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### 2022年美赛获奖C类论文解析 #### 概述本文献《2022年美赛获奖C类论文_2212336.pdf》是一篇关于数学建模在金融交易策略中的应用研究，具体探讨了如何通过数学模型预测资产价格并制定有效的交易策略。该研究基于2022年的美国数学建模竞赛（简称“美赛”）背景，是当年竞赛中获得C类奖项的优秀作品之一。 #### 研究背景与目的随着经济全球化的发展和信息技术的进步，金融活动已经成为人们日常生活的重要组成部分。在金融市场中，投资者通过不同的资产选择和交易策略来实现利润最大化或风险最小化的目标。本文聚焦于发展和评估有效的交易策略，旨在为投资者提供一种基于数学模型的决策支持工具。 #### 主要研究方法研究采用了两种主要的数学模型来构建交易策略： 1. **波动性资产价格预测模型(Model I)**：该模型主要用于预测资产价格的变化趋势。研究者首先尝试了自回归整合移动平均模型(ARIMA)进行预测，发现该模型能够较好地捕捉资产价格的线性变化趋势；随后，引入长短时记忆网络(LSTM)模型来捕捉非线性特征，最终通过结合两种模型的优点形成ARIMA-LSTM混合模型。该模型在预测精度上表现优异，其均方根误差(RMSE)仅为0.0342，平均绝对百分比误差(MAPE)为0.36，预测结果几乎与实际价格走势重合。 2. **策略改进模型(Model II)**：为优化原始策略并发现更多投资机会，研究者采用遗传算法进行策略优化。通过对交易组合、交易频率、买入时机等四个基因进行操作，模拟进化过程来寻找最佳的交易参数。这种方法能够帮助识别被传统方法忽视的投资机会。 #### 实证分析在实证部分，研究者运用动态规划方法将长期投资问题分解为一系列短期决策问题，并通过MATLAB编程计算出具体的投资回报率。以2021年9月10日为例，假设初始资金为$374,563.25，经过五年时间的价值折现计算后，该金额相当于2016年9月10日的$360,595.32。这意味着，在五年的时间里，该策略带来了高达37456.33%的投资回报率。 #### 研究结论与启示本研究表明，通过合理运用数学建模技术，可以在金融交易领域实现显著的盈利。ARIMA-LSTM模型和遗传算法的结合不仅提高了预测精度，还优化了交易策略，从而为投资者提供了强大的决策支持工具。此外，该研究也为未来类似领域的研究提供了有益的参考案例。 #### 关键词总结 - **美赛**：指美国数学建模竞赛，是一项国际性的数学建模竞赛活动。 - **数学竞赛**：指利用数学方法解决实际问题的比赛。 - **获奖论文**：指在特定竞赛中获得认可的研究成果。 - **A,B,C,D,E,F类**：可能指的是不同级别的奖项类别。 - **英文**：表明该论文是以英语撰写。《2022年美赛获奖C类论文_2212336.pdf》通过对资产价格预测模型和交易策略改进模型的构建与优化，展示了数学建模在金融交易领域的重要应用价值，为投资者提供了一种有效的决策辅助手段。

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The Queen of Strategy: The Road to Counterattack With $1,000

Summary

Nowdays, financial activities have become an important part of people’s lives. In trad-

ing, some people end up making profits but some people lose money. The differences be-

tween the losers and the profiteers are not only for the choice of asset variety, but also for the

trading strategy. In this paper, we focus on the development and evaluation of strategies. For

this purpose, we designed two models to develop our strategies: Model I: Volatile Asset Price

Forecasting Model; Model II: Strategy Improvement Model.

For Model I, as a forecasting model, the accuracy of its results is relevant to the formula-

tion of our strategy and its returns. It is prudent to predict only one day ahead in a single run.

We first use the ARIMA model to forecast the price and find that it can only predict well

about the linear part of bitcoin price. Second, we take the LSTM model to forecast, and just

the opposite of the ARIMA model, the LSTM model is able to capture the information of the

non-linear part of the asset price relatively well; finally, we combine them to form ARIMA-

LSTM model, and the final price prediction results almost overlap with the real results, with

RMSE and MAPE were 0.0342 and 0.36, respectively.

Second, to develop the strategy, we used the dynamic programming method to split the

five-year-long investment strategy problem into smaller problems of daily investment deci-

sions. After programming using MATLAB, we calculated the value of the initial money as

$374,563.25 on 9/10/21 and discounted this value to five years ago to obtain $360,595.32,

which means that the value of $374,563.25 on 9/10/21 is equivalent to the value of

$360,595.32 on 9/10/16. This results in a return of 37456.33%.

