n^i} {{\rm{e}}^{ - rT}}\left[ {{W_T}\left( {{t_n}} \right) + D_T^i\left( {{t_n}} \right)} \right] + \\ \;\;\;\;\;{{\tilde P}_{{x_0},T}}\sum\limits_{i = 1}^m {q_N^i} {{\rm{e}}^{ - rT}}\left[ {L_T^i(T + 1) + {W_T}(T + 1)} \right] \end{array}$ (6)

 ${\mathit{\boldsymbol{q}}_n} = \left[ {q_n^1,q_n^2, \cdots ,q_n^m} \right] = {\mathit{\boldsymbol{q}}_{n - 1}} \cdot \left[ {p_{ij}^{n - 1}} \right]$

3 数值结果

 图 2 合约价值随保费的变化 Fig.2 Computed variable annuity values with various insurance fee

 图 3 合约价值随利率的变化 Fig.3 Computed variable annuity values with various interest rate

 图 4 合约价值随跳跃强度的变化 Fig.4 Computed variable annuity values with various intensity λ
 图 5 合约价值随模型参数σ的变化 Fig.5 Computed variable annuity values with various σ

 图 6 Roll-up型合约价值随保费的变化 Fig.6 Computed variable annuity values with the roll-up type with respect to insurance fee
 图 7 Ratchet型合约价值随保费的变化 Fig.7 Computed variable annuity values with the ratchet type with respect to insurance fee

4 结语

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