Price Response Functions and Spread Impact in Correlated Financial Markets

Price response functions and spread impact in correlated ﬁnancial markets

Juan C. Henao-Londono a, Sebastian M. Krause, and Thomas Guhr FakultätfürPhysik, UniversitätDuisburg-Essen, Lotharstraße 1, 47048 Duisburg, Germany

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Abstract Recent research on the response of stock prices to trading activity revealed long lasting effects, even across stocks of different companies. These results imply non-Markovian effects in price formation and when trading many stocks at the same time, in particular trading costs and price correlations. How the price response is measured depends on data set and research focus. However, it is important to clarify, how the details of the price response definition modify the results. Here, we evaluate different price response implementations for the Trades and Quotes (TAQ) data set from the NASDAQ stock market and find that the results are qualitatively the same for two different definitions of time scale, but the response can vary by up to a factor of two. Further, we show the key importance of the order between trade signs and returns, displaying the changes in the signal strength. Moreover, we confirm the dominating contribution of immediate price response directly after a trade, as we find that delayed responses are suppressed. Finally, we test the impact of the spread in the price response, detecting that large spreads have stronger impact.

PACS. 89.65.Gh Econophysics – 89.75.-k Complex systems – 05.10.Gg Statistical physics

1 Introduction by a decrease as the time lag grows. In Ref. [17], Gerig found that larger sized transactions have a larger absolute Financial markets use order books to list the number of impact than smaller sized transactions but a much smaller shares bid or asked at each price. An order book is an relative impact. In Ref. [1], it is found that the impact of electronic list of buy and sell orders for a specific security small trades on the price is, in relative terms, much larger or financial instrument organized by price levels, where than that of large trades and the impact of trading on the agents can place different types of instructions (orders). price is quasi-permanent. In general, the dynamics of the prices follow a random For price cross-responses functions, Refs. [4,43] revealed walk. There are two extreme models that can describe this that the diagonal terms are on average larger than the behavior: the Efficient Market Hypothesis (EMH) and the off-diagonal ones by a factor ∼ 5. The response at posi- Zero Intelligence Trading (ZIT). The EMH states that all tive times is roughly constant, what is consistent with the available information is included in the price and price hypothesis of a statistically efficient price. Thus, the cur- changes can only be the result of unanticipated news, rent sign does not predict future returns. In Ref. [43] the which by definition are totally unpredictable [6,8,24,34]. trends in the cross-responses were found not to depend on On the other hand, the ZIT assumes that agents instead whether or not the stock pairs are in the same economic of being fully rational, have “zero intelligence” and ran- sector or extend over two sectors. domly buy or sell. It is supposed that their actions are Here, we want to discuss, based on a series of detailed arXiv:2010.15105v1 [q-fin.ST] 28 Oct 2020 interpreted by other agents as potentially containing some empirical results obtained on trade by trade data, that information [6,8,34,43]. In both cases the outcome is the the variation in the details of the parameters used in the same, the prices follow a random walk. Reality is some- price response definition modify the characteristics of the where in-between [8,34], and non-Markovian effects due results. Aspects like time scale, time shift, time lag and to stategies or liquidity costs are not contained either. spread used in the price response calculation have an in- There are diverse studies focused on the price response fluence on the outcomes. To facilitate the reproduction of [1,4,5,6,8,14,16,17,20,23,24,29,31,40,41,42,43]. In our opin- our results, the source code for the data analysis is avail- ion, a critical investigation of definitions and methods and able in Ref. [25]. how they affect the results is called for. We delve into the key details needed to compute the Regarding price self-response functions in Refs. [5,6,8], price response functions, and explore their correspond- Bouchaud et al. found an increase to a maximum followed ing roles. We perform a empirical study in different time scales. We show that the order between the trade signs a e-mail: [email protected] and the returns have a key importance in the price res- 2 J. C. Henao-Londono et al.: Price response functions and spread impact in correlated financial markets ponse signal. We split the time lag to understand the con- 3 Price response function definitions tribution of the immediate returns and the late returns. Finally, we shed light on the spread impact in the response In Sect. 3.1 we introduce the fundamental quantities used functions for single stocks. in the price response definitions. In Sect. 3.2 we describe The paper is organized as follows: in Sect.2 we present the physical time scale and the trade time scale. We in- our data set of stocks. We then analyze the definition of troduce the price response functions used in literature in the price response functions and describe the physical and Sect. 3.3. trade time in Sect.3. We implement different price responses for several stocks and pairs of stocks in Sect.4. In Sect.5 we show how the relative position between trade 3.1 Key concepts signs and returns has a huge influence in the results of the computation of the response functions. In Sect.6 we ex- Market orders are execute at the best available buy or sell plain in detail how the time lag τ behaves in the response price, limit orders set a maximum purchase price for a buy functions. Finally, in Sect.7 we analyze the spread impact order, or a minimum sale price for a sell order. If the limit in the price response functions. Our conclusions follow in price is not matched, the order will not be carried out Sect.8. [14,15,22,34]. Limit orders often fail to result in an immediate transaction, and are stored in a queue called the limit order book [1,9,15,22]. The order book also identifies 2 Data set the market participants behind the buy and sell orders, although some choose to remain anonymous. The order book is visible for all traders and its main purpose is to Modern financial markets, are organized as a double con- ensure that all traders have the same information on what tinuous auctions. Agents can place different types of or- is offered on the market. The order book is the ultimate ders to buy or to sell a given number of shares, roughly microscopic level of description of financial markets. categorized as market orders and limit orders. Buy limit orders are called “bids”, and sell limit or- In this study, we analyzed trades and quotes (TAQ) ders are called “asks”. At any given time there is a best data from the NASDAQ stock market. We selected NAS- (lowest) offer to sell with price a (t), and a best (highest) DAQ because it is an electronic exchange where stocks bid to buy with price b (t)[1,6,10,12,34]. The price gap are traded through an automated network of computers between them is called the spread s (t) = a (t) − b (t)[6,7, instead of a trading floor, which makes trading more ef- 8,10,14,21,34]. Spreads are significantly positively related ficient, fast and accurate. Furthermore, NASDAQ is the to price and significantly negatively related to trading vol- second largest stock exchange based on market capitaliza- ume. Companies with more liquidity tend to have lower tion in the world. spreads [2,3,10,18]. In the TAQ data set, there are two data files for each The average of the best ask and the best bid is the stock. One gives the list of all successive quotes. Thus, we midpoint price, which is defined as [1,6,8,14,21,34] have the best bid price, best ask price, available volume and the time stamp accurate to the second. The other data a (t) + b (t) m (t) = (1) file is the list of all successive trades, with the traded price, 2 traded volume and time stamp accurate to the second. Despite the one second accuracy of the time stamps, in As the midpoint price depends on the quotes, it changes both files more than one quote or trade may be recorded if the quotes change. The midpoint price grows if the best in the same second. ask or the best bid grow. On the other hand, the midpoint To analyze the response functions across different stocks price decreases if the best ask or the best bid decrease. in Sects.4,5 and6, we select the six companies with the Price changes are typically characterized as returns. If largest average market capitalization (AMC) in three eco- one denotes S (t) the price of an asset at time t, the return nomic sectors of the S&P index in 2008. Table1 shows the r(g) (t, τ), at time t and time lag τ is simply the relative companies analyzed with their corresponding symbol and variation of the price from t to t + τ [6,11,26,27,28,33], sector, and three average values for a year. To analyze the spread impact in response functions S (t + τ) − S (t) r(g) (t, τ) = (2) (Sect.7), we select 524 stocks in the NASDAQ stock mar- S (t) ket for the year 2008. The selected stocks are listed in AppendixA. It is also common to define the returns as [4,6,11,13,14, In order to avoid overnight effects and any artifact due 16,17,20,30,32] to the opening and closing of the market, we systemat- ically discard the first ten and the last ten minutes of S (t + τ) r(l) (t, τ) = ln S (t + τ) − ln S (t) = ln (3) trading in a given day [8,14,20,43]. Therefore, we only S (t) consider trades of the same day from 9:40:00 to 15:50:00 New York local time. We will refer to this interval of time Equations (2) and (3) agree if τ is small enough [6,11]. as the “market time”. The year 2008 corresponds to 253 At longer timescales, midpoint prices and transaction business days. prices rarely differ by more than half the spread. The J. C. Henao-Londono et al.: Price response functions and spread impact in correlated financial markets 3

Table 1. Analyzed companies.

