# The Effect of Limit Order Flows At The Best Quotes On Price Change

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The Effect of Limit Order Flows At The Best Quotes On Price Change ChongSeok Hyuna , Jeongsook Parka , Kiseop Leeb,∗ a Graduate Department of Financial Engineering, Ajou University, 206 World-cup Street, Woncheon-dong, Yeongtong-gu, Suwon, 443-749, Korea. b Department of Mathematics, University of Louisville, Louisville, KY 40292, USA Abstract We test the effect of order book events at the best quotes on price change with the model proposed by Cont, Kukanov and Stoikov (2012). The OFI (Order Flow Imbalance) measure in the model could reasonably explain the price change of the nearby KOSPI 200 futures contract, which is one of the most liquid exchange traded securities in the world. The model gets to fit less accurate when the sampling time interval become shorter; the adjusted R2 of the model drops down to 40% for the time interval of 1 second from around 70% for the cases with time interval longer than 1 minute. We postulate that this is related to the lead-lag effects between the OFI measure and price change for shorter time intervals . We use the vector auto-regressive model to verify this conjecture, which is supported by the empirical evidence. The result in the paper suggests that it is necessary to know the dynamic structure of limit order book for better understanding of high frequency price movement in a financial market. Keywords: Granger causality, KRDS (Korea Research Data Services), order flow imbalance, limit order book event, price impact I This research was supported by WCU(World Class University) program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (R31-20007). ∗ Corresponding author Email addresses: ordeeq@ajou.ac.kr (ChongSeok Hyun), jsbak@ajou.ac.kr (Jeongsook Park), kiseop.lee@louisville.edu (Kiseop Lee) Preprint. . . Please Do Not Circulate This Article Without Permission!!! April 23, 2013

1. Introduction The growing importance of limit order markets needs better understading the dynamics of limit order book (LOB).1 Limit order book contains the information on the supply and demand schedule of liquidity, so that it is natural to expect the effect of order book events on the price change. As buy (sell) orders accumulate near the top of the book traders may compete to submit more aggressive buy (sell) limit orders, even market orders, to get their orders executed first, which drives up (down) price. Biais, Hillion and Spatt (1995), Kaniel and Liu (2006) and Cao, Hansch, and Wang (2009) report empirical result supporting that price revisions move in the direction of previous limit order flows. But the possibility of information asymmetry leave the traders to go adversely and be picked off by a superior trader having more accurate information (Copeland and Galai (1983)). If majority of traders revise their orders to control the picking-off risk (or free-option risk as in Fong and Liu (2010)), limit order flows can give an impact on price in the opposite direction, as supported by Griffiths et al. (2000), Chordia, Roll, and Subrahmanyam (2002), and Chordia and Subrahmanyam (2004). Since we are mixed with the predictability of returns from the order book events, we need to compare recent studies like Cont, Kukanov and Stoikov (2012), Hautsch and Huang (2012), and Eisler, Bouchaud, and Kockelkoren (2012) on the market impact of order book events. Among the proposed, the model by Cont, Kukanov and Stoikov (2012), the CKS model from now on, is characterized by the most parsimoneous. It just define the order flow imbalance (OFI) measure simply defined with market orders and limit orders observed at best quotes to draw the linear relation between price change and the OFI measure. To the contrary Hautsch and Huang (2012) propose a cointegrated vector autoregressive (VAR) model for quotes and order book depth and estimate impulse response functions. Eisler, Bouchaud, and Kockelkoren (2012) propose a linear superposition model of the impact 1 We cite Parlour and Seppi (2008): “Now in 2007 most equity and derivative exchanges around the world are either pure electronic limit order markets or at least allow for customer limit orders in addition to an on-exchange market making.” See also Jain (2005), and Swan and Westerholm (2006). 2

