CH6 Pairs Selection in Equity Markets

Agenda Introduction Common trends cointegration model Common trends model and APT The distance measure Interpreting the distance measure Reconciling theory and practice

Introduction In this chapter , we focus on Identification of stock pairs. This book give us a method : Each pair is associated with a score/distance measure. - The higher the score , the greater the degree of comovement , and vice versa. A proper choice of the score measure become a key to the pairs selection process. If we find that a pair is unsuitable for pairs trading , then the score of other pairs which lower than this pair is also unsuitable. We will find the nexus between cointegration and arbitrage pricing theory (APT)

Common trends cointegration model Recall the CH5 : 𝑦 𝑡 = 𝑛 𝑦 𝑡 + 𝜀 𝑦 𝑡 𝑧 𝑡 = 𝑛 𝑧 𝑡 + 𝜀 𝑧 𝑡 if two series are cointegrated , 𝑛 𝑦 𝑡 =γ 𝑛 𝑧 𝑡 where γis cointegration coefficient. Let us examine some of the implications of the common trends model.

Common trends cointegration model Inference 1 : In a cointegrated system with two time series, the innovations sequences derived from the common trend components must be perfectly correlated. (Correlation value must be +1 or –1). <Explanation> The innovations sequences of common trend components is : 𝑛 𝑦 𝑡+1 − 𝑛 𝑦 𝑡 = 𝑟 𝑦 𝑡+1 𝑎𝑛𝑑 𝑛 𝑧 𝑡+1 − 𝑛 𝑧 𝑡 = 𝑟 𝑧 𝑡+1 We know that : 𝑛 𝑦 𝑡 =γ 𝑛 𝑧 𝑡 ⇒ 𝑛 𝑦 𝑡+1 =γ 𝑛 𝑧 𝑡+1 , So 𝑟 𝑦 𝑡+1 =γ 𝑟 𝑧 𝑡+1 . Then , 𝑐𝑜𝑟𝑟 𝑟 𝑦 𝑡+1 , 𝑟 𝑧 𝑡+1 = 𝑐𝑜𝑣 𝑟 𝑦 𝑡+1 , 𝑟 𝑧 𝑡+1 𝑣𝑎𝑟( 𝑟 𝑦 𝑡+1 ) 𝑣𝑎𝑟( 𝑟 𝑧 𝑡+1 ) = γ∗𝑣𝑎𝑟( 𝑟 𝑧 𝑡+1 ) γ 2 𝑣𝑎𝑟( 𝑟 𝑧 𝑡+1 ) 2 = γ γ 2 = γ γ if γ>0 , 𝑐𝑜𝑟𝑟 𝑟 𝑦 𝑡+1 , 𝑟 𝑧 𝑡+1 =1 ,γ<0 , 𝑐𝑜𝑟𝑟 𝑟 𝑦 𝑡+1 , 𝑟 𝑧 𝑡+1 =−1 So , the two variables are identical up to a scalar. 由一般趨勢項所推導出來的seq應該

Common trends cointegration model Inference 2 : The cointegration coefficient may be obtained by a regression of the innovation sequences of the common trends against each other. <Explanation> 𝑟 𝑦 𝑡 =𝛾 𝑟 𝑧 𝑡 ⇒𝐶𝑜𝑣 𝑟 𝑦 𝑡 , 𝑟 𝑧 𝑡 =𝐶𝑜𝑣 𝛾 𝑟 𝑧 𝑡 , 𝑟 𝑧 𝑡 =γ𝐶𝑜𝑣 𝑟 𝑧 𝑡 , 𝑟 𝑧 𝑡 =γ𝑉𝑎𝑟 𝑟 𝑧 𝑡 ∴𝛾= 𝐶𝑜𝑣 𝑟 𝑦 𝑡 , 𝑟 𝑧 𝑡 𝑉𝑎𝑟 𝑟 𝑧 𝑡