Company Symbol Sector Quotes1 Trades2 Spread3 Alphabet Inc. GOOG Information Technology (IT) 164489 19029 0.40$ Mastercard Inc. MA Information Technology (IT) 98909 6977 0.38$ CME Group Inc. CME Financials (F) 98188 3032 1.08$ Goldman Sachs Group Inc. GS Financials (F) 160470 26227 0.11$ Transocean Ltd. RIG Energy (E) 107092 11641 0.12$ Apache Corp. APA Energy (E) 103074 8889 0.13$ 1 Average number of quotes from 9:40:00 to 15:50:00 New York time during 2008. 2 Average number of trades from 9:40:00 to 15:50:00 New York time during 2008. 3 Average spread from 9:40:00 to 15:50:00 New York time during 2008. midpoint price is more convenient to study because it the time scales, the corresponding quantities of each time avoids problems associated with the tendency of trans- scale are not mixed. Thus physical time scale is measured action prices to bounce back and forth between the best in seconds and trade time scale is measured in trades. bid and ask [14]. We define the returns via the midpoint price as Due to the nature of the data, they are several options to define time for analyzing data. m (t + τ) − m (t) r (t, τ) = (4) In general, the time series are indexed in calendar time m (t) (hours, minutes, seconds, milliseconds). Moreover, tick-by- The distribution of returns is strongly non-Gaussian and tick data available on financial markets all over the world its shape continuously depends on the return period τ. is time stamped up to the millisecond, but the order of Small τ values have fat tails return distributions [6]. The magnitude of the guaranteed precision is much larger, usu- trade signs are defined for general cases as ally one second or a few hundreds of milliseconds [7,11]. In several papers are used different time definitions (calendar ε (t) = sign (S (t) − m (t − δ)) (5) time, physical time, event time, trade time, tick time) [11, 19,36]. The TAQ data used in the analysis has the char- where δ is a positive time increment. Hence we have acteristic that the trades and quotes can not be directly related due to the time stamp resolution of only one sec- +1, If S (t) is higher than the last m (t) ond [43]. Hence, it is impossible to match each trade with ε (t) = (6) −1, If S (t) is lower than the last m (t) the directly preceding quote. However, using a classification for the trade signs, we can compute trade signs in two ε(t) = +1 indicates that the trade was triggered by a scales: trade time scale and physical time scale. market order to buy and a trade triggered by a market order to sell yields ε(t) = −1 [6,8,20,29,37]. The trade time scale is increased by one unit each time It is well-known that the series of the trade signs on a a transaction happens. The advantage of this count is that given stock exhibit large autocorrelation. A very plausible limit orders far away in the order book do not increase the explanation of this phenomenon relies on the execution time by one unit. The main outcome of trade time scale is strategies of some major brokers on given markets. These its “smoothing” of data and the aggregational normality brokers have large transactions to execute on the account [11]. of some clients. In order to avoid market making move- The physical time scale is increased by one unit each ments because of an inconsiderable large order, they tend time a second passes. This means that computing the re- to split large orders into small ones [11]. sponses in this scale involves sampling [19,43], which has to be done carefully when dealing for example with several stocks with different liquidity. This sampling is made 3.2 Time definition in the trade signs and in the midpoint prices. A direct comparison between the trade time scale and the Facing the impossibility to relate midpoint prices and physical time scale is not possible. To compare them di- trade signs with the TAQ data in trade time scale, we will rectly we need to assume whether the midpoint price or use the midpoint price of the previous second with all the the trade signs are on the same scale. We assume the mid- trade signs of the current second. This will be our defini- point prices in the trade time scale to be the same as the tion of trade time scale analysis for the response function midpoint prices in physical time scale. Therefore, we have analysis. the time lag for both computations in seconds. This approximation allow us to directly compare both scales to For physical time scale, as we can sampling, we relate have an idea of the difference and similarities they have. the unique value of midpoint price of a previous second In the other sections, as we are not directly comparing with the unique trade sign value of the current second. 4 J. C. Henao-Londono et al.: Price response functions and spread impact in correlated financial markets

Trade time scale

We use the trade sign classification in trade time scale proposed in Ref. [43] and used in Refs. [38,39,42] that reads Figure 1. Sketch of data processing for trade time scale. In the midpoint price time line, the vertical lines represent the   sgn (S (t, n) − S (t, n − 1)) , if change in price of the quotes and the circles represent the last ε(t) (t, n) = S (t, n) 6= S (t, n − 1) (7) price change in a quote in a second. In the trade signs time line,  ε(t) (t, n − 1) , otherwise the squares represent the buy market orders and the triangles represent the sell market orders. The midpoint price time line and the trade sign time line are shifted in one second. ε(t) (t, n) = +1 implies a trade triggered by a market order to buy, and a value ε(t) (t, n) = −1 indicates a trade triggered by a market order to sell. In the second case of Eq. (7), if two consecutive trades with the same trading direction did not exhaust all the available volume at the best quote, the trades would have the same price, and in consequence they will have the same trade sign. With this classification we obtain trade signs for every single trade in the data set. According to Ref. [43], the average accuracy of the classification is 85% for the trade time scale. The TAQ time step is one second, and as it is impossible to find the correspondences between trades and midpoint prices values inside a second step, We used the last midpoint price of every second as the representative value of each second. This introduce an apparent shift between trade signs and returns. In fact, we set the last midpoint price from the previous second as the first midpoint price of the current second [43]. As we know the second in which the trades were made, we can relate the trade signs and the midpoint prices as Figure 2. Sketch of the return contributions from every mid- shown in Fig.1. For the trade time scale, there are in point price change in a second. The squares represent the rise general, several midpoint prices in a second. For each sec- of the price of the midpoint price and the triangles represent ond we select the last midpoint price value, and we relate the decrease of the price of the midpoint price. We illustrate three cases: (top left) the changes of the midpoint prices and it to the next second trades. In Fig.1, the last midpoint return are due to the rise of the prices, (top right) the changes price (circle) between the second −1 and 0 is related to of the midpoint prices and return are due to the decrease of all the trades (squares and triangles) in the second 0 to 1, the prices, and (bottom) the changes of the midpoint prices and so on. In the seconds when the quotes do not change, and return are due to a combination of rise and decrease of the the value of the previous second (vertical line over the prices. The blue vertical line represents the net return in each physical time interval) is used. Thus, all the seconds in case. the open market time have a midpoint price value, and in consequence returns values. We assume that as long as no changes occurred in the quotes, the midpoint price three different returns for these three midpoint price val- remains the same as in the previous second. ues. Furthermore, suppose that the volume of limit orders The methodology described is an approximation to with the corresponding midpoint prices are the same in compute the response in the trade time scale. A draw- the bid and in the ask (the returns have the same mag- back in the computation could come from the fact that nitude). In the case of the top left (top right) sketch, all the return of a given second is composed by the contribu- the changes are due to the rise (decrease) of the midpoint tion of small returns corresponding to each change in the price, that means, consumption of the best ask (bid), so midpoint price during a second. As we are assuming only all the contributions of the individual returns in the sec- one value for the returns in each second, we consider all ond are positive (negative), and in consequence, the net the returns in one second interval to be positive or neg- return is positive (negative). In the case of the bottom, ative with the same magnitude, which could not be the the changes are due to a combination of increase and de- case. This could increase or decrease the response signal crease of the midpoint price, so in the end, the individual at the end of the computation. returns sum up to a net return, which can be positive or Figure2 illustrate this point. Suppose there are three negative, depending of the type of midpoint price values in different midpoint prices in one second interval and thus, the interval. Thus, in this case, we are assuming in the end J. C. Henao-Londono et al.: Price response functions and spread impact in correlated financial markets 5 that all the returns were positive or negative, which prob- Fig.3, where we related the midpoint price of the previous ably was not the case, and in consequence will increase or second with the trade sign of the current second. Accord- decrease the real value of the net return. ing to Ref. [43], this definition has an average accuracy up In all cases, we choose the last change in the midpoint to 82% in the physical time scale. price in a second interval as described before in Fig.1. We use this method knowing that the variation in one second of the midpoint price is not large (in average, the 3.3 Response function definitions last midpoint price of a second differ with the average midpoint of that second in 0.007%), hence it can give us The response function measures price changes resulting representative information on the response functions. from execution of market orders. In Refs. [5,6,8], Bouchaud et al. use a self-response function that only depends on the time lag τ. This function measures how much, on av- Physical time scale erage, the price moves up (down) at time τ conditioned to a buy (sell) order at time zero. They found for France We use the trade sign definition in physical time scale pro- Telecom that the response function increases by a factor posed in Ref. [43] and used in Refs. [38,42], that depends 2 between τ = 1 and τ ≈ 1000 trades, before decreas- on the classification in Eq. (7) and reads ing back. For larger τ, the response function decreases, and even becomes negative beyond τ ≈ 5000. However, in ( sgn PN(t) ε(t) (t, n) , If N (t) > 0 some cases the maximum is not observed and rather the ε(p) (t) = n=1 (8) price response function keeps increasing mildly [6]. 0, If N (t) = 0 In Ref. [17], the price impact function, is defined as the average price response due to a transaction as a func- where N (t) is the number of trades in a second interval. tion of the transaction’s volume. Empirically the function (p) ε (t) = +1 implies that the majority of trades in second is highly concave [17]. The curvature of the price impact t were triggered by a market order to buy, and a value function is entirely due to the probability that a transac- (p) ε (t) = −1 indicates a majority of sell market orders. In tion causes a nonzero impact. The larger the size of the this definition, they are two ways to obtain ε(p) (t) = 0. transaction, the larger the probability. In Ref. [1], they One way is that in a particular second there are no trades, found that the response function for three French stocks and then no trade sign. The other way is that the addition first increases from τ = 10s to a few hundred seconds, and of the trade signs (+1 and −1) in a second be equal to then appears to decrease back to a finite value. zero. In this case, there is a balance of buy and sell market In Ref. [4] they defined a response function who mea- orders. sures the average price change of a contract i at time t+τ, Market orders show opposite trade directions to limit after experiencing a sign imbalance in contract j at time order executed simultaneously. An executed sell limit or- t. In this work τ is used in units of five minutes. der corresponds to a buyer-initiated market order. An ex- In later works [20,43], Grimm et al. and Wang et al. use ecuted buy limit order corresponds to a seller-initiated the logarithmic return for stock i and time lag τ, defined market order. via the midpoint price mi (t) to define a cross-response As the trade time scale, on the physical time scale we function. The response function measures how a buy or use the same strategy to obtain the midpoint price for sell order at time t influences on average the price at a later every second, so all the seconds in the open market time time t + τ. The physical time scale was chosen since the have a midpoint price value. Even if there is no change of trades in different stocks are not synchronous (TAQ data). quotes in a second, it still has a midpoint price value and They found that in all cases, an increase to a maximum is return value. followed by a decrease. The trend is eventually reversed. Finally, in Ref. [40], Wang et al. define the response function on a trade time scale (Totalview data), as the interest is to analyze the immediate responses. Here, the time lag τ is restricted to one trade, such that the price response quantifies the price impact of a single trade.