of past trades originated by Bouchaud et al. (2004). We wonder why one is simple and others are relatively more complex in discribing the same phenomenon. We choose the CKS model a base model to find out the differences among price impact models of order book events. The empirical test with the CKS model shows that it seems too simple to explain the real complexity and needs a careful interpretation when we compare it to such the classical price impact model by Kyle (1985) of trades. As in Eisler, Bouchaud, and Kockelkoren (2012), the assumption of high liquidity is critical in Cont, Kukanov and Stoikov (2012). If the limit order market often observes the bid-ask spreads and the gabs between quoted prices larger than the minimum tick size, the market would have to many venues of response to order book events.2 This will hinder the accuracy of such a model, which might be the reason that Hautsch and Huang (2012) assume a liquid market for their model. We test the CKS model in the KOSPI 200 index futures market, since it is has highly abundant liquidity for nearby futures contract but is lack of liquidity for second nearby products. The empirical result shows that the OFI measure the CKS model could explain the price change of the nearby futures contract. The adjusted R2 ’s of the linear regression go around 70% for the cases with time interval longer than 1 minute. This means that accumulation of bid (ask) orders at the best bid (ask) quotes is linearly related to the rise (decline) of the mid price, for the OFI measure is defined as the imbalance between the supply and the demand at the best bid and the best ask. By contrast, the model is not easy to fit the data for second nearby products. Even with nearby products we observe that the linear relation between the OFI measure and price change in the CKS model gets weakened when the sampling time intervals become shorter. The adjusted R2 of the linear regresssion drops down to 40 % for time interval of 1 second. A possibility explanation of this phenomena is the short term effects of both stale quotes and liquidity shocks. The accumulation of stale quotes would prevent the 2 Eisler, Bouchaud, and Kockelkoren (2012) use the term ‘a small (large) tick stock’ to refer a liquid (illiquid) stock. 3

fast response of mid-price on the one hand, the liquidity shock possibliy coming from granularity of the order book make the price change abruptly on the other hand. We observe the evidences showing this is really true. We give another possible explanation of low accuracy of the model by postulating the lead-lag effects between the OFI measure and price change for shorter time intervals. As argued in the above, traders could revise or cancel their orders by observing the current status of the order book. This process needs some minimum time interval to reach a stable status, so that the OFI measure could not explain price change enough for shorter intervals. We test the hypothesis by building a vector regression analysis is needed in the spirit of Hasbrouck (1991).3 The empirical findings in this paper suggest that such price impact model of order book events like Cont, Kukanov and Stoikov (2012) needs an extension that accommodate the dynamics of an order book.4 Actually Cont, Kukanov and Stoikov (2012) intended to apply the OFI based model to high frequency world, but it is proven to be not accurate in a world of seconds. This limit occurs mainly due to that the model could not explain the level of resiliency in such a high frequency area and just accept the existence of hidden liquidity in the market. The remainder of the paper is organized as follows. We first review the CKS model. Then we describe the data we used in this paper. Next two sections report the empirical result obtained by fitting the CKS model and testing the linear relation between the OFI 3 Brown, Walsh and Yuen (1997) use a VAR model in the Australian Stock Exchange to find bi- directional causality between order imbalance and stock price return. Hautsch and Huang (2012) use a cointegrated VAR model to test the case in Euronext Amsterdam. Wuyts (2012) simulates a VAR model to test resiliency of aggressive orders in an order-driven market. 4 The general theoretical works on modeling a dynamic equilibrium in an order book market include Parlour (1998), Foucault (1999), Foucault, Kadan and Kandel (2005), Goettler, Parlour and Rajan (2005), Goettler, Parlour and Rajan (2009) etc. Parlour and Seppi (2008) is an excellent review of this issue. For empirical evidence on the importance of dynamic order book structure, on ma refer to Obizhaeva and Wang (2013) and references therein. 4

measure and price change. We conclude the paper in the last section. 2. The Price Impact Model and Data 2.1. The CKS Model In this section, we briefly summarize the price impact model proposed by Cont, Kukanov and Stoikov (2012). Let us consider a time interval [tk−1 , tk ]. Let Lbk be the total size of buy orders that arrived to the current best bid during that time interval. Similarly, let Ckb be the total size of buy orders that were canceled from the current best bid during that time interval. We denote by Mkb the total size of marketable buy orders that arrived to the current best ask. The quantities Lsk , Cks , and Mks for sell orders are defined analogously. Pkb and Pka denote the bid price and the ask price at time tk respectively. The order flow imbalance measure for each k is defined as follows: (1) OF Ik = Lbk − Ckb − Mks − Lsk + Cks + Mkb . We consider the normalized mid price Pkb + Pka Pk = , 2δ where δ denotes the tick size. We assume that an order book can be approximated locally with uniform distributions, in the sense that the number of shares (depth) at each price level beyond the best and ask is constant. Let us denote this value on the time interval [Ti−1 , Ti ] by Di . We also assume that order arrivals and cancelations occur only at the best bid and ask. Moreover, when the bid (or ask) size reaches Di , the next passive order arrives on tick above (or below) the best quote, initializing the new best level. Then, we expect the following linear relation between price changes and OFI on each short intervals of time [tk−1,i , tk,i ] ⊂ [Ti−1 , Ti ]: (2) ∆Pk,i = βi OF Ik,i + k,i , 5