Common trends cointegration model Discussion: The correlation of the innovation sequences of common trend component and whole series are difference. If cointegration exist, the specific component is absolutely necessary to be stationary. The first difference of the specific component must not be white noise, because if the differenced series were white noise, then the specific series would be a random walk, a nonstationary series. 前面只有提到RW項的相關係數,現在要計算整條Series的相關係數:1.一般趨勢項的IS跟整條的Inno-Series,他們的correlation計算方法不同 2.沒有限制的時候CT它可以是定態也可以是非定態,但卻不是影響共整合的最大原因,所以最重要的因素是特別向的部分必須是定態 根據1跟2 我們可以知道說 特別項的一階差分不可以為白噪音數列 不然原本的特別項就會是RW 有為被原本 特別項是定態的假設

Common trends model and APT We know that logarithm of stock price was modeled as a random walk. Common trends model: log 𝑝𝑟𝑖𝑐𝑒 𝑡 = 𝑛 𝑡 + 𝜀 𝑡 Differencing it we get the sequence of return , so log 𝑝𝑟𝑖𝑐𝑒 𝑡 − log 𝑝𝑟𝑖𝑐𝑒 𝑡−1 = 𝑛 𝑡 − 𝑛 𝑡−1 +( 𝜀 𝑡 − 𝜀 𝑡−1 ) ⇒ 𝑟 𝑡 = 𝑟 𝑡 𝑐 + 𝑟 𝑡 𝑠 So , 𝑛 𝑡 − 𝑛 𝑡−1 = 𝑟 𝑡 𝑐 is the same with innovation derived from the common trend component. By 1) and 2), If two stocks are cointegrated, the returns from their common trends must be identical up to a scalar. 如果我們要回到股票資料的問題 唯一可行的應用就是我們要把時間序列拆成Sta跟Nonsta 所以必須建立 APT跟CTM的關係

Common trends model and APT Why in the world should stocks ever have common returns? APT include that : - The stock return = common factor returns + specific returns. - If two stocks share the same risk factor exposure profile, then the common factor returns for both the stocks must be the same. The specific return 𝑟 𝑡 𝑠 should not be white noise , but APT do not give us any guarantees on the time series of specific returns . - When running the cointegration tests and pairs where the specific component is nonstationary are eliminated. Now , we assume that specific return is not white noise and interpret the inferences from the common trends model in APT. APT是一個單一時點的架構而且也不能給時間序列的Spec R 擔保.必須假設Spec r部份不時白噪音 但可以透過共整合檢定

Common trends model and APT Observation 1: A pair of stocks with the same risk factor exposure profile satisfies the necessary conditions for cointegration. <Condition 1> Stock A 𝛾𝑥= 𝛾 𝑥 1 ,𝛾 𝑥 2 ,…,𝛾 𝑥 𝑛 Stock B 𝑥=( 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑛 ) Let 𝑏=( 𝑏 1 , 𝑏 2 ,…, 𝑏 𝑛 ) is factor returns vector . 𝑟 𝐴 =𝛾 𝑥 1 𝑏 1 + 𝑥 2 𝑏 2 +…+ 𝑥 𝑛 𝑏 𝑛 + 𝑟 𝐴 𝑠𝑝𝑒𝑐 𝑟 𝐵 = 𝑥 1 𝑏 1 + 𝑥 2 𝑏 2 +…+ 𝑥 𝑛 𝑏 𝑛 + 𝑟 𝐵 𝑠𝑝𝑒𝑐 𝑟 𝐴 𝑐𝑓 =𝛾 𝑥 1 𝑏 1 + 𝑥 2 𝑏 2 +…+ 𝑥 𝑛 𝑏 𝑛 𝑟 B 𝑐𝑓 = 𝑥 1 𝑏 1 + 𝑥 2 𝑏 2 +…+ 𝑥 𝑛 𝑏 𝑛 → 𝑟 𝐴 𝑐𝑓 = 𝛾.𝑟 B 𝑐𝑓 The factor exposure are identical up to scalar. 滿足共整合的第一個條件