Figure 3. Sketch of data processing for physical time scale. 4 Price response function implementations In the midpoint price time line, the horizontal lines between seconds represent the midpoint prices. In the trade signs time The main objective of this work is to analyze the price line, the horizontal lines between seconds represent the trade response functions. In general we define the self- and cross- sign values. The midpoint price time line and the trade sign response functions in a correlated financial market as time line are shifted in one second. (scale) D (scale) (scale) E Rij (τ) = ri (t − 1, τ) εj (t) (9) average In this case we do not compare every single trade sign in a second, but the net trade sign obtained for every sec- where the index i and j correspond to stocks in the mar- (scale) ond with the definition, see Eq. (8). This can be seen in ket, ri is the return of the stock i in a time lag τ 6 J. C. Henao-Londono et al.: Price response functions and spread impact in correlated financial markets

(scale) in the corresponding scale and εj is the trade sign where the superscript t refers to the trade time scale. We of the stock j in the corresponding scale. The superscript explicitly calculate the average in Eq. (12), scale refers to the time scale used, whether physical time scale (scale = p) or trade time scale (scale = t). Fi- (t) 1 Rij (τ) = nally, The subscript average refers to the way to average Pdf Ptf d=d t=t Nj,d (t) the price response, whether relative to the physical time 0 0 df tf N(t) scale (average = P ) or relative to the trade time scale X X X (p) (t) (average = T ). ri,d (t − 1, τ) εj,d (t, n) (13) d=d0 t=t0 n=1 We use the returns and the trade signs to deﬁne three df tf PN(t) (t) response functions: trade time scale response, physical time X X (p) n=1 εj,d (t, n) = ri,d (t − 1, τ) scale response and activity response. Pdf Ptf N (t) d=d0 t=t0 d=d0 t=t0 j,d

To compare the three response functions, we define the df tf following quantities X X (p) (t) = ri,d (t − 1, τ) sgn (Ej,d (t)) wj,d (t) d=d0 t=t0 N(t) (14) X (t) E (t) = ε (t, n) = sgn (E (t)) |E (t)| (10) j,d j,d j,d j,d where n=1 (t) |Ej,d (t)| (p) w (t) = (15) j,d df tf εj,d (t) = sgn (Ej,d (t)) (11) P P N (t) d=d0 t=t0 j,d is a weight function that depends on the normalization of where the subscript d refers to the days used in the res- the response. ponse computation. We use Eq. (10) to make easier the To compute the response functions on trade time scale, comparison between the results, as all the defined responses we used all the trade signs during a day in market time. use the trade sign term. As we can not associate an individual midpoint price with their corresponding trade signs, all the trade signs in one In Sect. 4.1 we analyze the responses functions in trade second are associated with the midpoint price of the pre- time scale, in Sect. 4.2 we analyze the responses functions vious second. As τ depends on the midpoint price, even if in physical time scale and in Sect. 4.3 we define a response we are using trade signs in trade time scale, the value of function to analyze the influence of the frequency of trades τ is in seconds. in a second. The results of Fig.4 show the self- responses of the six stocks used in the analysis and the cross-responses for pairs of stocks representing three different economic sectors. The self-response functions increase to a maximum 4.1 Response functions on trade time scale and then slowly decrease. In some stocks this behavior is more pronounced than in others. For our selected tick- ers, a time lag of τ = 103s is enough to see an increase to a maximum followed by a decrease. Thus, the trend We define the self- and cross-response functions in trade in the self-response functions is eventually reversed. On time scale, using the trade signs in trade time scale. For the other hand, the cross-response functions have smaller the returns, we select the last midpoint price of every sec- signal strength than the self-response functions. For our ond and compute them. We use this strategy with the cross-response functions of stocks in the same sectors, some couples exhibit the increase-decrease behavior in- TAQ data set considering that the price response in trade 3 time scale can not be directly compared with the price side a time lag of τ = 10 s. Other couples seems to need response in physical time scale. In this case we relate each a larger time lag to reach the decrease behavior. trade sign in one second with the midpoint price of the previous second. Then, to compute the returns, instead of using trades as the time lag (it would make no sense as 4.2 Response functions on physical time scale all the midpoint price are the same in one second) we use seconds. Thus, we force the response in trade time scale One important detail to compute the market response to have a physical time lag, and then, be able to compare in physical time scale is to define how the averaging of with the physical time scale response. This approximation the function will be made, because the response functions (p) is feasible considering the discussion in Sect.3. The price highly differ when we include or exclude εj (t) = 0 [43]. response function in trade time scale is defined as (p) The price responses including εj (t) = 0 are weaker than the excluding ones due to the omission of direct influence D E of the lack of trades. However, either including or exclud- (t) (p) (t) p Rij (τ) = ri (t − 1, τ) εj (t, n) (12) ing ε (t) = 0 does not change the trend of price reversion T j J. C. Henao-Londono et al.: Price response functions and spread impact in correlated financial markets 7

(t) Figure 4. Self- and cross-response functions Rij (τ) in 2008 versus time lag τ on a logarithmic scale in trade time scale. Self-response functions (left) of individual stocks and cross-response functions (right) of stock pairs from the same economic sector. versus the time lag, but it does affect the response func- is a weight function that depends on the normalization tion strength [42]. For a deeper analysis of the influence of of the response. (p) the term εj (t) = 0, we suggest to check Refs. [42,43]. We The results showed in Fig.5 are the self- and cross- will only take in account the response functions excluding response functions in physical time scale. For the self- p εj (t) = 0. response functions we can say again that in almost all the We define the self- and cross-response functions in phys- cases, an increase to a maximum is followed by a decrease. ical time scale, using the trade signs and the returns in Thus, the trend in the self- and cross-response is eventu- physical time scale. The price response function on phys- ally reversed. In the cross-response functions, we have a ical time scale is defined as similar behavior with the previous subsection, where the

(p) D (p) (p) E time lag in some pairs was not enough to see the decrease Rij (τ) = ri (t − 1, τ) εj (t) (16) of the response. P Compared with the response functions in trade time where the superscript p refers to the physical time scale. scale, the response functions in physical time scale are The explicit expression corresponding to Eq. (16) reads stronger.

(p) 1 Rij (τ) = h i Pdf Ptf η ε(p) (t) d=d0 t=t0 j,d

df tf X X (p) (p) h (p) i 4.3 Activity response functions on physical time scale ri,d (t − 1, τ) εj,d (t) η εj (t) (17) d=d0 t=t0 (p) h (p) i df tf ε (t) η ε (t) X X (p) j,d j,d = ri,d (t − 1, τ) h i Finally, we deﬁne the activity self- and cross-response func- Pdf Ptf (p) t=t η ε (t) d=d0 0 d=d0 t=t0 j,d tions in physical time scale, using the trade signs and the

df tf returns in physical time scale. We add a factor Nj,d (t) X X (p) (p) to check the inﬂuence of the frequency of trades in a sec- = r (t − 1, τ) sgn (Ej,d (t)) w (t) i,d j,d ond in the response functions. The activity price response d=d0 t=t0 (18) function is deﬁned as where 1, If x 6= 0 η (x) = (19) 0, otherwise (p,a) D (p) (p) E Rij (τ) = ri (t − 1, τ) εj (t) N (t) (21) P take only in account the seconds with trades and

(p) η [sgn (Ej,d (t))] where the superscript a refers to the activity response wj,d (t) = (20) Pdf Ptf η [sgn (E (t))] function. The corresponding explicit expression reads d=d0 t=t0 j,d 8 J. C. Henao-Londono et al.: Price response functions and spread impact in correlated ﬁnancial markets

(p) (p) Figure 5. Self- and cross-response functions Rij (τ) excluding εj (t) = 0 in 2008 versus time lag τ on a logarithmic scale in physical time scale. Self-response functions (left) of individual stocks and cross-response functions (right) of stock pairs from the same economic sector.

