#### 期刊菜单

ARIMA族汇率挂钩型触发式理财产品定价研究
A Study on the Pricing of ARIMA Family-Linked Triggering Financial Products
DOI: 10.12677/AAM.2019.81013, PDF, HTML, XML, 下载: 1,141  浏览: 1,442  科研立项经费支持

Abstract: In this paper, triggering financial products are studied based on the data of dollar/yen exchange rate from 20 November to 25 November in 2017. Firstly, statistical analysis was conducted on the exchange rate data which is changed to stationary panel data by logarithmic difference method. Secondly, model MA(1) is chosen as the optimum time series model using AIC criterion and maxi-mum likelihood. Noting the fact that the data of residual do not follow normal distributions, we use inverse transformation to fit the residual data which is used to predict the price of risk assets. Finally, the value of triggering financial products is analyzed by the Monte-Carlo simulation method.

1. 引言

Table 1. Hook-type triggered financial products related information

2. 数据收集、整理及金融市场建模

Figure 1. Timing chart of USD/JPY spot exchange rate sequence

Figure 2. Sequence timing diagram after exchange rate data logarithmic difference

Figure 3. Autocorrelation plot and partial autocorrelation plot

${X}_{t}=\mu +{\epsilon }_{t}-{\theta }_{1}{\epsilon }_{t-1}-\cdots -{\theta }_{q}{\epsilon }_{t-q}$ (1.1)

Figure 4. EACF fixed result graph

Table 2. MA(1), MA(7) Comparison of model fixed index

${X}_{t}={\epsilon }_{t}+0.5588{\epsilon }_{t-1}$ (1.2)

3. 基于拟合残差分布的逆变换方法

${x}_{1},{x}_{2},\cdots ,{x}_{n}$

Figure 5. Residual data density fitting and box plot

Figure 6. Corrected residual density fitting and empirical distribution function graph

${F}_{n}\left(x\right)=\left\{\begin{array}{l}0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x<{x}_{1}\\ \frac{i}{n},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{\left(i\right)}\le x<{x}_{\left(i+1\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,3,\cdots ,n-1\\ 1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\ge {x}_{\left(n\right)}\end{array}$ (2.1)

${F}_{n}\left(x\right)=\left\{\begin{array}{l}\frac{i}{n},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x={x}_{\left(i\right)}\\ \frac{i}{n}\left(i+1-x\right)+\frac{i+1}{n}\left(x-i\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}i (2.2)

(2.2)式为插值后的经验分布函数，采用此逼近的反函数，用修正后的经验分布代替分布函数，并用逆变换的方法产生随机数进行拟合(如图6右)，进而对相应的理财产品进行分析。

4. 理财产品价格的Monte-Carlo模拟

${F}_{\text{pay-off}}=\left(1+{r}_{1}T\right){I}_{\left\{\underset{\tau \in \left[t,T\right]}{\mathrm{max}}S\left(\tau \right)>{p}_{1}\right\}}+\left(1+{r}_{2}T\right){I}_{\left\{{p}_{2}\le \underset{\tau \in \left[t,T\right]}{\mathrm{max}}S\left(\tau \right)<{p}_{1}\right\}}+\left(1+{r}_{3}T\right){I}_{\left\{\underset{\tau \in \left[t,T\right]}{\mathrm{max}}S\left(\tau \right)<{p}_{2}\right\}}$ (3.1)

$\upsilon =\mathrm{exp}\left\{\left(-rT\right)\right\}E\left[{F}_{\text{pay-off}}\right]$ (3.2)

Table 3. Financial products revenue statement

1) 汇率数据的模拟

${s}_{j}\left({t}_{1}\right),{s}_{j}\left({t}_{2}\right),{s}_{j}\left({t}_{3}\right),\cdots ,{s}_{j}\left(tn\right)$

2) 计算路径 ${S}_{j}\left({t}_{0}\right),{S}_{j}\left({t}_{1}\right),{S}_{j}\left({t}_{2}\right),\cdots ,{S}_{j}\left({t}_{n}\right)$ 下的理财产品的收益

${F}_{\text{pay-off}}=\left(1+{r}_{1}T\right){I}_{\left\{\underset{\tau \in \left[t,T\right]}{\mathrm{max}}S\left(\tau \right)>{p}_{1}\right\}}+\left(1+{r}_{2}T\right){I}_{\left\{{p}_{2}\le \underset{\tau \in \left[t,T\right]}{\mathrm{max}}S\left(\tau \right)<{p}_{1}\right\}}+\left(1+{r}_{3}T\right){I}_{\left\{\underset{\tau \in \left[t,T\right]}{\mathrm{max}}S\left(\tau \right)<{p}_{2}\right\}}$

3) 重复以上步骤(1)和步骤(2)，共N次(不少于10,000次)

4) 通过以下公司计算理财产品定价的期望收益率

$E\left[{F}_{\text{pay-off}}\right]=\frac{1}{N}\underset{j=1}{\overset{N}{\sum }}{F}_{\text{pay-off}}\left(j\right)$ (3.3)

5) 按照无风险利率r进行贴现，获得理财产品在到期时刻的价格为

$\upsilon =\mathrm{exp}\left\{\left(-rT\right)\right\}E\left[{F}_{\text{pay-off}}\right]$

5. 理财产品价值分析

Table 4. Price of wealth management products linked to the spot exchange rate value of USD/JPY

6. 总结

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