Common trends model and APT <Condition 2> 𝑟 𝐴 −𝛾 𝑟 𝐵 = 𝑟 𝐴 𝑐𝑓 −𝛾 𝑟 B 𝑐𝑓 + 𝑟 𝐴 𝑠𝑝𝑒𝑐 −𝛾 𝑟 B 𝑠𝑝𝑒𝑐 This is a portfolio return that long 1 unit stock A and short 𝛾 stock B. 𝑟 𝑝𝑜𝑟𝑡 = 𝑟 𝑝𝑜𝑟𝑡 𝑐𝑓 + 𝑟 𝑝𝑜𝑟𝑡 𝑠𝑝𝑒𝑐 So, if stock A and B are cointegration , 𝑟 𝑝𝑜𝑟𝑡 𝑐𝑓 is zero. If we view as the differencing spread, 𝑠𝑝𝑟𝑒𝑎𝑑 𝑡 𝑐𝑓 − 𝑠𝑝𝑟𝑒𝑎𝑑 𝑡−1 𝑐𝑓 = 𝑟 𝑝𝑜𝑟𝑡 𝑐𝑓 𝑠𝑝𝑟𝑒𝑎𝑑 𝑡 𝑠𝑝𝑒𝑐 − 𝑠𝑝𝑟𝑒𝑎𝑑 𝑡−1 𝑠𝑝𝑒𝑐 = 𝑟 𝑝𝑜𝑟𝑡 𝑠𝑝𝑒𝑐 𝑠𝑝𝑟𝑒𝑎𝑑 𝑡 𝑝𝑜𝑟𝑡 = 𝑠𝑝𝑟𝑒𝑎𝑑 𝑡 𝑐𝑓 + 𝑠𝑝𝑟𝑒𝑎𝑑 𝑡 𝑠𝑝𝑒𝑐 So, if stock A and B are cointegration , 𝑠𝑝𝑟𝑒𝑎𝑑 𝑡 𝑐𝑓 is zero. 𝑠𝑝𝑟𝑒𝑎𝑑 𝑡 𝑝𝑜𝑟𝑡 is stationary if 𝑠𝑝𝑟𝑒𝑎𝑑 𝑡 𝑠𝑝𝑒𝑐 is stationary.

The distance measure Now , we will go into the distance measure , we have already know: The innovation sequences derived from the common trends must be perfectly correlated. The common factor return of the APT model might be interpreted as the innovations derived from the common trends. We will explain that the distance measure is the absolute value of the correlation of the common factor returns.

The distance measure The distance measure formula : 𝑑 𝐴,𝐵 = 𝜌 = 𝐶𝑜𝑣( 𝑟 𝐴 , 𝑟 𝐵 ) 𝑉𝑎𝑟( 𝑟 𝐴 ) 𝑉𝑎𝑟( 𝑟 𝐵 ) In APT terms , 𝑥 𝐴 𝑥 𝐵 are factor exposure vector and 𝐹 is covariance matrix 𝑑 𝐴,𝐵 = 𝜌 = 𝑥 𝐴 𝐹 𝑥 𝐵 𝑇 𝑥 𝐴 𝐹 𝑥 𝐴 𝑇 𝑥 𝐵 𝐹 𝑥 𝐵 𝑇 (Only common factor term)

Interpreting the distance measure In this section , we will show that the correlation measure 𝑑 𝐴,𝐵 = 𝜌 = 𝑥 𝐴 𝐹 𝑥 𝐵 𝑇 𝑥 𝐴 𝐹 𝑥 𝐴 𝑇 𝑥 𝐵 𝐹 𝑥 𝐵 𝑇 can be interpreted as the cosine of the angle between transformed versions of the factor exposure vectors corresponding to the two stocks.

Interpreting the distance measure Calculating the Cosine of the Angle between Two Vectors : 𝑒 𝐴 =( 𝑒 1 𝐴 , 𝑒 2 𝐴 ,…, 𝑒 𝑁 𝐴 ) 𝑒 𝐵 =( 𝑒 1 𝐵 , 𝑒 2 𝐵 ,…, 𝑒 𝑁 𝐵 ) Inner product: 𝑒 𝐴 𝑒 𝐵 𝑇 =( 𝑒 1 𝐴 𝑒 1 𝐵 + 𝑒 2 𝐴 𝑒 2 𝐵 +…+ 𝑒 𝑁 𝐴 𝑒 𝑁 𝐵 ) 𝑙𝑒𝑛𝑔𝑡ℎ 𝑒 𝐴 = 𝑒 𝐴 𝑒 𝐴 T Then we calculate unit vector and the cosine is inner product between two unit vector : c𝑜𝑠𝜃= 𝑒 𝐴 𝑒 𝐵 T (𝑒 𝐴 𝑒 𝐴 T )( 𝑒 𝐵 𝑒 𝐵 T )