We propose a methodology to directly compare price response functions in trade time scale and physical time (p,a) 1 scale. Additionally, we suggest a new definition to measure Rij (τ) = Pdf Ptf N (t) the impact of the number of trades in physical time scale. d=d0 t=t0 j,d In the three curves in the figure can be seen the increase- df tf X X (p) (p) r (t − 1, τ) ε (t) N (t) (22) decrease behavior of the response functions. i,d j,d j,d Our results are consistent with the current literature, d=d t=t 0 0 where the results differ about a factor of two depending on df tf (p) X X (p) εj,d (t) Nj,d (t) the time scale. We note that the activity response function = ri,d (t − 1, τ) Pdf Ptf implementation is only a test and was never defined in Nj,d (t) d=d0 t=t0 d=d0 t=t0 previous works. That is why the difference of a factor of (p,a) df tf two can not be seen in Fig.6 for R (τ). X X (p) (a) ij = ri,d (t − 1, τ) sgn (Ej,d (t)) wj,d (t) d=d0 t=t0 (23) 5 Time shift response functions where (a) Nj,d (t) The relative position between returns and trade signs di- wj,d (t) = (24) Pdf Ptf rectly impact the result of the response functions. Shifting Nj,d (t) d=d0 t=t0 the values to the right or to the left either in trade time is a weight function that depends on the normalization of scale or physical time scale have approximately the same the response. effect. As Ej,d (t) is the sum of +1 and −1 in one second and To test this claim, we used the definition of the res- Nj,d (t) is the number of trades in a second, Nj,d (t) ≥ ponse function from Ref. [43] and add a parameter ts that Ej,d (t). Nj,d (t) = Ej,d (t) only when all the trades in a shifts the position between returns and trade signs. To see second are buys (+1). the impact of the time shift we analyzed the stocks showed (t) in Table1 in the year 2008. We used different time shifts The trade weight wj,d (t) reduces noises, The physi- (p) in the response function cal weight wj,d (t) gives every step the same weight, and (a) D E the activity weight w (t) emphasizes seconds with large (scale,s) (scale) (scale) j,d Rij (τ) = ri (t − ts, τ) εj (t) activity. average In Fig.6, we can see how the three responses have (25) where the index i and j correspond to stocks in the mar- approximately the same shape, but the strength of the (scale) signal varies depending on the definition. The frequency ket, ri is the return of the stock i in a time lag of trades have a large influence in the responses. τ with a time shift ts in the corresponding scale and (scale) As predicted by the weights, the event response is εj is the trade sign of the stock j in the correspond- weaker than the physical response, and the activity res- ing scale. The superscript scale refers to the time scale ponse is the strongest response. used, whether physical time scale (scale = p) or trade J. C. Henao-Londono et al.: Price response functions and spread impact in correlated financial markets 9

(scale) (p) Figure 6. Self- and cross-response functions Rij (τ) excluding εj (t) = 0 in 2008 versus time lag τ on a logarithmic scale. Self-response functions (left) of Goldman Sachs Group Inc. stock and cross-response functions (right) of Goldman Sachs Group Inc.-CME Group Inc. stocks.

(scale,s) time scale (scale = t). Rij is the time shift price res- 10 trades (left) and 100 trades (right). In both, self- and ponse function, where the superscript s refers to the time cross-response results are qualitatively the same. It can be shift. Finally, The subscript average refers to the way to seen that the response functions have a zero signal before average the price response, whether relative to the physi- the time shift. After the returns and trade signs find their cal time scale (average = P ) or relative to the trade time corresponding order the signals grow. In comparison with scale (average = T ). the values obtained in Fig.4, it looks like the response We compute the response functions according to two function values with large time shift are stronger. How- cases. In one case we set τ to a constant value and vary ever, this is an effect of the averaging of the functions. As ts, and in the other case we set ts to a constant value and the returns and trade signs are shifted, there are less val- vary τ. ues to average, and then the signals are stronger. Anyway, In Sect. 5.1 we analyze the influence of the time shift the figure shows the importance of the position order be- between the trade signs and returns in trade time scale tween the trade signs and returns to compute the response and in Sect. 5.2 we analyze the influence of the time shift function. between the trade signs and returns in physical time scale.

5.2 Physical time scale shift response functions 5.1 Trade time scale shift response functions

On the trade time scale we compute the response function In the physical time scale we compute the response function (t,s) D (t) (t) E Rij (τ) = ri (t − ts, τ) εj (t) (26) t (p,s) D (p) (p) E Rij (τ) = ri (t − ts, τ) εj (t) (27) (t) t In this case for ri , we associate all the trade signs to a return value and create pseudo midpoint price values in Similar to the results in Subsect. 5.1, Fig.9 shows the trade time scale. Then, we shift the trade signs and the responses functions for fixed τ values while ts is variable. returns by trades. Hence, the time lag and time shift are Again, the response functions are zero if the time shift in trade time scale. is larger than the time lag, or if the time shift is smaller In Fig.7, we show the response functions results for than zero. For every τ value, there is a peak. The peak fixed τ values while ts is variable. In the different τ val- grows and decay relatively fast. The response signal usu- ues figures, the results are almost the same. The response ally starts to grow in zero or a little bit earlier and grows functions are zero either if the time shift is larger than τ, to a value around to τ. In this zone the response functions or if the time shift is smaller than zero. However, related are different to zero. to the time lag, there is a zone where the signal is different The results for fixed time shift values and variable time from zero. For values between zero and τ there is a peak lag (ts = 10s and ts = 100s) are shown in Fig. 10. The self- in a position related to τ. The response function grows and cross-response results are qualitatively the same com- and decreases relatively fast. pared with the previous subsection. The response func- We tested the response function for fixed time shift tions are zero before the time shift value. After the re- values while τ is variable. In Fig.8 we use a time shift of turns and the trade signs reach their order, the signals 10 J. C. Henao-Londono et al.: Price response functions and spread impact in correlated financial markets

(t,s) Figure 7. Self-response functions Rii (τ) in 2008 versus shift for the Transocean Ltd. stock (top) and cross-response functions (t,s) Rij (τ) in 2008 versus shift for the Transocean Ltd.-Apache Corp. stocks (bottom) in trade time scale.

(t,s) Figure 8. Self- and cross-response functions Rij (τ) in 2008 versus time lag τ on a logarithmic scale for different shifts in trade time scale. Self-response functions (top) of individual stocks and cross-response functions (bottom) of stocks pairs from the same economic sector. We use time lag values τ = 10 trades (left) and τ = 100 (right). grow. The same effect of the apparent stronger signal can and the relation between them. When we shift the trade be seen here, and again, it is due to the averaging values. signs and returns, this order is temporarily lost and as The results in trade time scale and physical time scale outcome the signal does not have any meaningful infor- can be explained understanding the dynamics of the mar- mation. When the order is recovered during the shift, the ket. A trade can or can not change the price of a ticker. signal grows again, showing response function values dif- Therefore, when a change in price happens, a change in ferent to zero. In this section we were interested only in midpoint price, and consequently in returns happens. Thus, the order (shift) and not in the responses values, which it is extremely important to keep the order of the events were analyzed in Sect.4. A time shift smaller than zero J. C. Henao-Londono et al.: Price response functions and spread impact in correlated financial markets 11

(p,s) (p) Figure 9. Self-response functions Rii (τ) excluding εi (t) = 0 in 2008 versus shift for the Transocean Ltd. stock (top) (p,s) (p) and cross-response functions Rij (τ) excluding εj (t) = 0 in 2008 versus shift for the Transocean Ltd.-Apache Corp. stocks (bottom) in physical time scale. does not have any useful information about the response. estimated by the immediate response on an event time If the time shift is equal to zero, the signal is weak, due scale. In our case, we check all over the range of the time to the time needed by the market to react to the new lag in the price response function on a physical time scale. information. On the other hand, a time shift larger than To use the short and long time lag, we rewrite the two steps shows the information is lost and the signal only returns in physical time scale as grows when the original order is resumed. Then the question is what is the ideal time shift to compute the response functions. Our approach in Sect.4 (p,sl) mi (t + τ) ri (t, τ) = ln takes in account that the changes in the quotes are the mi (t) 0 ones that attract the agents to buy or sell their shares. mi (t + τ) mi (t + τ ) Hence, they directly impact the trade signs. According to = ln 0 mi (t + τ ) mi (t) the results, the response can take up to two time steps in 0 the corresponding scale to react to the change in quotes. mi (t + τ) mi (t + τ ) = ln 0 + ln Thus, a time shift larger than two time steps makes no mi (t + τ ) mi (t) 0 0 sense. On the other hand, in the case of the physical time mi (t + τ) − mi (t + τ ) mi (t + τ ) − mi (t) scale, where a sampling is used, to assure the selection of ≈ 0 + mi (t + τ ) mi (t) a midpoint price at the beginning of a second, it is a good (29) strategy to use the last midpoint of the previous second as the first midpoint price of the current second. In this where the superscript sl refers to short-long and the case an apparent one second shift is used between returns second term of the right part is constant with respect to and trade signs. τ. Replacing Eq. (29) in the price response function in physical time scale (Eq. (16)) we have 6 Time lag analysis (p,sl) D (p,sl) (p) E Rij (τ) = ri (t − 1, τ) εj (t) Regarding Equation (2), we use a time lag τ in the returns P 0 to see the gains or loses in a future time. However, the mi (t − 1 + τ) − mi (t − 1 + τ ) (p) ≈ 0 εj (t) strength of the return in the time lag should not be equal mi (t − 1 + τ ) P along its length. Then, we divide the full range time lag τ 0 mi (t − 1 + τ ) − mi (t − 1) (p) in an immediate time lag and in a late time lag as show + ε (t) m (t − 1) j in Fig. 11, where i P (30) τ = τ 0 + (τ − τ 0) (28) Where the first term in the right side of Eq. (30) is the for τ 0 < τ. This distinguish the returns depending on the long response and the right term is the short response. time lag as the short (immediate) return τ 0 with the long Again, the right term of Eq. (30) is independent of τ. return τ −τ 0. This approach is similar to the concept used The results in Fig. 12 show the short response, the in Ref. [40], where the price impact for a single trade is long response, the addition of the short response and long 12 J. C. Henao-Londono et al.: Price response functions and spread impact in correlated financial markets