where ∆Pk,i = Pk,i −Pk,i−1 and k,i is the measurement error. βi is a price impact coefficient for an i-th time interval, which is allowed to change with index i. Cont, Kukanov and 1 Stoikov (2012) interpret 2βi as an implied estimation of the order book depth Di .5 2.2. Data The KOSPI 200 index futures market is a purely order-driven market where liquidity and prices are formed by by traders through the open competition. Since the opening in 1996, it has been one of the most liquid exchange traded markets. As of 2011, it is ranked the sixth by the number of contracts traded among the equity index futures markets in the world (Figure 1). Figure 1: The Worldwide Rankings of the KOSPI 200 Index Futures Market Source: Futures Industry Association. The KOSPI 200 index futures market has two kinds of auction – the single price call auction and the continuous auction. On every 9:00 AM, it runs the single price call auction 5 Cont, Kukanov and Stoikov (2012) claims that the price impact coefficient is a better estimate of Kyle’s λ than traditional estimates based on trade data. This is obviously not the case, for Kyle (1985) consider an informed trader who uses market orders to utilize private information to generate profits. Since the OFI measure involves limit orders by a trader regardless of informativeness, we can not compare two price impact coefficient directly. 6

with orders accumulated from 8:00 AM to the auction time. The continuous auction starts just after the first single price call auction and stops at 15:05 PM.6 The second single price call auction clears the market at 15:15 PM with orders accumlated until then. Order execution during the continuous auction is subject to the priority of price and time. During the continuous auction, traders can observe the accumulated order amounts at the five consecutive price level on both sides from the best quote. A trader can submit an order among 4 types of order: market order, limit order, conditional limit order, and best limit order. A market order has higher priority than other types of order. A limit order, a conditional limit order, and a best limit order require the same information: the quote price and order amount. A conditional limit order is a limit order, but it will be converted into a market order if is not executed or cancelled until the second call auction in the same trading day starts. A best limit is also a limit order whose limit price is determined by the best price of the opposite side when the order is submitted.The Korea Exchange (KRX) allows market order, conditional limit order and best limit order only for the nearby month contract to prevend price distortion of other month contracts. Besides, an order can have either the Fill-Or-Kill(FOK) condition or the Immediate-Or-Cancel(IOC) condition. An order with FOK condition will be cancelled if all contracts are not executed immediately. An order with IOC condition must be executed immediately. If the market does not provide the full amount, the order will be filled with only the available amount, and any portion that cannot be filled immediately will be cancelled. Our data set consists of the time-stamped sequence of transactions and quotes for KOSPI 200 futures traded on the Korea Exchange provided by the Korea Research Data Services (KRDS).7 It spans between March 11, 2011 and March 8, 2012 We extract market orders and limit orders within best five quotes by matching the transaction data and the 6 The stocks market in the Korea Exchange closes the continuous time call auction 15 minutes earlier than the futures market at 14:50 PM. 7 KOSCOM, a sub-company of the Korea Exchange, has launched the KRDS since 2012, which provides the transaction and quote data for stocks, derivatives, ELW, and ETF. 7