Common trends model and APT Example : 𝐴= 0,2 𝐵=(3,0) 𝑙𝑒𝑛𝑔𝑡ℎ 𝐴 = 0 2 + 2 2 =2 𝑙𝑒𝑛𝑔𝑡ℎ 𝐵 = 3 2 + 0 2 =3 So , unit vectors of A and B is 𝑎= 1 𝑙𝑒𝑛𝑔𝑡ℎ 𝐴 𝐴=(0,1) b= 1 𝑙𝑒𝑛𝑔𝑡ℎ 𝐵 𝐵=(1,0) Then 𝑐𝑜𝑠𝜃=𝑎𝑏=0∗1+1∗0=0 It means that the angle between the two vectors is 90 ° . i.e. The two vectors are orthogonal to each other.

Common trends model and APT Geometric Interpretation : We have to know Eigenvalue Decomposition : λ and 𝑣 are eigenvalue and eigenvector , if A is square matrix satisfy 𝐴𝑣=λ𝑣 A n × n square matrix may have n eigenvalue λ 1 , λ 2 ,…, λ 𝑛 and n corresponding eigenvectors 𝑣 1 , 𝑣 2 ,…, 𝑣 𝑛 , so 𝐴𝑣 1 = λ 1 𝑣 1 , 𝐴𝑣 2 = λ 2 𝑣 2 , …, 𝐴𝑣 𝑛 = λ 𝑛 𝑣 𝑛 ; If 𝐷= λ 1 0 0 λ 2 … 0 ⋱ ⋮ ⋮ ⋱ ⋱ 0 0 … 0 λ 𝑛 U= 𝑣 1 𝑣 2 … 𝑣 𝑛 ⇒𝐴𝑈=𝑈𝐷⇒𝐴=𝑈𝐷 𝑈 −1 .

Common trends model and APT Then we back to 𝐹 covariance matrix : 𝐹 is a symmetric matrix , its eigenvector 𝑈 has the property that 𝑈 −1 = 𝑈 𝑇 . So , 𝐹=𝑈𝐷 𝑈 −1 =𝑈𝐷 𝑈 𝑇 Let us consider a transformation of the two vectors 𝑒 𝐴 = 𝑥 𝐴 𝑈 𝐷 1/2 𝑒 𝐵 = 𝑥 𝐵 𝑈 𝐷 1/2 This is the transformation from the factor exposure space to the factor return space. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑒 𝐴 = 𝑥 𝐴 𝐹 𝑥 𝐴 𝑇 = 𝑉𝑎𝑟( 𝑟 𝐴 ) 𝑙𝑒𝑛𝑔𝑡ℎ 𝑒 𝐵 = 𝑥 𝐵 𝐹 𝑥 𝐵 𝑇 = 𝑉𝑎𝑟( 𝑟 𝐵 ) 𝑒 𝐴 𝑒 𝐵 𝑇 = 𝑥 𝐴 𝐹 𝑥 𝐵 𝑇 =𝐶𝑜𝑣( 𝑟 𝐴 , 𝑟 𝐵 ) So, 𝑐𝑜𝑠𝜃= 𝑒 𝐴 𝑒 𝐵 𝑇 𝑙𝑒𝑛𝑔𝑡ℎ 𝑒 𝐴 𝑙𝑒𝑛𝑔𝑡ℎ 𝑒 𝐵 = 𝑥 𝐴 𝐹 𝑥 𝐵 𝑇 𝑥 𝐴 𝐹 𝑥 𝐴 𝑇 𝑥 𝐵 𝐹 𝑥 𝐵 𝑇 = 𝐶𝑜𝑣( 𝑟 𝐴 , 𝑟 𝐵 ) 𝑉𝑎𝑟( 𝑟 𝐴 )𝑉𝑎𝑟( 𝑟 𝐵 ) =𝜌

Reconciling theory and practice Stationarity of integrated specific returns: Using cointegration test to evaluate the common factor and specific returns for each time period. Cointegration test : Estimation of the cointegration coefficient Ensuring that the spread series of the long–short portfolio constructed with this ratio is indeed stationary. 說完定理之後 我們接這要把定理與實際操作做一個結合,