(p,s) (p) Figure 10. Self- and cross-response functions Rij (τ) excluding εj (t) = 0 in 2008 versus time lag τ on a logarithmic scale for diﬀerent shifts in physical time scale. Self-responses functions (top) of individual stocks and cross-response functions (bottom) of stocks pairs from the same economic sector. We use time lag values τ = 10 trades (left) and τ = 100 (right).

line) has the same shape to the addition of the short and long response (green line). For the response functions that show the increase-decrease 3 Figure 11. τ value divided in short and long time lag. behavior in between the time lag τ = 10 , the peak is usually between τ = 101 and τ = 102. In these cases the long response are always negative after the τ 0 value and is comparable in magnitude with the random signal. On the response (Sum), the original response, a random response 0 other hand, the response functions that requires a bigger and the value of τ . time lag to show the increase-decrease behavior, have non The main signal of the response function come from negative long responses, but still they are comparable in the short response. Depending on the stock and the value magnitude with the random signal. of τ 0 the long response can increase or decrease the short According to our results, price response functions show response signal, but in general the long response does not a large impact on the first instants of the time lag. We pro- give a significant contribution to the complete response. posed a new methodology to measure this effect and evalu- Before τ 0, the short response and long response are the ate their consequences. The short response dominates the same, as the self and cross-response definition do not de- signal, and the long response vanishes. fine values smaller than τ 0, so it is computed as the original response. In the figure, the curves of the short and long response are under the curve of the original response. After 7 Spread impact in price response functions τ 0, the short response is a strong constant signal. On the other hand, the long response immediately fades, showing When we calculate the price response functions, the sig- the small contribution to the final response. To compare nal of the response depends directly on the analyzed stock. the significance of the long response, We added a random Thus, even if the responses functions are in the same scale, response made with the trade signs used to compute the their values differ from one to another. We choose the response but with a shuffle order. The long response and spread [35] to group 524 stocks in the NASDAQ stock the random response are comparable, and show how the market for the year 2008 in physical time scale, and check long response is not that representative in the final res- how the average strength of the price self-response func- ponse. If we add the short and long response, we obtain tions in physical time scale behaved for this groups. For the original response. In Fig. 12, the original response (red each stock we compute the spread in every second along J. C. Henao-Londono et al.: Price response functions and spread impact in correlated financial markets 13

