Grid the profit: Adaptive Periodic Grid Model with ARIMA Prediction
网格带来利润采用 ARIMA 预测的自适应周期网格模型

Contents  目录

1 Introduction  1 引言

1.1 Problem Background  1.1 问题背景

1.2 Restatement of the problem
1.2 问题的重述

• Using the officially provided dataset, develop a mathematical model to describe the price change trend and give the best daily strategy based only on price data up to the current day. In addition, the model should use the strategy which maximize the final profit as of 2021/9/10 with an initial $1000 investment.
  使用官方提供的数据集，建立一个数学模型来描述价格变化趋势，并仅根据截至当日的价格数据给出最佳每日策略。此外，该模型应使用初始投资额为 1000 美元、截至 2021/9/10 年最终利润最大化的策略。
• Compare the performance of different trading strategies using a noisy dataset to illustrate the advantage of our model.
  使用噪声数据集比较不同交易策略的性能，以说明我们模型的优势。
• Test our model with different transaction cost rates and compare the final results for different cost rates and evaluate the stability of our model.
  用不同的交易费率测试我们的模型，比较不同费率下的最终结果，评估模型的稳定性。
• Write a two-page memorandum. This memorandum is used to communicate the strategy, model and results with the trader.
  撰写一份两页的备忘录。这份备忘录用于与交易员交流策略、模型和结果。

1.3 Literature Review  1.3 文献综述

1.4 Our work && model overview
1.4 我们的工作&&模式概述

The following is a flow chart of our model framework.
以下是我们的模型框架流程图。

2 Assumptions  2 假设

• Assumption 1. We can trade any amount of gold or Bitcoin on any day with the given price.
  假设 1.我们可以在任何一天以给定的价格交易任意数量的黄金或比特币。
• Reason 1. Following the requirement in the question sheet, we have only one price on certain trading. The initial funding is $\mathrm{\$}1000$$\mathrm{\$}1000$$1000\$ 1000, which is so little that has almost no effect on the price of gold or Bitcoin in the exchange, even though we have earned some times profit.
  原因 1.根据问题单的要求，我们在某些交易中只有一个价格。初始资金是 $\mathrm{\$}1000$$\mathrm{\$}1000$$1000\$ 1000 ，它是如此之少，以至于对交易所中黄金或比特币的价格几乎没有影响，尽管我们已经赚取了一些利润。
• Assumption 2. The trend of daily price has periodicity to a certain degree.
  假设 2.每日价格趋势具有一定的周期性。
• Reason 2. There exist economic cycles in years, because of the production cycle in the whole society.
  原因 2.由于整个社会的生产周期，每年都存在经济周期。
• Assumption 3. The prices around a certain day have some similarities so that we can give a reasonable prediction from the price a few days ago.
  假设 3.某一天前后的价格具有一定的相似性，因此我们可以根据几天前的价格做出合理的预测。
• Reason 3. The market has inertia to keep the trend unless some unpredictable incidents occur. Hence, this prediction might be inaccurate.
  原因 3.市场有保持趋势的惯性，除非发生一些不可预测的事件。因此，这种预测可能不准确。

3 Notations  3 备注

• Time series  时间序列

• Vanilla Grid Strategy  香草网格战略

• Adaptive Periodic Grid Model
  自适应周期网格模型

4 Vanilla Grid Strategy
4 香草网格战略

4.1 The fundamental of Grid Strategy
4.1 电网战略的根本

Vanilla Grid Strategy, which is known as the Forex grid trading strategy[1], is a technique that seeks to make a profit on the natural movement of the market by positioning buy stop orders and sell stop orders. This is performed on a predefined market distance with a preset size of take-profit and no stop-loss. The grid is divided by a sequence of assessments of current prices, from high to low, expressing expectations for long-term future prices.
香草网格策略（Vanilla Grid Strategy），即外汇网格交易策略[1]，是一种通过定位买入止损单和卖出止损单，寻求在市场自然波动中获利的技术。这是在预先确定的市场距离上进行的，有预设的止盈规模，没有止损。网格由当前价格从高到低的评估序列划分，表达了对未来长期价格的预期。