For Model II, we decided to use genetic algorithm to improve the original strategy as we

found many opportunities were missing. We created 4 genes, namely trading portfolio, trad-

ing frequency, buying strategy, and selling strategy, hoping to improve our strategy from

these four dimensions. After MATLAB solving, we learned that the initial capital was valued

at $442,697.25 on 9/10/21 and after discounting it was $426,188.16, creating a total yield of

44269.73% ,an annualized return of 338.08%.

Again, we demonstrate that our strategy is optimal in two ways, one by changing the

strategy itself and the other by observing different market environments. To change the strat-

egy, we reset gene4 in the genetic algorithm. After running the results, we found that the

yield after the change was reduced to 273%. To observe the strategy performance in differ-

ent market environments, we extracted the bull, bear and oscillator markets that existed in

five years, using bitcoin trading as an example. Comparing the strategy performance and the

market performance during the periods respectively, we again find that the strategy outper-

forms the market performance by a wide margin.

Finally, we wrote a memorandum to investors detailing our modeling process, introduc-

ing our strategy, analyzing the strategy results, and informing about the risks and shortcom-

ings of the strategy.

Keywords: ARIMA-LSTM Model, Dynamic Programming, Genetic Algorithm

Problem Chosen

2022

MCM/ICM

Summary Sheet

Team Control Number

2212336

更多数模资讯和学习资料，请关注b站/公众号：数学建模BOOM

b站主页：https://space.bilibili.com/350975620

Team # 2212336 Page 2 of 25

Contents

1 Introduction ...................................................................................................... 3

1.1 Problem Background ....................................................................................................... 3

1.2 Restatement of the Problem ............................................................................................. 3

1.3 Our Work .......................................................................................................................... 3

2 Assumptions and Justifications ....................................................................... 4

3 Definitions and Notations ................................................................................ 4

3.1 Definitions ........................................................................................................................ 4

3.2 Notations .......................................................................................................................... 5

4 Model I: Volatile Asset Price Forecasting Model .......................................... 5

4.1 Volatile Asset Price Forecasting Theory .......................................................................... 5

4.2 Volatile Asset Price Forecasting Methods ........................................................................ 5

4.3 Volatile Asset Prediction Results ..................................................................................... 8

4.4 Comparative Analysis of Model Forecasting Performance ............................................. 9

5 Dynamic Planning Based Trading Strategy Determination ...................... 10

5.1 Dynamic Programming to Determine the Trading Strategy Process ............................. 10

5.2 Dynamic Programming Results ..................................................................................... 11

6 Model II: Strategy Improvement Model...................................................... 11

6.1 Introduction to Genetic Algorithm ................................................................................. 11

6.2 Process of Genetic Algorithm Optimization Trading Strategy ...................................... 12

6.3 Genetic algorithm to optimize trading strategy results .................................................. 14

7 Effectiveness Evaluation of The Trading Strategy ..................................... 16

7.1 Bull and Bear Market Strategies .................................................................................... 16

7.2 Stop Loss on Capital ...................................................................................................... 17

8 Transaction Cost Sensitivity Analysis .......................................................... 18

9 Memorandum to Investor ............................................................................. 19

10 Model Evaluation ......................................................................................... 20

10.1 Strengths ...................................................................................................................... 20

10.2 Weaknesses .................................................................................................................. 20

11 Conclusion ..................................................................................................... 20

References .......................................................................................................... 21

Appendices ......................................................................................................... 22

Team # 2212336 Page 3 of 25

1 Introduction

1.1 Problem Background

In real world of finance, all buying and selling behaviors are based on people's expecta-

tions for the future, so how to predict the future price of assets as accurately as possible has

become one of the most important research contents in finance. The more accurate the in-

vestor's prediction of asset prices, the easier it is to achieve higher expected returns.

Of course, it is rogue to talk about returns without risk. Theoretically, in the market, va-

rieties with higher risk tend to have greater profit. In order to adapt to people's different risk

preferences, a wealth of financial products have emerged. Among them, Bitcoin is favored by

investors due to its limited total amount, safety, free transaction, wide circulation range and

so on. However, due to high price volatility of bitcoin, gold, which is considered as a safe-

haven capital, is paired with it to form an investment portfolio.

The "best daily trading strategy" described in the subject must have a specific measure-

ment. It is often said in finance that if investor A is a risk averter and investor B is a risk

seeker, we cannot consider investor B's strategy to be superior one even if investor A's return

is lower than B's return. Therefore, in this question, our strategy must be based on a given

risk preference so that our strategy comparable.

1.2 Restatement of the Problem

Considering the background information and restricted conditions identified in the prob-

lem statement, we need to solve the following problems:

 If you have $1000 on 9/10/2016, with known historical prices of the asset, create a

model to make an optimal strategy, calculate how much is it worth on 9/10/2021.