Table 1: Summary of the Trades and Orders # Market Order # Limit Order Price Spread Bid Ask Bid Ask Bid Ask Nearby Month 37673.03 36720.57 158315.62 152337.77 255.154 255.10 0.00523 Second Month 315.54 299.5 22168.03 20911 248.950 248.78 0.169 The column 2 and 3 (4 and 5) contain the average number of market orders (limit orders) in a day. Column 6 (7) contains the average bid (ask) price of the sample. The last column contains the average bid-ask spread. quote data. We explain the more details on the matching algorithm with trades and quotes of KRDS data in the appendix. We we devide the time interval of continuous auction into sub-intervals following the description in the previous. The order flow imbalance measure on each sub-interval is calculated based on the transaction and limit order data excepted from the data set provided by the KRDS. Table 1 gives the summary statistics of the traders and orders in our data set. 3. The Regression Analysis As explained in the past, the nearby KOSPI 200 index futures is one of the most liquid exchange traded product in the world. We have the equation (2) between the OFI measure and price change expecting the linear relation to fit well for some coefficient. Actually, Figure 2 shows that the OFI measure could reasonably explain the price change of the nearby products. The averaged adjusted R2 (the left axis) are measure for different sampling time scales and it increases as the time scale increases. The predictability matures as the time scale reaches around 3 minutes staying near 70%. The estimation of the price impact coefficient (the right axis) is stable for time scales longer than 1 minutes, which shows that the nearby KOSPI 200 index futures market is highly liquid in other sense. We suspect that the result in Figure 2 occurs for the market highly liquid. This lead us to test the same relation with the second month KOSPI 200 index futures and to 8

Figure 2: The Adjusted R2 and the Price Impact Coefficient (Nearby Product) compare the result with that of the nearby product. Figure 3 reports the regression result of the second month product. Comare it with Figure 2, the adjusted R2 the second month product is lower than that of the nearby product: the values are not higher than 15% for every choice of time interval. It has the hump at 1 minute, which is similar to Figure 12 in Cont, Kukanov and Stoikov (2012). The price impact coefficient of Figure 3 decreases relatively faster as time interval gets longer after 10 seconds. Figure 3: The Adjusted R2 and the Price Impact Coefficient (Second Nearby Product) We interpret the the lower price predictability and the hump shape in Figure 3 are due to the difference of liquidity characteristics between the nearby product and the second 9

month product. To verify the idea described in the above, we plot scatter diagrams in Figure 4 and Figure 5. Figure 4 is for the nearby product with the sampling time interval of 3 minutes and Figure 5 is for the second month product with the time intervale of 3 minutes. Compared with Figure 4, we observe many points far deviated in the vertical direction from the regression line in Figure 5. This means that the second month product tends to have large price change without accompanying the limit order flow at the best quotes. This could happen when the limit order book is shallow in depths and has large price gaps between the current best price and the current next best price (See e.g. Farmer et al. (2004) or Weber and Rosenow (2006)). Figure 4: The Scatter Plot (Nearby: 3 Minutes Scale) Figure 5: The Scatter Plot (Second Month: 3 Minutes scale) If a market is not liquid, the price impact model like CKL may face events that it has no 10

way to handle: observations with a negative (positive) OFI value and positive (negative) price change in Figure 5. A shallow limit order book would allow a small amount of market order to bite limit orders at the best quote successively. If this event attract a number of more aggressive limit orders on the opposite sides from sensitive liquidity traders to execution probability, the negative (positive) OFI value and positive (negative) price change can happen (Ranaldo (2004), Fong and Liu (2010) or Roşu (2009)).8 The shallow limit order market has too many venues and rooms to restrict the response to order book events. So we claim that a price impact model in an illiquid market tends to be more complex to accommodate those possibility. One example is given in Eisler, Bouchaud, and Kockelkoren (2012, section 7), where the dynamics of price gaps in the limit order book is modeled using a linear regression on the past order flow. We recall that the linear relation (2) fits less accurate when the sampling time interval become shorter both in Figure 2 and in Figure 3. We relate this with another dimension of liquidity: resiliency.9 Even though it is relatively highly liquid, the KOSPI 200 index nearby futures market may still face a slow resiliency speed or stale quotes in the high- frequency world, where events occur within seconds. Since the next month product has lower liquidity than the nearby product, we focus on the market for nearby product in the next section to highlighten the effect of the dynamic behavior of order book events on price change. 4. The Order Book Dynamics and the CKS Model 4.1. The Response Time Effect Figure 6) shows the the scatter plot of the nearby product for the sampling interval of 10 seconds. Compared this with Figure 4, we have relatively many points far from the regression line in Figure 6. One possibility that cause this phenomenon is the discrepancy 8 An extreme case is so-called ‘fleeting orders’. An interested reader could refer to Hasbrouck and Saar (2009) for this issue. 9 Kyle (1985) describes the liquidity of a market with the bid-ask spread, the depth, and the resiliency. 11