Reconciling theory and practice Deviations from ideal conditions : - In APT’s views , two stocks will be cointegration if the common factor correlation between them must be +1 or –1. (very difficult!!) - If not +1 or -1 , 𝑟 𝐴 −𝛾 𝑟 𝐵 = 𝑟 𝐴 𝑐𝑓 −𝛾 𝑟 B 𝑐𝑓 + 𝑟 𝐴 𝑠𝑝𝑒𝑐 −𝛾 𝑟 B 𝑠𝑝𝑒𝑐 𝑟 𝑝𝑜𝑟𝑡 = 𝑟 𝑝𝑜𝑟𝑡 𝑐𝑓 + 𝑟 𝑝𝑜𝑟𝑡 𝑠𝑝𝑒𝑐 𝑠𝑝𝑟𝑒𝑎𝑑 𝑡 𝑝𝑜𝑟𝑡 = 𝑠𝑝𝑟𝑒𝑎𝑑 𝑡 𝑐𝑓 + 𝑠𝑝𝑟𝑒𝑎𝑑 𝑡 𝑠𝑝𝑒𝑐 實際上很難找到CF會全正相關跟負相關 除非是同一家公司的AB類股,如果沒有完全正負相關 會違背價差必需為定態的的假設 Nonzero Nonstationary Nonstationary!!

Reconciling theory and practice - We still make do with less than perfect conditions of cointegration? 𝜎 𝑠𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑟𝑦 2 : The variance of the stationary component 𝜎 𝑛𝑜𝑛𝑠𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑟𝑦,𝑇 2 : The variance of the nonstationary component is specified for a time horizon T. Signal-to-noise ratio , SNR : 𝑆𝑁𝑅= 𝜎 𝑠𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑟𝑦 2 𝜎 𝑛𝑜𝑛𝑠𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑟𝑦,𝑡 2 , t is trading horizon. (1)If 𝜎 𝑛𝑜𝑛𝑠𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑟𝑦,𝑡 2 →0 , 𝑆𝑁𝑅→∞ . (2)it make our assumption of cointegration reasonable if SNR is big! (3) 𝜎 𝑛𝑜𝑛𝑠𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑟𝑦,𝑡 2 increases linearly with t , t ↓ then close to the ideal condition of cointegration. 說完定理之後 我們接這要把定理與實際操作做一個結合, 所以我們可以知道 如果CF項不是定態 整個spread就像有一個隨機drift項一樣

Reconciling theory and practice Numerical example: Consider three stocks A, B, and C with factor exposures in a two factor model as follows: 𝑥 𝐴 =[ 1 1 ] 𝑥 𝐵 =[ 0.75 1 ] 𝑥 𝐶 =[ 1 0.75 ] Let the factor covariance matrix 𝐹= 0.0625 0.0225 0.0225 0.1024 說完定理之後 我們接這要把定理與實際操作做一個結合,

Reconciling theory and practice Step1: Calculate the common factor variance and covariance 𝑉𝑎𝑟 𝐴 = 1 1 0.0625 0.0225 0.0225 0.1024 1 1 =0.2099, Volatility = 45.8% 𝑉𝑎𝑟 𝐵 = 0.75 1 0.0625 0.0225 0.0225 0.1024 0.75 1 =0.171, Volatility = 41.3% 𝑉𝑎𝑟 𝐶 = 1 0.75 0.0625 0.0225 0.0225 0.1024 1 0.75 =0.1539, Volatility = 39.2% 𝐶𝑜𝑣 𝐴,𝐵 = 1 1 0.0625 0.0225 0.0225 0.1024 0.75 1 =0.1887 𝐶𝑜𝑣 𝐴,𝐶 = 1 1 0.0625 0.0225 0.0225 0.1024 1 0.75 =0.1787 說完定理之後 我們接這要把定理與實際操作做一個結合,