Algorithm 1: Brief algorithm framework for grid strategy
        Input : A price sequence \(\left\{p^{\left\{T_{\text {start }}\right\}}, \cdots, p^{\left\{T_{\text {start }}\right\}}\right\}\) from \(T_{\text {start }}\) to \(T_{\text {end }}\) and parameters of grid strategy
        \(p_{\text {max }}, p_{\text {min }}, r_{0}\)
    Output: A decision sequence \(\left\{a^{\left\{T_{\text {start }}\right\}}, \cdots, a^{\left\{T_{\text {start }}\right\}}\right\}\)
    1. Calculate maximum number of grids num \(_{0} \leftarrow\) CalculateMaxGrid \(\left(p_{\max }, p_{\min }, r_{0}\right)\)
    2. \(n и т \leftarrow \min \left\{\right.\) num \(_{\text {in }}\), num \(\left._{0}\right\}\)
    3. Calculate the price of each grid
    4. Initialize the property portfolio in the first day
    5. Set \(t \leftarrow T_{\text {start }}+1\)
    6. If \(p^{\{t\}}>p^{\{t-1\}} \& \&\) Cross \(k\) grids then \(a^{\{t\}} \leftarrow\) Sell \(k\) units
    7. Else If \(p^{\{t\}}<p^{\{t-1\}} \& \&\) Cross \(k\) grids then \(a^{\{t\}} \leftarrow\) Buy \(k\) units
    9. Else \(a^{\{t\}} \leftarrow\) Hold
    8. \(t \leftarrow t+1\)
    10. If \(t \leqslant T_{\text {end }}\) then Go back to Step 6 .
    11. End

4.2 Isometric Grid and Proportional Grid
4.2 等距网格和比例网格

Then, the following Figure 2 shows an example of Isometric Model and Proportional Model.
下图 2 展示了等距模型和比例模型的示例。

Figure 2: The difference between ISO and Pro
图 2：ISO 和 Pro 之间的区别
Finally, we attempt to use the basic grid strategy to trade Bitcoin for six months. Both results of the isometric and proportional grid are displayed in the following Figure 3.
最后，我们尝试使用基本网格策略进行为期六个月的比特币交易。等距网格和比例网格的结果都显示在下图 3 中。

5 Adaptive Periodic Grid Model
5 自适应周期网格模型

This model is similar to the moving average filter in the Fourier transform, which smoothes sharply changing time series. Furthermore, the advantage of MA is that this model not only removes the noise part but also preserves the overall trend, which can avoid meaningless trades. The parameter $N$$N$NN represents the lag-time of the MA model and we initialize $N$$N$NN to 30 in our model. The following Figure 4 provides a demo for moving average line and the adaptive movement of the grid through time.
这种模型类似于傅立叶变换中的移动平均滤波器，可以平滑急剧变化的时间序列。此外，MA 模型的优势还在于它不仅能去除噪声部分，还能保留整体趋势，从而避免无意义的交易。参数 $N$$N$NN 代表 MA 模型的滞后时间，我们在模型中将 $N$$N$NN 初始化为 30。下图 4 演示了移动平均线和网格随时间的自适应移动。

5.1 ARIMA

5.1.1 Model Settings of ARIMA
5.1.1 ARIMA 的模型设置

5.1.2 ACF and PACF function - p,q selection
5.1.2 ACF 和 PACF 函数 - p,q 选择

5.1.3 $$$$quad\quad q-test  5.1.3 $$$$quad\quad Q 测试

5.1.4 Results in ARIMA prediction
5.1.4 ARIMA 预测的结果

Following the above theory, we apply ARIMA to predict the price of Bitcoin. To satisfy the requirement in this question, we only use the nearest 30 -day data to train our model with the parameter vector $(p,q,d)=(6,9,1)$$(p,q,d)=(6,9,1)$(p,q,d)=(6,9,1)(p, q, d)=(6,9,1) and predict the price of the next day. Our result of prediction is shown as the following Figure 5.
根据上述理论，我们应用 ARIMA 来预测比特币的价格。为了满足这个问题的要求，我们只使用最近 30 天的数据来训练模型，使用参数向量 $(p,q,d)=(6,9,1)$$(p,q,d)=(6,9,1)$(p,q,d)=(6,9,1)(p, q, d)=(6,9,1) ，并预测第二天的价格。我们的预测结果如下图 5 所示。

5.2 APGM  5.2 亚太性别平等主流化协会