of response time between the mid-price change and the order flow imbalance. According to Hautsch and Huang (2012, Fig. 13), the impulse response of mid-price to trade is far faster than the impulse response to an equal-size limit order shock at the best quotes, which is consistent with the prediction by Roşu (2012). If the time length of sampling interval is too short, mid-price could not fully reflect the effect of limit order events. Figure 6: The Scatter Plot (10 Seconds Scale) We claim that the level of limit order activity gets higher than that of trading activity in the high frequency domain, so the discrepancy of response time get exacerbated more. We introduce the limit order activity ratio (LOAR) for comparing with the trading activity in a market as follows: 0 if Lbt − Ctb − Lst + Cts = 0 (3) LOARt , |Lbt − Ctb − Lst + Cts | otherwise. |Mtb − Mts | + |Lbt − Ctb − Lst + Cts | The nominator of limit order activity ratio corresponds to the subtraction of the trade imbalance from OFI measure in Cont, Kukanov and Stoikov (2012). So high (low) limit order activity ratio means relatively large (small) contribution of limit orders in the ob- served order flow imbalance. The black (red) line in Figure 7 shows the probability distribution function of the limit order activity ratio of the nearby product for the time interval of 10 seconds (3 minutes). The distribution of the time interval of 10 seconds is shifted to the right from that of the 12

time interval 3 minutes. This means that we have higher probability to get the shock generated by limit orders than by market orders given the same OFI measure. Figure 7: Limit Order Activity Ratio The black (red) line represents the probability distribution function of the nearby product for the time interval of 10 seconds (3 minutes). 4.2. The Lead-Lag Effect As we see in Figure 2, 6 and 4, the observed pair of the OFI value and price change in the market for the nearby product tend to get closer to the regressions line fitting the equation (2). This leads us to suspect that there is the lagged price change induced by the order flow imbalance or the causality to the other direction. We can verify this idea by performing the causality test à la Granger (1969). We first introduce a vector autoregressive model (VAR model) as follows: X ∆Pt (4) xt = A 0 + Ai xt−i , xt = , i=1 OFIt where Ai ’s are coefficient matrices (i = 0, 1, . . .). We estimate the VAR model (4) and investigate whether the coefficients are different from zero or not. If the coefficients are 13

different enough from zero, we reject the null hypothesis of no causality from one variable to the other. Before performing the causality test, we use the augmented Dickey-Fuller test to check whether the time series admit a unit root. We do not report the result, but both the series of the OFI value and the series of mid-price change do not have a unit root each day and each subinterval at the significance level of 1%. Then we use the Schwarz information criterion (SIC) to choose the maximum time lag. This test lead us to set 3 for the short timescale and 1 for the longer timescale. Finally, we count the number of days on which we can reject the null hypothesis of no causality from one variable to the other or in total. Figure 8 shows the ratio of the number of days on which we can reject the hypothesis of no causality with at the significance level of 10%. The green line is the case in which the price change causes the order flow imbalance, and the red line is for the other direction. We see that the OFI causes the price change for time intervals less than 20 seconds. This could explain the observations for which the OFI measure is too large relative to small price return in absolute values. Similarly the price change causes the change of the OFI for time intervals less than 10 seconds. The figure also shows that the causality from the OFI to the price change tends to be stronger (weaker) than that from the price change to the change of OFI for smaller (longer) time intervals than 1 minute. 5. Conclusion We test the effect of order book events at the best quotes on price change with the model proposed by Cont, Kukanov and Stoikov (2012). The OFI measure could reasonably explain the price change of the nearby KOSPI 200 futures contract for the longer sampling time. The CKS model miss to fit well with the second month KOSPI 200 futures. This seems to happen because the one market is not liquid but the other is highly liquid. We examine the scatter diagrams to see that a shallow limit order market has too many venues and rooms to restrict the response to order book events. 14