(p,sl) (p) Figure 12. Self- and cross-response functions Rij (τ) excluding εj (t) = 0 in 2008 versus time lag τ on a logarithmic scale using a τ 0 = 40 in physical time scale. Self-response functions (left) of Alphabet Inc. stock and cross-response functions (right) of Alphabet Inc.-Mastercard Inc. stocks. the market time. Then we average the spread during the 253 business days in 2008. With this value we group the stocks. We used three intervals to select the stocks groups (s < 0.05$, 0.05$ ≤ s < 0.10$ and 0.10$ ≤ s < 0.40$). The detailed information of stocks, the spread and the groups can be seen in AppendixA. With the groups of the stocks defined, we averaged the price response functions of each group. In Fig. 13 we show the average response functions for the three groups. The average price response function for the stocks with smaller spreads (more liquid) have in average the weakest signal in the figure. On the other hand, Figure 13. Average price self-response functions R(p) (τ) ex- the average price response function for the stocks with ii cluding ε(p) (t) = 0 in 2008 versus time lag τ on a logarithmic larger spreads (less liquid) have in average the strongest i signal. According to the results described in Sect.3 and scale in physical time scale for 524 stocks divided in three rep- 4, the average price response functions for all the groups resentative groups. follow the increment to a maximum followed by a decrease in the signal intensity. 8 Conclusion The strength of the price self-response function signals grouped by the spread can be explained knowing that the We went into detail about the response functions in corre- response functions directly depend on the trade signs. As lated financial markets. We define the trade time scale and long as the stock is liquid, the number of trade signs grow. physical time scale to compute the self- and cross-response Thus, at the moment of the averaging, the large amount functions for six companies with the largest average mar- of trades, reduces the response function signal. Therefore, ket capitalization for three different economic sectors of the response function decrease as long as the liquidity the S&P index in 2008. Due to the characteristics of the grows, and as stated in the introduction the spread is neg- data used, we had to classify and sampling values to ob- atively related to trading volume, hence, firms with more tain the corresponding quantities in different time scales. liquidity tend to have lower spreads. The classification and sampling of the data had impact on Finally, an interesting behavior can be seen in Fig. 13. the results, making them smoother or stronger, but always Despite each stock has a particular price response accord- keeping their shape and behavior. ing to the returns and trade signs, the average price res- The response functions were analyzed according to the ponse function for the different groups seems to be quite time scales. We proposed a new approach to compare price similar. As all the analyzed stocks come from the same response functions from different scales. We used the same market we can infer that the general behavior of the mar- midpoint prices in physical time scale with the correspond- ket affect all the stocks, influencing in average a group ing trade signs in trade time scale or physical time scale. response. This assumption allowed us to compare both price res- 14 J. C. Henao-Londono et al.: Price response functions and spread impact in correlated financial markets ponse functions and get an idea of how representative was We thank S. Wang for fruitful discussions. One of us (JCHL) the behavior obtained in both cases. For trade time scale, acknowledges financial support from the German Academic the signal is weaker due to the large averaging values from Exchange Service (DAAD) with the program “Research Grants all the trades in a year. In the physical time scale, the - Doctoral Programmes in Germany” (Funding programme response functions had less noise and their signal were 57381412) stronger. We proposed an activity response to measure how the number of trades in every second highly impact the responses. As the response functions can not grow in- Appendix A NASDAQ stocks used to definitely with the time lag, they increase to a peak, to analyze the spread impact then decrease. It can be seen that the market needs time to react and revert the growing. In both time scale cases depending on the stocks, two characteristics behavior were We analyzed the spread impact in the response functions shown. In one, the time lag was large enough to show the for 524 stocks from the NASDAQ stock market for the complete increase-decrease behavior. In the other case, the year 2008. In Tables2,3,4 and5, we listed the stocks in time lag was not enough, so some stocks only showed the their corresponding spread groups. growing behavior. We modify the response function to add a time shift parameter. With this parameter we wanted to analyze References the importance in the order of the relation between returns and trade signs. In trade time scale and physical 1. More statistical properties of order books and price im- time scale we found similar results. When we shift the or- pact. Physica A: Statistical Mechanics and its Applica- der between returns and trade signs, the information from tions, 324(1):133 – 140, 2003. Proceedings of the Interna- the relation between them is temporarily lost and as out- tional Econophysics Conference. come the signal does not have any meaningful information. 2. Hee-Joon Ahn, Jun Cai, Yasushi Hamao, and Richard Y.K When the order is recovered, the response function grows Ho. The components of the bid–ask spread in a limit-order again, showing the expected shape. We showed that this market: evidence from the tokyo stock exchange. Journal is not an isolated conduct, and that all the shares used in of Empirical Finance, 9(4):399 – 430, 2002. our analysis exhibit the same behavior. Thus, even if they 3. Yakov Amihud and Haim Mendelson. The effects of beta, are values of time shift that can give a response function bid-ask spread, residual risk, and size on stock returns. signal, empirically we propose this time shift should be a The Journal of Finance, 44(2):479–486, 1989. value between ts = (0, 2] time steps. 4. M Benzaquen, I Mastromatteo, Z Eisler, and J-P We analyzed the impact of the time lag in the response Bouchaud. Dissecting cross-impact on stock markets: an functions. We divided the time lag in a short and long time empirical analysis. Journal of Statistical Mechanics: The- lag. With this division we adapted the price response func- ory and Experiment, 2017(2):023406, Feb 2017. tion in physical time scale. The response function that 5. J. P. Bouchaud, J. Kockelkoren, and M. Potters. Random depended on the short time lag, showed a stronger res- walks, liquidity molasses and critical response in financial markets, 2004. ponse. The long response function vanish, and depending 6. Jean-Philippe Bouchaud. The subtle nature of financial on the stock could take negative and non-negative values random walks. Chaos: An Interdisciplinary Journal of comparable to a random signal. Nonlinear Science, 15(2):026104, 2005. Finally, we checked the spread impact in price self- 7. Jean-Philippe Bouchaud, J. Doyne Farmer, and Fabrizio response functions. We divided 524 stocks from the NAS- Lillo. How markets slowly digest changes in supply and DAQ stock market in three groups depending on the year demand. 2008. average spread of every stock. The response functions sig- 8. Jean-Philippe Bouchaud, Yuval Gefen, Marc Potters, and nal were stronger for the group of stocks with the larger Matthieu Wyart. Fluctuations and response in financial spreads and weaker for the group of stocks with the smaller markets: the subtle nature of “random” price changes. spreads. A general average price response behavior was Quantitative Finance, 4(2):176–190, Apr 2004. spotted for the three groups, suggesting a market effect 9. Jean-Philippe Bouchaud, Marc Mézard,and Marc Potters. on the stocks. Statistical properties of stock order books: empirical results and models. Quantitative Finance, 2(4):251–256, Aug 2002. 10. Carolyn Callahan, Charles Lee, and Teri Yohn. Account- 9 Author contribution statement ing information and bid-ask spread. Accounting Horizons, 11:50–60, 01 1997. 