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研究生： 盧晉偉
Lu, Jin-Wei
論文名稱： 最小平方法、卡爾曼濾波器與其在配對交易上的應用—數學理論及數值模擬
Least Squares Method and Kalman Filter with Its Application to Pairs Trading — Mathematical Theory and Numerical Simulation
指導教授： 許瑞麟
Sheu, Ruey-Lin
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系應用數學碩博士班
Department of Mathematics
論文出版年： 2024
畢業學年度： 112
語文別： 英文
論文頁數： 106
中文關鍵詞： 線性迴歸模型 、最小平方法 、加權最小平方法 、遞迴最小平方法 、資產定價模型 、單一指標模型 、卡爾曼濾波器 、配對交易
外文關鍵詞： Linear regression model, Least squares method, Weighted least squares method, Recursive least squares method, Capital asset pricing model, Single-index model, Kalman filter, Pairs trading
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    • 本篇論文中，我們對 Elliott 的價差模型 (spread model) [14, 2005] 感到興趣。在金融上，如果有兩個歷史走勢相近的股票，這兩檔股票就得以配對 (pair) 去進行交易，而它們的價格差距稱為價差(spread)。當兩檔股票走勢改變時，做空價格高的那檔股票同時做多價格低的另一檔股票，等待價差回歸至預設價格時獲利，這便是配對交易策略 (pairs trading strategy)。Elliott 的價差模型運用卡爾曼濾波器(Kalman filter) [26, 1960] 來預估價差走勢。為了充分了解 Elliott 價差模型的數學基礎，我們詳細介紹最小平方法、權重最小平方法、遞迴最小平方法與卡爾曼濾波器等資料分析工具的數學理論，並寫下演算法提供本文數值模擬測試。本文同時介紹資產定價模型 (capital asset pricing model)[39, 1964]、單一指標模型 (single-indexmodel)[35, 1952] 等重要金融模型，並特別研究包括 Sharpe 指標[39, 1964]、beta 指標 [39, 1964] 等參數之數學原理。我們以 2012-2019 的百事可樂與可口可樂交易資料為例，以 Elliott 價差模型模擬這兩檔股票的價差行為，並以我們寫下的卡爾曼濾波器演算法在 R 語言上計算價差的預估價格，並分析卡爾曼濾波器的預估效果。由於Elliott 價差模型沒有說明兩檔股票要具有何種特性才能形成有效的配對交易，在本文中我們也替如何尋找配對標的給出初步想法，並提供數據測試結果，作為未來進一步研究之參考。

    In this thesis, we are interested in Elliott’s spread model [14, 2005]. In finance, if two stocks are roughly of the same historical trend, they can be paired to trade and the price difference is called the spread. When the trend of this pair happens to deviate, by longing the underperforming stock and shorting the outperforming one until the spread returns to the predetermined price, one can make a profit by such a pairs trading strategy. Elliott’s spread model uses the Kalman filter [26, 1960] to estimate the spread. In order to fully understand the mathematical basis of the Elliott spread model, we introduce the data analysis tools in detail, including the least squares method, the weighted least squares method, the recursive least squares method, and the Kalman filter. The Kalman filter algorithm is later implemented for numerical simulation purpose in this paper. In addition, we introduce two classical financial models, the capital asset pricing model (CAPM) [39, 1964] and the single-index model [35, 1952], especially the important financial parameters such as the Sharpe ratio [39, 1964] and the market beta [39, 1964] therein with their respective mathematical basis. We take the Pepsi and Coca-Cola trading data from 2012 to 2019 as an example. We incorporate Elliott spread model with the Kalman filter to simulate the spread and analyse the accuracy of Kalman estimation. Since the Elliott spread model did not explain how to pick a pair for effective trading, we propose some preliminary idea as to how to choose such a pair. We hope the initial numerical results serve the purpose for future researches in this direction.

    摘要 I Abstract II 誌謝 III Table of Contents IV List of Tables VI List of Figures VII 1 Introduction 1 2 Preliminary 13 2.1 Probability 13 3 Linear regression model 18 3.1 Least squares method 18 3.2 Weighted least squares method 28 3.3 Recursive least squares method 36 4 Discrete Kalman filter 43 4.1 State space model 43 4.2 Kalman filter algorithm 45 5 Capital asset pricing model and single-index model 51 5.1 Financial market assumptions 51 5.2 Return rate, risk and Sharpe ratio 52 5.3 Capital Asset Pricing Model 55 5.4 Single-index model 55 6 Mathematical model of spread 59 6.1 Two approaches to find the valid pair 59 6.1.1 Zero beta approach 60 6.1.2 Mean reversion approach 63 6.2 Spread model and its simulation 63 6.3 State space model for spread 71 6.4 Numerical Example: Pepsi and Coca-Cola 72 6.4.1 Finding weight w from the mean reversion approach 76 6.4.2 Finding weight w from the zero beta approach 84 7 Conclusion and Future Research 90 Bibliography 92

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