 Prove that your strategy is optimal in different situations.

 Perform a sensitivity analysis of strategy results between trading costs and de-

rive how trading costs affect strategy and investment results.

 Writing a letter with traders to summarize your strategy, model, and trading results.

1.3 Our Work

The flow chart of our idea is shown in Figure 1.

Figure 1: Flow Chart

Team # 2212336 Page 4 of 25

2 Assumptions and Justifications

Assumption 1: Investor who will use this strategy is a risk-seeker who willing to take

arbitrary risk for the sake of return.

Justification: As explained in Section 1.1: strategies with different risks are not compa-

rable. In order to evaluate strategies better, we must assume the risk of strategy, that is, the

risk preferences of the investors implementing the strategy. However, since individual risk

preferences are difficult to quantify, we make the above assumption that investors are ex-

treme risk seekers, that is, investor will ignore risk to achieve higher expected return.

Assumption 2: Once we decide to trade, we can achieve the deal instantly, regardless of

the volume we need to achieve.

Justification: In the real world, if there are no enough sellers, then even investors trad-

ing at current prices may still not be able to close the deal. To simplify the circumstance, we

need to make this assumption to rule out this contingency.

Assumption 3: No more positions are allowed to be added to an asset while it is held.

Justification: Position control is a major challenge in investment strategy formulation

and is difficult to implement in code. Since Assumption 1 assumes that the investor is a risk-

seeker, it is reasonable for us to have a perception that such investor will tend to invest as

much money as possible in profitable assets in order to maximize profits, regardless of the

risk involved in doing so.

Assumption 4: Assume that the minimum transaction unit for bitcoin is 0.1 bitcoins,

and the minimum transaction unit for gold is 1 ounce.

Justification: In order to match reality as much as possible, and to simplify the diffi-

culty of programming, we change the minimum transaction unit for bitcoin from 0.01 bit-

coins to 0.1 bitcoins, and the minimum transaction unit for gold to be consistent with reality.

3 Definitions and Notations

3.1 Definitions

Bull Market：A market with a long-term upward price trend. The overall trend is up-

ward, with some declines, but one wave higher than the other.

Bear Market：A market that has a long-term downward price trend. The overall trend

is downward, although there are rallies, but one wave is lower than the other,.

Oscillating Market: A market situation in which share prices fluctuate and the future of

the market is uncertain. It is characterized by an increase in short term investment, unstable

market popularity and large ups and downs in prices.

Position control: It is your control over the size of your position, and your control over

opening, adding, reducing and cutting positions.

Risk-seeker: People who prefer to receive the expected income from a risk rather than

the expected value of the risk. For a risk taker, the utility of the expectation is greater than the

expected utility of the risk itself.

time value of money: A certain amount of money currently held has a higher value

Team # 2212336 Page 5 of 25

than an equivalent amount of money acquired in the future.

3.2 Notations

The key mathematical notations used in this paper are listed in Table 1.

Table 1: Notations used in this paper

Symbol Description

Asset price in time T

Predicted asset price at time T

The autoregressive term

The number of differences when the time series is stationary

The number of moving average



The constant term



The autocorrelation coefficient



The error term



The coefficient of error term

real

real value by dynamic programming

min

no al

nominal value by dynamic programming

risk-free yield

4 Model I: Volatile Asset Price Forecasting Model

4.1 Volatile Asset Price Forecasting Theory

According to the question, we need to develop a strategy based on price movements

to maximize returns by buying in a lower price and selling in a higher price. Therefore, we

need to forecast the daily price in the future first.

In the financial market, different asset prices have their own fluctuation trends and

patterns. In order to obtain higher returns, people often build reasonable time series models

to predict the future development of volatile assets. Price series forecasting is something that

based on historical data, using some scientific method to estimate the price of assets in a

certain period in the future. Knowing the time series

1 2 3

{Y ,Y ,Y , ,Y }

, find the asset price

1 2 3

, , , ,

T T T T m

Y Y Y Y

+ + + +

. The formula is shown in equation 1.

1 2 3 m 1 2 3

ˆ ˆ ˆ ˆ

, , , , ( , , , , )

T T T T T

Y Y Y Y f Y Y Y Y

+ + + +

(1)

4.2 Volatile Asset Price Forecasting Methods

The forecasting methods for volatile assets can be divided into linear forecasting models

and nonlinear forecasting. Financial markets contain many uncertainties and can be influ-

enced by economic, political, social factors and so on. The changes caused by these factors

have a strong disorderly nature, so it is difficult to say exactly whether it is a purely linear or

nonlinear system. Therefore, we need to build a model that contains both linear and nonlinear

features.

Traditional time series models can extract linear features, while neural network models

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