11. Anirban Chakraborti, Ioane Toke, Marco Patriarca, and TG proposed the research. SMK and JCHL developed the FrédéricAbergel. Econophysics: Empirical facts and agent- method of analysis. The idea to look the time shift and to based models. arXiv.org, Quantitative Finance Papers, 09 analyze the spread impact was due to JCHL, and the idea 2009. of the time lag analysis was due to SMK. JCHL carried 12. Kee H Chung, Bonnie F [Van Ness], and Robert A [Van out the analysis. All the authors contributed equally to Ness]. Limit orders and the bid–ask spread. Journal of analyze the results and write the paper. Financial Economics, 53(2):255 – 287, 1999. J. C. Henao-Londono et al.: Price response functions and spread impact in correlated financial markets 15 1 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ 02$ ...... Symbol Company Spread 1 01$01$01$ AA01$ KR01$ MRK01$ Alcoa Corp. NSM01$ Kroger Merck Company WMT &01$ Company Inc. FITB Nationstar01$ Mortgage Holdings Walmart Inc. EK01$ 0 Fifth PHM Third01$ Bancorp JPM01$ 0 Eastman PulteGroup Kodak WAG01$ Inc. Company 0 SCHW JP01$ Morgan Chase Walgreen Co. & 0 RF01$ Co. Charles Schwab Corp. 0 ADBE01$ 0 MAT01$ 0 Adobe Regions Inc. PAYX Financial01$ Corp. 0 Mattel PGR01$ Inc. Paychex Inc. 0 HPQ 0 01$ DOW Progressive01$ Corp. 0 TE HP01$ Inc. 0 Dow Inc. BBBY01$ JNJ01$ Bed TECO Bath IP Energy &01$ Inc. Beyond Inc. RSH01$ 0 Johnson & Johnson MER01$ 0 0 0 HAL International01$ Respiri Paper Ltd. Company 0 Mears KO01$ Group PLC 0 PBCT Halliburton01$ Company 0 WU01$ People’s Coca-Cola United Company 0 USB Financial01$ 0 0 Inc. MCHP01$ Western 0 Union Company SO01$ U.S. Microchip Bancorp Technology 0 Inc. BJS 0 01$ MAS01$ 0 0 Southern 0 NWL Company01$ 0 BJ’s Wholesale Club M Masco01$ Holdings Corp. Newell CVS Brands01$ 0 Inc. CTSH02$ GNW02$ Macy’s CVS Inc. Health Cognizant Corp. Tech. DFS Solutions 0 02$ 0 Genworth Financial KEY Inc. ADI Discover Financial 0 Services 0 SYY KeyCorp 0 Analog Devices 0 Inc. 0 Sysco 0 Corp. 0 0 0 0 01$01$01$ TSN CA XEL Tyson Foods Inc. CA Inc. Xcel Energy Inc. 0 0 0 ...... Information of the stocks in Group 1. Table 2. LUV Southwest Airlines Company 0 1 01$01$01$ BRCM01$ TXN01$ Broadcom Inc. TER01$ MYL Texas01$ Instruments Inc. HCBK Teradyne01$ Inc. Mylan SPLS01$ N.V. Hudson City Bancorp SGP01$ Staples 0 HST Inc.01$ AES 0 01$ Siamgas and Petrochemicals KFT01$ Host Hotels & 0 NTAP Resorts 0 01$ Aes Inc. Corp. BAC Kraft01$ 0 Foods NetApp Inc. Inc. HD01$ 0 0 SOV Bank01$ of America Corp. QLGC01$ Home 0 Depot T Life01$ Inc. QLogic Storage Corp. Inc. GPS01$ DIS01$ 0 0 GM01$ Gap AT&T Inc. Inc. 0 JNPR 0 01$ Walt Disney Company LOW01$ General Juniper Motors 0 Networks CBS Company01$ Inc. 0 Lowe’s CAG Companies01$ Inc. 0 LLTC01$ CBS 0 0 Corp. ConAgra DTV01$ Brands Inc. Linear 0 Technology HBAN Corp.01$ 0 DirecTV KG01$ Group Huntington 0 Bancshares Inc. CMS01$ 0 0 ODP01$ 0 0 Kinross Gold NVLS CMS01$ Corp. Energy Corp. WFC Office01$ Depot Inc. Nivalis Therapeutics MO01$ Inc. 0 Wells VZ Fargo01$ & 0 Company DHI01$ Altria 0 Group 0 SNDK Inc.01$ 0 Verizon Communications CCE01$ D. 0 Sandisk R. Corp. Horton NI01$ Inc. 0 EXPE Coca-Cola01$ Enterprises 0 AIG01$ Expedia Group NiSource INTU Inc.01$ Inc. 0 LTD01$ American International Intuit Inc. CNP 0 0 QCOM Limited Brands 0 Inc. JBL CenterPoint Qualcomm Energy Inc. Inc. 0 0 Jabil Inc. 0 0 0 0 0 0 ...... Group 1 Symbol Company Spread Average spread from 9:40:00 to 15:50:00 New York time during 2008. 1 SymbolF CompanyQETFCPFE Ford E-Trade MotorMOT Financial Company Corp. Qwest CommunicationsAMD Int. Pfizer Inc.TLAB Motus 0 GI Holdings 0 INTC Inc. Advanced Micro 0 Devices TellabsTWX Inc. Spread CSCO Intel Corp. 0 THC 0 Time Warner Inc. CiscoLSI Systems Inc.MU Tenet Healthcare Corp.EMC 0 LifeMSFT Storage Inc. MicronNOVL Technology EMC Inc. 0 Corp. 0 0 MicrosoftJAVA 0 Corp. Novell 0 ORCL Inc. SunS 0 Microsystems Inc. OracleDELL Corp. 0 AMAT DellSLE Technologies Inc. Applied 0 Sprint 0 MaterialSBUX Nextel Inc. Corp. 0 DYN Spark Starbucks EnergyDUK Corp. Inc. 0 0 CMCSA Dynergy Inc. 0 0 IPG Comcast Duke Corp. Energy 0 Corp.BSXGE Interpublic GroupSYMC 0 of Boston Co. 0 ScientificC Corp. SymantecCPWR Corp. General 0 Electric Company 0 NVDA 0 0 Ocean ThermalYHOO 0 Energy 0 Citigroup Nvidia Inc. Corp.BMY Yahoo Inc.ALTR 0 GLW Bristol-Myers Squibb 0 Co. AltairJDSU Engineering Inc.NCC Corning Inc. 0 EBAY JDS Uniphase Corp.EP 0 NCC 0 Group 0 eBayXRX Inc.WIN 0 XLNX El Xerox Paso 0 Holdings Corp. Corp. Windstream Holdings Xilinx Inc. Inc. 0 0 0 0 0 0 0 16 J. C. Henao-Londono et al.: Price response functions and spread impact in correlated financial markets 1 03$ 03$ 03$ 03$ 03$ 03$ 03$ 03$ 03$ 03$ 03$ 03$ 03$ 03$ 03$ 03$ 03$ 03$ 03$ 03$ 03$ 03$ 03$ 03$ 03$ 03$ 03$ 03$ 03$ 04$ 04$ 04$ 04$ 04$ 04$ 04$ 04$ 04$ 04$ 04$ 04$ 04$ 04$ 04$ 04$ ...... Symbol Company Spread 1 02$ MI Marshall and Lisley Corp. 0 02$02$02$ AEP02$ NEM02$ MRO American02$ Electric Power Co. Newmont ITW02$ Corp. Marathon FFIV Oil02$ Corp. CVX Illionois02$ Tool 0 Works F5 Inc. PCAR Networks02$ Inc. OMC Chevron02$ Corp. PACCAR Inc. XRAY02$ Omnicom COP Group02$ DENTSPLY Inc. SIRONA Inc. PCS 0 02$ 0 0 RHI ConocoPhillips03$ SEE03$ MetroPCS Communications COST03$ 0 Robert 0 Half International FIS03$ Inc. Sealed Costco Air Wholesale 0 Corp. PKI Corp.03$ 0 0 0 BMC03$ 0 Fidelity National RRD Info.03$ PerkinElmer Services Inc. BMC UTX03$ Software Inc. 0 R.R. D03$ 0 Donnelley 0 & Sons PBI United03$ Co. Technologies Corp. ACAS03$ K 0 03$ Dominion Pitney American Energy Bowes Capital Inc. Inc. JCP Ltd.03$ 0 0 AMT03$ 0 0 ALL03$ Kellogg J. Company C. Penney American MWV Company03$ Tower Corp. HRB03$ Allstate Corp. MeadWestvaco Corp. 0 0 NYT03$ RDC H&R03$ Block 0 Inc. PTV New03$ York times Company 0 FISV Redcape03$ 0 Hotel Group EXPD Pactiv03$ 0 Company Fiserv BBT03$ Inc. 0 Expeditors Int.of PCG Washington03$ 0 BIG BB&T03$ Corp. 0 KMX Pacific03$ 0 Gas & 0 Electric TSS 0 03$ Co. Big Lots CarMax Inc. BK Inc.03$ TEL03$ Total 0 System Services KSS03$ Inc. The 0 Bank CAT Tellurian of Inc. N. Y. HSY Mellon Kohl’s Corp. Corp. 0 GIS Caterpillar Inc. 0 0 0 The Hershey Company General Mills Inc. 0 0 0 0 0 0 0 ...... Information of the stocks in Group 1. Table 3. TGT Target Corp. 0 VLO Valero Energy Corp. 0 1 $ 02 02$02$02$ MMC02$ CPB02$ Marsh TYC &02$ McLennan Co. MCD Campbell02$ Soup Company HON Tyco02$ International PLC McDonald’s DF02$ Corp. 0 Honeywell NWSA02$ International Inc. 0 URBN02$ News Corp. Dean 0 Foods RX02$ 0 Company Urban Outfitters Inc. BBY02$ CCL02$ Recylex 0 WMI Best02$ Buy Inc. POM02$ Carnival 0 Corp. WMI ERTS02$ Investment 0 Corp. Polymet MBI02$ Mining Corp. Electronic Arts ADM02$ Inc. 0 PEP02$ MBIA Inc. Archer-Daniels-Midland CBG Co. 0 02$ IR02$ PepsiCo Inc. 0 0 0 HNZ CBRE02$ 0 Group Inc. CTX02$ 0 0 Ingersoll TSO Heinz02$ Rand Company Inc. IGT Qwest02$ Corp. WYN02$ Tesoro Corp. GT02$ International Game Wynnstay Tech. 0 JCI Group02$ 0 PLC JWN02$ 0 The 0 Goodyear FRX02$ Tire & Johnson 0 Rubber Controls 0 Nordstrom FHN Int.02$ Inc. 0 ABT Fennec02$ 0 Pharmaceutical Inc. ADP 0 First02$ Horizon National Corp. 0 PBG Abbott02$ Laboratories 0 ROST Automatic02$ Data 0 0 Processing KBH Pacific02$ Brands Ross Ltd. Stores YUM Inc.02$ 0 0 KB MAR02$ Home Yum! STJ Brands 0 02$ Inc. Marriott FDO02$ International ED02$ St Jude 0 Medical UNM Family02$ Inc. Dollar Stores Inc. ORLY 0 Consolidated Unum Edison SVU Group 0 Inc. O’Reilly Automotive Inc. EMR 0 0 SUPERVALU Inc. 0 Emerson 0 Electric 0 Company 0 0 0 0 ...... Group 1 Symbol Company Spread Average spread from 9:40:00 to 15:50:00 New York time during 2008. 