﻿ 跳扩散下汇率变动的外商直接投资问题研究
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Study of a foreign investor's investment with fluctuations of exchange rate under jump-diffusion
FEI Wei-yin, HE Dan-dan, ZHANG Wei
Department of Financial Engineering, Anhui Polytechnic University, Wuhu 241000, China
Abstract:This paper studies a foreign investor's direct investment with fluctuations of exchange rate under jump-diffusion environment. First, we obtain the dynamics of asset price denoted by native currency by using Itô formula. Then, under maximizing the expected utility of the terminal wealth, through using HJB equation, we obtain the optimal allocation strategy, and derive an approximate solution of the optimal dynamic asset allocation. Finally, we analyze the impacts of the jump and the fluctuations of exchange rate on the optimal allocation strategy of an investor through a numerical simulation.
Key words: jump-diffusion process     fluctuations of exchange rate     optimal portfolio     HJB equation     stochastic calculus

0 引言

 $\frac{{ d}H(t)}{H(t)}=h{ d}t+\delta { d}Z(t),H(0)=H_0$ (1)

 $\frac{{ d}P_t}{P_t}=(\mu_t-\lambda g){ d}t+\sigma { d}B(t)+({ e}^q-1){ d}Q(\lambda)$ (2)

 ${ Pr}(\tau \mbox{期限内$n$个跳})={ e}^{-\lambda\tau}\frac{(\lambda \tau)^n}{n!}$ (3)

 $\begin{array}{rll} { d}\hat{P}(t)&={ d}(H(t)P(t)) =H(t){ d}P(t)+P(t){ d}H(t)+{ d}H(t)P(t)\\ &=\hat{P}(t)(\mu_t-\lambda g+h+\delta\sigma\rho)+\hat{P}(\delta { d}Z(t)+\sigma { d}B(t))+\hat{P}(t)({ e}^q-1){ d}Q(\lambda)\\ &=\hat{P}(t)(\mu_t-\lambda g+h+\delta\sigma\rho)+\hat{P}(t)\sqrt{\sigma^2+\delta^2+2\delta\sigma\rho}{ d}\hat{B}(t)+\hat{P}(t)({ e}^q-1){ d}Q(\lambda), \end{array}$

 ${ d}P_0=rP_0{ d}t$ (4)

 $\eta_t=\frac{\mu_t-\lambda g+h+\delta\sigma\rho-r}{\sqrt{\sigma^2+\delta^2+2\delta\sigma\rho}}$ (5)

 $\begin{array}{rll} { d}W_t&=(1-\theta_t)W_t r{ d}t+\theta_tW_t[(\mu_t-\lambda g+h+\delta\sigma\rho-r){ d}t+\hat{\sigma}{ d}\hat{B}(t)+({ e}^q-1){ d}Q(\lambda)]\\ &=rW_t { d}t+\theta_tW_t[\hat\sigma\eta_t+\hat\sigma { d}\hat{B}(t)+({ e}^q-1){ d}Q(\lambda)] \end{array}$ (6)

 $J(W,H,\tau)=J(W_t,H_t,\tau)=\max\limits_{\theta_t}E_t[{ e}^{-r\tau}U(W_T)]$ (7)

 $\begin{array}{rll} 0=&\max\limits_{\theta_t}\bigg\{-J_\tau+\lambda E_t[J(W',H,\tau)-J(W,H,\tau)]+J_W rW+J_W \theta\hat{\sigma}\eta W+\\ &\frac{1}{2}J_{WW}\theta^2\hat{\sigma}^2W^2+J_H h+\frac{1}{2}J_{HH}\delta^2+J_{WH}\theta \hat{\sigma}\delta\hat{\rho}W\bigg\} \end{array}$ (8)

 $E[{ d}\hat{B}(t)Z(t)]=\hat{\rho}{ d}t=\frac{\delta+\sigma\rho}{\sqrt{\sigma^2+\delta^2+2\delta\sigma\rho}}$ (9)
(8)式中的$W'_t=W_t[1+\theta_t({ e}^q-1)]$是一个跳发生下的财富水平,当完全违约或破产的特殊情形时(即$q=-\infty$),投资者的财富减少到$W'_t=W_t[1-\theta_t]$. 假设投资者的效用函数满足$U'(0)=\infty$,$U'(\infty)=0$,则为了保证投资者的财富为正值,投资者不会将所有的资产全部配置在风险资产中,即在任何时候$\theta_t<100\%$.

 $\theta^*(W,H,\tau)=\bigg(\frac{-J_W}{J_{WW} W_t}\bigg)\frac{\eta_t}{\hat{\sigma}}+\bigg(\frac{-J_{WH}}{J_{WW} W_t}\bigg)\frac{\delta\hat{\rho}} {\hat{\sigma}}+\frac{\lambda E_t[J_W'(W',H,\tau)({ e}^q-1)]}{-J_{WW} W\hat{\sigma}^2}$ (10)

 $U(W)=\frac{W^{1-\alpha}}{1-\alpha},\alpha>0$ (11)

 $J(W,H,\tau)\approx\Phi(H,\tau)U({ e}^{r\tau}W)$ (12)

 $\Phi(H,\tau)=\mathrm{exp}(A(\tau)+B(\tau)H+C(\tau)H^2 /2)$ (13)

 $\frac{{ d}C}{{ d}\tau}=aC^2,\frac{{ d}B}{{ d}\tau}=aBC+bC,\frac{{ d}A}{{ d}\tau}=\frac{a}{2}B^2+(b+c\hat{\rho}\delta\eta_t)B+\frac{1}{2}\delta^2C+D_t$ (14)

 $\theta^*(W,H,\tau)=\frac{\eta_t\hat{\sigma}+\lambda\hat{g}_t+\hat{\sigma}\delta\hat{\rho}[B(\tau)+C(\tau)H]} {\alpha\hat{\sigma}^2}$ (15)

 $C(\tau)=-\frac{1}{a(\tau+1)},\ \ B(\tau)=-\frac{1-b(\tau+1)}{a(\tau+1)}$ (16)
1. 2 模型结论

 $\theta^*(W,H,\tau)=\frac{\eta_t\hat{\sigma}+\lambda\hat{g}_t+\hat{\sigma}\delta\hat{\rho}[B(\tau)+C(\tau)H]} {\alpha\hat{\sigma}^2},$

 图 1 即期预期汇率$h$对最优投资组合$\theta^*$的影响

 图 2 汇率波动率$\delta$对最优投资组合$\theta^*$的影响

 图 3 跳强度$\lambda$对最优投资组合$\theta^*$的影响

 图 4 跳大小$\mu_q$对最优投资组合$\theta^*$的影响

 图 5 跳波动率$\nu_q$对最优投资组合$\theta^*$的影响

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#### 文章信息

FEI Wei-yin, HE Dan-dan, ZHANG Wei

Study of a foreign investor's investment with fluctuations of exchange rate under jump-diffusion

Systems Engineering - Theory & practice, 2015, 35(2): 283-290.