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The Relationship Between Trading Volume and JumpProcesses in Financial MarketsPongpitch AmatyakulProfessor George Tauchen, Faculty Advisor The Duke Community Standard was upheld in the completion of this paper.Honors Thesis to be submitted in partial fulfillment of the requirements forGraduation with Distinction in Economics in Trinity College, Duke University.Duke University, Durham, North Carolina, 2010

AcknowledgementsI would like to thank my thesis advisor, Professor George Tauchen, for hisguidance and advice throughout this research process. I am grateful for feedback Ireceived from Professor Tim Bollerslev and Bruno Feunou throughout Econ 201. I wouldlike to thank my classmates Sam Lim, Derek Song, Matt Rognlie, and Ab Sawant fortheir comments and suggestions.2

AbstractThis paper explored the relationship between volume and jumps in stock pricesusing high frequency, minute by minute, data from the S&P 500 stocks. It has beenknown that stock prices compose of a continuous part that follows geometric Brownianmotion and a jump part. There have been jump-detection tests developed by variousfinancial economists. These jump tests were performed, analyzed, and regressions hadbeen run to find possible relationships between the trading volume and jump detection.The relationship tends to be slightly negative. Economically, this gives evidence towardsjumps being a result of common knowledge shocks. Common knowledge shocks areinformation interpreted the same way resulting in price movements without heavytrading.3

1. IntroductionJumps are rare financial events in which asset prices move drastically from onemoment in time to the next. This paper aims to find what type of information flow causesthese jumps to occur in the financial market. Jumps can play an important role in risk andportfolio management, and jump detection has been a hot area of research in the pastdecade due to more accessible high frequency data. Although continuous models remainprevalent decades later, potentials for jumps were first noticed as early as 1976 byMerton; he figured out that pricing of assets should not just incorporate the continuousprocess, but should also take into account the jumps that occur. If asset prices followpurely continuous processes, then it should be possible to draw a smooth line betweenprices at different times without lifting a pen or pencil, but that is not the case fromobservations of real data. Since then, we have fully established that jumps do exist, andresearchers have been trying to study and gather more information on them in order tofind better ways to manage risk.Trading volume is defined to be the number of shares traded over a given day.Trading volume is easily observable in financial data, and it can provide insight to thestructure of financial markets. It can tell us the amount of market activity at any giventime interval. However, the volume alone doesn‟t give away much information, so it isnecessary to investigate the relationship between volume and other observable data in themarket, such as price. The relationship of price and volume can give us the rate ofinformation flow in a market. According to the model developed by Tauchen and Pitts(1983), volume and price changes are related. Price changes form the basis of estimatorsused in jump tests, so there could be some relationship between jumps and volume. Thisrelationship can be explored by implementing different jump tests from literature.This paper is the first to explore the economic interpretation of jumps byobserving the relationship between trading volume and jumps. It is known that volume isindicative of information flow. Epps and Epps (1976) built a model in which a largeramount of disagreement among traders results in a higher volume. An example of thiswould be a rumor of a merger between company A and company B. If a group ofinvestors believe that this is beneficial to company A, then they will price company A‟sstock at a higher price, and they would be willing to buy stocks at the original price or4

lower. They would buy company A‟s stocks from those investors who believe that themerger is going to harm company A. This results in high volume of trading. If jumps areindeed a result of disagreements, then it should have a positive relationship with volume.On the other hand, jumps can result from common knowledge shocks. A commonknowledge shock is a result of information being interpreted exactly the same way bymost of the traders. Following the example above, if all investors believe that the mergerbetween company A and company B will help company A operate more efficiently, theneverybody suddenly values company A‟s stocks at a higher price, the price will jump upinstantaneously without much change in volume. If we assume that regular price changeswithout jumps have some positive relationship with volume, this common knowledgeshock could lower that trend between jumps and volume. If jumps occur on a particularday, the volume on that day could be lower than other days with similar price changes. Sothere are two ways that a jump can occur; the relationship between the jump componentand volume can give us key insights as to which of the two theories dominate the jumpprocess.On a similar note, this paper will also discuss and test the relationship of volumeand realized variance. Realized variance is the best way to estimate the variance of stockprice movements. Realized variance can be accurately approximated using highfrequency data. There has been literature such as Tauchen and Pitts (1983) and Andersen(1996) that include models and empirical data that indicated a positive relationshipbetween the variance and volume at least for longer time-spans. This paper will testwhether those relationships exist in high frequency data covering the last 12 years.The next logical step is to place the all the variables into the same regression.Since we already have the individual relationships, it would be interesting to see howthey interact. This can separate out their individual effects and could yield more insightsto the dynamics of volume, jumps, and variance in general.This paper is organized in the following way: Section 2 lays out the backgroundfor the model of equity prices which the jump tests are based upon. Section 3 describeseach of the three jump tests and how to calculate the daily jump test statistics, which willeventually be used in the regressions. Section 4 discusses the market microstructurenoise, the effects it has on high frequency data, and how to minimize those effects.5