1 SymbolWMB CompanySE Williams CompaniesPG Inc.CIEN 0 SeaLEN Ltd American Procter Dep. Ciena &AKAM Corp. Gamble Co.UNH Lennar Akamai Technologies Corp. 0 DD Spread 0 UnitedHealthKLAC Group Inc.CIT 0 KLA DuPont Corp. 0 deLEG Nemours Inc.WFMI 0 CIT Group 0 Inc.MS Leggett 0 Whole & Foods Platt Market Inc.VRSN Inc.AN 0 VeriSign Morgan Inc. StanleyRHTCTXS 0 0 AutoNation 0 Inc.MDT Red Hat Citrix Inc. SystemsNBR Inc. MedtronicMOLX plc. 0 GILD Nabors Molex Industries Inc. 0 CHK 0 0 GileadTJX Sciences Inc. ChesapeakeAMGN Energy 0 SWY Amgen TJX 0 0 Inc. Companies Inc.XOM 0 SafewaySTZ Inc. 0 ExxonADSK Mobil 0 Corp. 0 LLY Constellation Autodesk Brands Inc.CTASLIZ Eli Lilly Cintas and Corp. 0 0 GCI Company 0 AXP 0 Liz ClaiborneTIE Inc. 0 Gannett Co. Inc.SAI American 0 Express Co.PDCO Titanium Metals Corp.WYE Patterson SAIC Companies 0 Inc. 0 COH 0 Wyeth 0 CVG Inc. 0 CochlearWDC Ltd. 0 ConvergysAVP Corp. Western DigitalA Corp.JNY Avon Products Inc.SLM 0 0 Jones Agilent Apparel Technologies 0 0 Group 0 SLM Corp. 0 0 0 0 J. C. Henao-Londono et al.: Price response functions and spread impact in correlated financial markets 17 1 06$ 07$ 06$ 06$ 06$ 06$ 06$ 06$ 06$ 06$ 06$ 06$ 06$ 06$ 06$ 06$ 06$ 06$ 06$ 06$ 06$ 06$ 07$ 07$ 07$ 07$ 07$ 07$ 07$ 07$ 07$ 07$ 07$ 07$ 07$ 07$ 07$ 07$ 07$ 07$ 07$ 07$ 07$ 07$ 07$ ...... Symbol Company Spread 1 05$ CAM Corporate Actions Middleware 0 06$ NOV National Oilwell Varco Inc. 0 05$ RAI Reynolds American Inc. 0 06$ SUN Sunoco LP 0 05$05$05$ NOC05$ PLL05$ Northrop SYK05$ Grumman Corp. SRE05$ Piedmont Lithium Ltd. TIF Stryker05$ Corp. NUE05$ Sempra 0 Energy OXY05$ Tiffany & Co. FPL Nucor05$ Corp. 0 Occidental ESRX05$ Petroleum Corp. AIV05$ First Trust Express New Scripts BHI Opportunities05$ Holding Co. 0 FCX05$ 0 Apartment Investment Co. APOL 0 05$ Boulevard 0 0 Holdings Inc. SCG Freeport-McMoRan05$ Inc. Apollo Group CB Inc.05$ 0 0 IBM05$ Scentre Group 0 Ltd. GME05$ 0 Chubb 0 Limited DE05$ Int. Business Machines GameStop LNC Corp.05$ Corp. AVY05$ Deere 0 0 & MCK Lincoln05$ Company National Corp. Avery PLD05$ 0 Dennison Corp. McKesson DGX Corp.05$ CERN05$ Prologis Inc. Quest 0 IRM05$ Diagnostics Cerner Inc. 0 Corp. 0 COL05$ 0 NU 0 05$ Irom Mountain Inc. CVH Rockwell05$ Collins Inc. WFR05$ NeutriSci 0 0 International AGN Coventry05$ Inc. Health Care Inc. MEMC GAS Electronic05$ Materials Allergan APH PLC. 0 0 EXC 0 GAS 0 0 0 0 WEC Amphenol Corp. Exelon06$ Corp. WEC Energy06$ Group Inc.06$ BTU06$ ROK06$ 0 0 PCL Peabody06$ Energy Corp. 0 Rockwell JOYG Automation Inc. AMP Plum Creek Joy Global Timber TAP Inc. Co. 0 Inc. Ameriprise 0 Financial 0 Inc. 0 Molson 0 Coors Beverage Co. 0 0 0 06$ AFL AFLAC Inc. 0 ...... Information of the stocks in Group 1 and 2. Table 4. DOV Dover Corp. 0 WLP WellPoint Inc. 0 1 1 04$ VIAB Viacom Inc. 0 05$ TWC TWC Entreprises Ltd. 0 04$04$04$ ABC04$ APC04$ CBE AmerisourceBergen04$ Corp. FAST Anadarko04$ Petroleum Corp. MHP Cooper04$ Industries Fastenal Company 0 AMZN04$ 0 McGraw-Hill EQR04$ Companies Amazon.com Inc. Inc. CL04$ 0 ECL Equity04$ Residential MKC04$ Colgate-Palmolive 0 AOC Company 0 04$ Ecolab Inc. McCormick TXT &04$ Company 0 Inc. Aon CSX04$ 0 Corp. 0 PNW Textron04$ Inc. 0 KIM04$ CSX Corp. Pinnacle WPI West04$ Capital Corp. MET04$ Kimco Realty Corp. 0 UST04$ Watson Pharmaceuticals Metlife TMO04$ Inc. 0 HCP04$ ProShares Ultra Thermo SIAL Fisher04$ 0 Scientific 0 Inc. 0 FLIR HCP04$ 0 Inc. 0 Sigma-Aldrich EL04$ Corp. 0 FLIR AYE04$ Systems Inc. CI04$ Estee NFLX Allegheny Lauder04$ Energy Companies Inc. 0 CHRW 0 04$ Netflix 0 Inc. Cigna GENZ04$ C.H. Corp. Robinson Worlwide Inc. DDR04$ 0 Genzyme 0 Corp. WFT 0 04$ 0 DDR MHS 0 04$ Corp. West DTE Fraser Timber Co. Ltd. TRV Medco Health Solutions Inc. NYX DTE 0 Energy Company The05$ Travelers 0 Companies 0 0 NYSE 0 05$ Group IPO05$ SWN 0 0 05$ NSC 0 05$ Southwestern CMA05$ Energy Co. NKE Norfolk Southern Corp. 0 Comerica EQ Inc. CLX Nike 0 Inc. Equillium Inc. 0 Clorox Company 0 0 0 0 ...... Group 1 Symbol Company Spread Group 2 COF Capital One Financial Corp. 0 Average spread from 9:40:00 to 15:50:00 New York time during 2008. 1 SymbolHAS Company Hasbro Inc. Spread 0 SymbolHOT Company Hot Topic Inc. Spread 0 XTOPPL XTOHOG Energy Inc.UPS PPL Corp. Harley-DavidsonHSP Inc.CTL United Parcel ServiceTDC Hospira Inc. 0 0 BA CenturyLink Inc.BAX Teradata Corp. 0 PGN Boeing CompanyDRI Baxter 0 International Inc.JNS Progress Energy Inc. 0 0 BMS Darden Restaurants 0 Inc.RTN 0 Janus Capital GroupCVC Bristol-Myers Inc. 0 Squibb 0 0 PWR Raytheon Company 0 LXK Cablevision Systems QuantaCELG 0 Services Inc.MMM Lexmark International Celgene 0 Corp.RSG 0 3M CompanyDNR 0 0 MTW Republic Services Inc. DenburyNE Resources Inc. Manitowoc CompanyNDAQCINF 0 0 Nasdaq 0 Inc. Noble Corp.BIIB 0 0 CincinnatiKMB FinancialEIX Biogen Inc. Kimberly-ClarkNRG Corp.IVZ 0 Edison International NRGAEE Energy Inc. 0 0 AAPL 0 Invesco Ltd.CAH Ameren Corp. Apple 0 Inc.MCO 0 Cardinal Health Inc. 0 Moody’s Corp.GPC 0 0 0 PEGEFX Genuine Parts Co 0 COV Public 0 Service Ent.AET Equifax Inc. CovidienXL Ltd. Aetna Inc. 0 0 XL Group Ltd. 0 0 0 0 18 J. C. Henao-Londono et al.: Price response functions and spread impact in correlated financial markets 1 13$ 13$ 13$ 13$ 14$ 14$ 14$ 14$ 15$ 15$ 15$ 15$ 16$ 16$ 16$ 17$ 17$ 17$ 17$ 18$ 18$ 19$ 19$ 20$ 20$ 20$ 21$ 21$ 23$ 24$ 25$ 27$ 28$ 38$ 38$ 38$ 13$ ...... Symbol CompanyETR Entergy Corp. Spread 0 1 1 09$09$09$ MUR ROP Murphy JEC Oil10$ Corp. Roper10$ Technologies Inc.10$ Jura Energy Corp. MON10$ 0 DV10$ 0 Monsanto VFC Company10$ GWW10$ Dolly Varden 0 WHR V.F.10$ Silver Corp. Corp. W. W. MOS Grainger10$ Inc. 0 0 Whirlpool SPG Corp.10$ EOG Mosaic10$ Company VMC10$ Simon Property 0 Group X EOG10$ Resources Inc. Vulcan SHLD Materials10$ Co LLL 0 10$ 0 0 Sears Holding 0 WYNN10$ United States Steel BXP 0 Wynn11$ Corp. L3 Resorts Technologies Ltd. 0 Inc. PSA11$ 0 PCP Boston11$ Properties Inc. VNO11$ Public Storaga DNB11$ Precision 0 Castparts 0 Corp. 0 Vornado CLF11$ Realty 0 FLR Dun 0 11$ & Bradstreet Corp. PCLN11$ Cleveland-Cliffs Inc. MTB11$ Fluor Corp. Priceline 0 Group AVB 0 Inc.12$ M&T BEN12$ Banc 0 Corp. DO 0 AvalonBay12$ Comm. Inc. CF Franklin12$ Resources Inc. 0 AZO12$ Diamond 0 Offshore FLS12$ Drilling CF 0 0 ICE Industries AutoZone12$ 0 0 Holdings Inc. MA12$ Flowserve Corp. FSLR12$ Intercontinental Exc. 0 Inc. CMG12$ Mastercard First Inc. Solar13$ Inc. 0 Chipotle13$ Mexican Grill13$ 0 0 13$ 0 13$ 0 0 ...... Information of the stocks in Group 2 and 3. ATI Allegheny Technologies Inc. 0 Table 5. 1 09$09$09$ NTRS09$ APD09$ Northern Trust ANR Corp.09$ NBL Air09$ Products and Chemicals Alpha APA Natural Resources FMC 0 Noble Energy Inc. 0 BDK Apache Corp. 0 FMC Corp. Black and Decker Corp. 0 0 0 0 07$07$07$ ETN07$ VTR07$ Eaton07$ Corp. Symbol Ventas07$ Inc. HAR Company07$ NFX07$ Harman EQT07$ Int. Industries Inc. HRS Nasdaq07$ Futures COG EQT07$ 0 Corp. PPG08$ Harris Corp. 0 Group 3 Cabot MEE08$ Oil & 0 Gas HP PPG08$ Corp. Industries Inc. Massey DVN08$ Energy Spread Company IFF08$ 0 Helmerich & Devon LH 0 08$ Energy Payne Inc. Corp. 0 UNP08$ Int. Flavors 0 0 & EW08$ Fragrances 0 Laboratory STT Union Corp.08$ 0 Pacific Corp. RRC08$ 0 Edwards 0 Lifesciences SJM Corp.08$ State Street HES Corp. SNA Range08$ Resources 0 Corp. PNC08$ J. M. Smucker 0 CEG Hess Company08$ Snap-On 0 Corp. Inc. LMT PNC08$ Fin. Services 0 Group FTI Constellation09$ Energy 0 Group 0 Lockheed CNX09$ Martin Corp. 0 ARG 0 09$ TechnipFMC PLC DVA CNX09$ Resources Corp. Amerigo GS09$ Resources Ltd. 0 DaVita TMK09$ Inc. 0 0 EMN09$ Goldman Torchmark RIG Corp. Sachs 0 09$ 0 Group 0 Inc. Eastman PX09$ Chemical Co. RL09$ 0 Transocean Ltd. R Pelangio Exploration MIL Inc. 0 Ralph CRM Lauren Corp. 0 0 Millipore Ryder 0 System Corp. Salesforce.com Inc. 0 0 0 0 0 ...... Group 2 Symbol Company Spread Average spread from 9:40:00 to 15:50:00 New York time during 2008. 1 PHACEBNI Parker-Hannifin Corp.ACS ACE Ltd.PXD Burlington Northern SantaAIZ Affiliated Fe Computer ServicesWAT Pioneer 0 Natural Resources 0 0 Assurant Inc. Waters Corp. 0 0 0 0 SymbolGD CompanyCSCFII General DynamicsHIG Corp. Computer Sciences Corp.ITT Federated InvestorsTROW Inc. Hartford Fin. ServicesMDP T. Group 0 ITT Rowe Price Inc. 0 GroupGR Inc. 0 MeredithAKS Corp. 0 Spread PFG Goodrich 0 Corp.HUM AK Steel Holding Corp.STI Principal Financial Group HumanaCMI Inc.STR SunTrust BanksZMH Inc. Cummins 0 0 Inc. 0 FO Questar Corp. 0 ZimmerSII Holdings Inc. 0 HRL Fortune BrandsOI Inc. 0 SprottHCN Hormel Inc. Foods Corp. 0 ANF 0 O-ILM Welltower Inc. Glass Inc. 0 FDX Abercrombie & 0 Fitch Co.PRU 0 Legg MasonDHR FedEx Inc. Corp. 0 WY Prudential Financial Inc. 0 DanaherOKE Corp.SRCL Weyerhaeuser 0 Company ONEOKLUK 0 Inc. Stericycle 0 0 Inc.BLLFE Leucadia National 0 Corp. 0 SWK Ball Corp. 0 ESV FirstEnergy StanleySHW Corp. 0 Black & Decker 0 Inc.BDX Ensco PLC Sherwin-WilliamsVAR Company 0 0 TEG Becton, 0 Dickinson and Co. Varian 0 Medical Systems Inc. Ten Ent. Group 0 PLC 0 0 0 0 0 J. C. Henao-Londono et al.: Price response functions and spread impact in correlated financial markets 19

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