Reconciling theory and practice Step2: Calculate the correlation (absolute value of correlation is the distance measure). 𝐶𝑜𝑟𝑟 𝐴,𝐵 = 𝐶𝑜𝑣(𝐴,𝐵) 𝑉𝑎𝑟 𝐴 𝑉𝑎𝑟(𝐵) = 0.1887 0.2099∗0.171 =0.9957 𝐶𝑜𝑟𝑟 𝐴,𝐶 = 𝐶𝑜𝑣(𝐴,𝐶) 𝑉𝑎𝑟 𝐴 𝑉𝑎𝑟(𝐶) = 0 1787 0.2099∗0.1539 =0.9431 Base on the distance measure , we choose pair( A , B ) 說完定理之後 我們接這要把定理與實際操作做一個結合,

Reconciling theory and practice Step3: Calculate the cointegration coefficient. λ 𝐴𝐵 = 𝐶𝑜𝑣(𝐴,𝐵) 𝑉𝑎𝑟(𝐵) =1.1032 λ 𝐴𝐶 = 𝐶𝑜𝑣(𝐴,𝐶) 𝑉𝑎𝑟(𝐶) =1.1613 Step4: Calculate the residual common factor exposure in the paired portfolio. This is the exposure that causes mean drift. 𝑒𝑥𝑝 𝐴𝐵 = 𝑥 𝐴 − λ 𝐴𝐵 𝑥 𝐵 = 1 1 −1.1032 0.75 1 =[ 0.1726 −0.1032 ] 𝑒𝑥𝑝 𝐴𝐶 = 𝑥 𝐴 − λ 𝐴𝐶 𝑥 𝐶 = 1 1 −1.1613 1 0.75 =[ −0.1613 0.129 ] 說完定理之後 我們接這要把定理與實際操作做一個結合,

Reconciling theory and practice Step5: Calculate the common factor portfolio variance/variance of residual exposure. 𝑉𝑎𝑟 𝑟 𝐴𝐵 𝑐𝑓 = 0.1726 −0.1032 0.0625 0.0225 0.0225 0.1024 0.1726 −0.1032 =0.0021 𝜎 𝐴𝐵 𝑐𝑓 = 0.0021 =0.046 𝑉𝑎𝑟 𝑟 𝐴𝐶 𝑐𝑓 = −0.1613 0.129 0.0625 0.0225 0.0225 0.1024 −0.1613 0.129 =0.0024 𝜎 𝐴𝐵 𝑐𝑓 = 0.0024 =0.049 說完定理之後 我們接這要把定理與實際操作做一個結合,

Reconciling theory and practice Step6: Calculate the specific variance of the portfolio. Let us assume the specific variance for all of the stocks to be 0.0016. 𝑉𝑎𝑟 𝑟 𝐴𝐵 𝑠𝑝𝑒𝑐 =𝑉𝑎𝑟 𝑟 𝐴 𝑠𝑝𝑒𝑐 + λ 𝐴𝐵 2 𝑉𝑎𝑟 𝑟 𝐵 𝑠𝑝𝑒𝑐 =0.0016+ 1.1032 2 ×0.0016=0.0035 𝜎 𝐴𝐵 𝑠𝑝𝑒𝑐 = 0.0035 =0.059 𝑉𝑎𝑟 𝑟 𝐴𝐶 𝑠𝑝𝑒𝑐 =𝑉𝑎𝑟 𝑟 𝐴 𝑠𝑝𝑒𝑐 + λ 𝐴𝐶 2 𝑉𝑎𝑟 𝑟 𝐶 𝑠𝑝𝑒𝑐 =0.0016+ 1.1613 2 ×0.0016=0.0037 𝜎 𝐴𝐶 𝑠𝑝𝑒𝑐 = 0.0037 =0.061 說完定理之後 我們接這要把定理與實際操作做一個結合,

Reconciling theory and practice Step7: Calculate the SNR ratio with white noise assumptions for residual stock return. 𝑆𝑁𝑅 𝐴𝐵 = 𝜎 𝐴𝐵 𝑠𝑝𝑒𝑐 𝜎 𝐴𝐵 𝑐𝑓 = 0.059 0.046 =1.282 𝑆𝑁𝑅 𝐴𝐶 = 𝜎 𝐴𝐶 𝑠𝑝𝑒𝑐 𝜎 𝐴𝐶 𝑐𝑓 = 0.061 0.049 =1.245 By SNR , we still choose pair( A , B ) A higher specific variance means higher stock volatility, indicating that a high volatility environment is conducive for pairs trading. 說完定理之後 我們接這要把定理與實際操作做一個結合,