Section 5 introduces the data that was used, and section 6 describes the regressions thatwere performed in order to determine the relationship between jumps and volume. Theresults of the regressions are explained in section 7, and interpreted in section 8. Finally,section 9 summarizes the findings and highlights important conclusions. The tables andfigures that accompany the paper can be found in section 10 and the appendix.2. Movements of Equity PricesBefore discussing jump tests, it is important to fully comprehend the theory ofequity prices and returns. This section will first introduce the necessary variables (2.1)and will then go on to discuss the theoretical processes that underlie stock prices andreturns (2.2). This will provide a basis for what is used to determine jump test statistics insection 3.2.1 Basic VariablesThe basic variables that are necessary to understand the movement of stock priceswill be defined here. The first variable will be the logarithmic price.p(t i ) log( s(t i ))(1)Where 𝑠 𝑡𝑖 is the real stock price at time 𝑡𝑖 .Following that equation, the returns at each time period will be defined by:ri p(t i ) p(t i 1 )(2)2.2 Model of Equity PricesThe theory behind security price movements will be developed here. Prior toMerton (1976), stock price returns were thought to have followed Brownian motion withmean μ and standard deviation σ. The stochastic equation for the stock price return r isthe following:dr (t ) (t )dt (t )dW (t )(3)where W is the normal distribution.The model above is deemed insufficient by Merton (1976). He claimed that thestock price returns are a mixture of continuous stochastic processes and Poisson-driven„jump processes.‟ Thus, the original model has been replaced by the continuous time6

jump diffusion model. This model introduces one more term to the equation in the formof a jump process. A jump may or may not occur in any given time interval. The model isas follows:dr (t ) (t )dt (t )dW (t ) (t )dq(t )(4)where κ is the size of the jump, and 𝑞 𝑡 is the number of jumps up until time t.3. Jump Tests and Jump Test statisticsIn recent years, with readily-accessible high frequency data and greaterprocessing power, several tests have been developed to assess whether stock price returnscontain discontinuous processes. The most notable ones include Mancini (2006), Lee andMykland (2006), Barndorff-Nielsen and Shephard (2006), Jiang and Oomen (2008), andAït-Sahalia and Jacod (2008). The last three were selected for this paper to explore therelationship between volume and jumps.3.1 Barndorff-Nielsen and Shephard TestThe Barndorff-Nielsen and Shephard test (2004, 2006) is one of the mostcommonly used jump tests in literature. It uses the relationship between realized varianceand bipower variation in order to detect rare jumps in stock prices. If there are no jumps,bipower variation and realized variance asymptotically approach the integrated varianceas the sampling frequency reaches infinity.If each sampled data is represented by the letter i and there are M number of sampleddata per day, the daily realized variance is defined as the following.MRVt rt 2,i(5)i 1In the limit, the realized variance goes to the integrated variance plus the jumpcomponent, κ.tMt 1i 1lim( M ) RVt 2 ( s)ds 2 (t i )(6)Bipower variation is defined to be the following.BVt M M ri 1 ri2 M 1 i 2(7)7

In the limit, this bipower variation goes just to the integrated variance.tlim( M ) BVt 2 ( s)ds(8)t 1These asymptotic properties only hold at high sampling frequencies, which is the case forthe data used in this paper. Our interest lies in the jump component and so we will try toisolate this component. This could be achieved by obtaining the difference between therealized variance and the bipower variation. Huang and Tauchen (2005) used thisvariable called relative jump.RJ t RVt BVtRVt(9)It is necessary to studentize this value so that we can compare the statistic to thenormal distribution. After studentizing, the numbers will be scale-free and can be used inhypothesis testing. As a result, it is necessary to find the standard deviation. BarndorffNielsen and Shephard (2006) recommended using the quadpower quarticity term, definedhere. M MQPt M 1 4 rt , j 3 rt , j 2 rt , j 1 rt , j M 3 j 4where 1 (10)2 In the limit, this becomes the integrated quarticity.tlim( M )QPt s4 dst 1(11)With this information, Barndorff-Nielsen and Shephard (2006) suggested the ratio maxadjusted test statistic which is a one sided normal with variance one.z BNS RJ t 1QPt 5 max( 1,)BVt 2 4 M2(12)This is the test statistic that will be used later on in order to find the relationship betweenvolume and jump days; the higher the statistic, the higher probability that the daycontains a jump.8

3.2 Jiang and Oomen TestBased partially on the work of Barndorff-Nielsen and Shephard, Jiang and Oomen(2008) devised their own method to test for jump days. It is based on a variable in whichthey called „swap-variance.‟ It is motivated by an observation in finance that, in theabsence of jumps, the difference between simple return and log returns captures one halfof the integrated variance. This observation is well known in financial literature and is thebasis of the variance swap replication strategy, a strategy that will only work in theabsence of jumps. Specifically, this is the equivalent of a delta hedged log contract whenthere is no discontinuity.Let us define some variables. Capitalized R is the geometric return of the real price.Rt ,i st ,i st ,i 1(13)st ,i 1Swap variance is defined to beMSwVt 2 Rt ,i 2 lni 1STS0(14)Where ST is the price at the end of the day and S0 is the opening price.In the probability limit, swap variance and realized variance (defined in equation5) should be equal if there is no discontinuity. In the absence of jumps, the onlydifference between RV and SwV should come as a result of discretization, but this value islikely to be small and negligible. With jumps, a noticeable difference should be observed.The sign of the difference between swap variance and realized variance is alsosignificant. A negative result means that a negative jump occurred and a positive resultmeans that a positive jump occurred.The task is again to studentize the test statistic so it is distributed normally andthus the test statistic is easy to interpret. Jiang and Oomen came up with three simple teststatistics that are distributed as standard normals. This paper picked only one of the three:the ratio test. The formula of the ratio test is as follows:Z JO BVt M RVt 1 SwV SwVt (15)where M is the number of sampled data per day.9

The formula uses the fact that the ratio of realized variance over the swapvariance is supposed to be 1. Studentization was completed by multiplying the bipowervariation and dividing through by standard deviation, which corresponds to the squareroot of the omega term, defined below SwV t 693 ( u ) du 2 6 M 3 3 /429 M 3M ri 43/ 2t ,irt ,i 13/ 2rt ,i 23/ 2rt ,i 33/ 2(16)awhere μa E ( x ) and x is normally distributed with mean 0 and variance 1.The test statistic that will be used to find the relationship between volume andjump will be the absolute value of equation 15. Since a jump will occur if the absolutevalue of