Figure 8: The Ratio of the Days On Which the Hypothesis of No Causality Is Rejected Even a market admits narrow bid-ask spreads and high depths, the price impact model like the CKS model may fail to explain the high-frequency behavior of price change. As sampling period gets shorter the delay of information due to the limit of response time may hinder the accuracy of the CKS model. In a high-frequency region, the activity of limit orders given a order flow imbalance tends to get stronger than that of transactions. This may be a reason that we observe the delayed response between price change and the order flow imbalance verified by the Granger-causality test with a VAR model. We conclude that the lack of liquidity in a market of the information of the dynamic in a high-frequency region of a market could eode the validity of the CKS model. A price impact model of order booke events need to be sophisticated to accommodate an illiquid market or high-frequency price behavior of a market. In this sense the valididy of the CKS model should be restricted only to a liquid market for a med-frequency or a low- frequency region as opposed Cont, Kukanov and Stoikov (2012, p. 3) intend to model the instantaneous impact of order book events. It is also not surprising that Hautsch and Huang (2012) report the difference between the short-run and long-run impact of an incoming limit orders and that Eisler, Bouchaud, and Kockelkoren (2012) try to model the dynamics of price gaps in the limit order book using a linear regression on the past order flow. 15

Appendix A. KRDS Data Processing The states of limit order book in KRDS are recorded whenever a trade occurs or a submission, cancelation or reorder of limit order within the five best quotes occurs. In particular, if the trader submits a market order and it is executed, it is recorded to transaction data with transaction price and transaction amount. Quote data are consisting of date, time, futures code(name), prices and amounts of 5 consecutive outstanding orders from the best prices, and the number of contracts. Transaction data include information like date, time, futures code, transaction price, transaction amount, and the cumulative transaction amounts and trading volume (See Table A.2). Since we can observe the state of the limit order book at that time from quote data, we cannot tell a cancelation order and a market order with the same function to decrease the total outstanding amounts. But by matching the message serial numbers10 from transaction data and quote data, we can distinguish those two. Table A.2 explains this way easily. The upper onen of Table A.2 is the part of quote data refleting the state of the limit order book changed by an order submitted at 10:17:43 on March 11 in 2011. The lower one of Table 1 shows information of a transaction by a market order submitted at that time. Because a market order is submitted and executed at 10:17:43:13611 , the transaction price, transaction amount, and the direction of trade(buy or sell) are recorded to transaction data simultaneously. But the decrease in the amount at 10:17:43:139 is due to a cancelation order, not a market order. This is why there is no serial number 189055 in transaction data. The row with time 10:17:43:125 tells us that a sell limit order with 3 contracts at the best ask price is submitted since the outstanding amounts in the ask side and the best 10 This is the number increasing by 1 when each order is submitted and is useful to distinguish a cance- lation order and a market order. If a market order with larger amount than outstanding amounts at the best price is submitted, then the message serial number is the biggest tick which is touched to execute the market order. 11 We think that the reason the time in transaction data is different to that in quote data for a market order is due to the data transmission time. 16

Date Name TP TA Time CTA BS SN 20110311 KR4101F60005 258.35 2 101743086 125072 1 189053 20110311 KR4101F60005 258.35 5 101743095 125077 1 189054 .. .. .. .. .. .. .. .. . . . . . . . . Panel A: sample transaction data. Abbreviations denote as follows. TP: Transaction Price; TA: Transaction Amount; CTA: Cumulative Transaction Amount; BS: Buy or Sell; SN: Serial Number. Date Time SN Name TSA BAP OA SBAP OA ... 20110311 101743125 189051 KR4101F60005 8313 258.35 23 258.3 50 ... 20110311 101743135 189052 KR4101F60005 8316 258.35 26 258.3 50 ... 20110311 101743136 189053 KR4101F60005 8314 258.35 24 258.3 50 ... 20110311 101743137 189054 KR4101F60005 8309 258.35 19 258.3 50 ... 20110311 101743139 189055 KR4101F60005 8305 258.35 15 258.3 50 ... .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . Panel B: sample order data. Abbreviations denote as follows. SN: serial number; TSA: Total Sell Amounts; BAP: Best Ask Price; OAM: Outstanding Amount; SBAP: Second Best Ask Price. Table A.2: Sample Transaction Data and Quote Data ask price increase by 3. In this way, we can obtain information about limit orders at the specific price level and tell between a cancelation order and a market order. In the case of price revision we can consider this as the simutaneous occurance of a limit order and a cancelation. This event is recorded to quote data without the change of the total amounts, but with a decrease in the amounts at the specific price level and a increase at the another specific price level. 17

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