# Equilibrium Market Prices are Unique – Convexity and Concavity Utility Functions on a Linear Case. Note Quote + Didactics.

Consider a market consisting of a set B of buyers and a set A of divisible goods. Assume |A| = n and |B| = n′. We are given for each buyer i the amount ei of money she possesses and for each good j the amount bj of this good. In addition, we are given the utility functions of the buyers. Our critical assumption is that these functions are linear. Let uij denote the utility derived by i on obtaining a unit amount of good j. Thus if the buyer i is given xij units of good j, for 1 ≤ j ≤ n, then the happiness she derives is

j=1nuijxij —— (1)

Prices p1, . . . , pn of the goods are said to be market clearing prices if, after each buyer is assigned an optimal basket of goods relative to these prices, there is no surplus or deficiency of any of the goods. So, is it possible to compute such prices in polynomial time?

First observe that without loss of generality, we may assume that each bj is unit – by scaling the uij’s appropriately. The uij’s and ei’s are in general rational; by scaling appropriately, they may be assumed to be integral. By making the mild assumption that each good has a potential buyer, i.e., a buyer who derives nonzero utility from this good. Under this assumption, market clearing prices do exist.

It turns out that equilibrium allocations for Fisher’s linear case are captured as optimal solutions to a remarkable convex program, the Eisenberg–Gale convex program.

A convex program whose optimal solution is an equilibrium allocation must have as constraints the packing constraints on the xij’s. Furthermore, its objective function, which attempts to maximize utilities derived, should satisfy the following:

1. If the utilities of any buyer are scaled by a constant, the optimal allocation remains unchanged.
2. If the money of a buyer b is split among two new buyers whose utility functions are the same as that of b then sum of the optimal allocations of the new buyers should be an optimal allocation for b.

The money weighted geometric mean of buyers’ utilities satisfies both these conditions:

max (∏i∈Auiei)1/∑iei —– (2)

then, the following objective function is equivalent:

max (∏i∈Auiei) —– (3)

Its log is used in the Eisenberg–Gale convex program:

maximize, ∑i=1n’eilogui

subject to

ui = ∑j=1nuijxij ∀ i ∈ B

i=1n’ xij ≤ 1 ∀ j ∈ A

xij ≥ 0 ∀ i ∈ B, j ∈ A —– (4)

where xij is the amount of good j allocated to buyer i. Interpret Lagrangian variables, say pj’s, corresponding to the second set of conditions as prices of goods. Optimal solutions to xij’s and pj’s must satisfy the following:

1. ∀ j ∈ A : p≥ 0
2. ∀ j ∈ A : p> 0 ⇒ ∑i∈A xij = 1
3. ∀ i ∈ B, j ∈ A : uij/pj ≤ ∑j∈Auijxij/ei
4. ∀ i ∈ B, j ∈ A : xij > 0 ⇒ uij/pj = ∑j∈Auijxij/ei

From these conditions, one can derive that an optimal solution to convex program (4) must satisfy the market clearing conditions.

For the linear case of Fisher’s model:

1. If each good has a potential buyer, equilibrium exists.
2. The set of equilibrium allocations is convex.
3. Equilibrium utilities and prices are unique.
4. If all uij’s and ei’s are rational, then equilibrium allocations and prices are also rational. Moreover, they can be written using polynomially many bits in the length of the instance.

Corresponding to good j there is a buyer i such that uij > 0. By the third condition as stated above,

pj ≥ eiuij/∑juijxij > 0

By the second condition, ∑i∈A xij = 1, implying that prices of all goods are positive and all goods are fully sold. The third and fourth conditions imply that if buyer i gets good j then j must be among the goods that give buyer i maximum utility per unit money spent at current prices. Hence each buyer gets only a bundle consisting of her most desired goods, i.e., an optimal bundle.

The fourth condition is equivalent to

∀ i ∈ B, j ∈ A : eiuijxij/∑j∈Auijxij = pjxij

Summing over all j

∀ i ∈ B : eij uijxij/∑j∈Auijxij = pjxij

⇒ ∀ i ∈ B : ei = ∑jpjxij

Hence the money of each buyer is fully spent completing the proof that market equilibrium exists. Since each equilibrium allocation is an optimal solution to the Eisenberg-Gale convex program, the set of equilibrium allocations must form a convex set. Since log is a strictly concave function, if there is more than one equilibrium, the utility derived by each buyer must be the same in all equilibria. This fact, together with the fourth condition, gives that the equilibrium prices are unique.

# Momentum of Accelerated Capital. Note Quote.

Distinct types of high frequency trading firms include independent proprietary firms, which use private funds and specific strategies which remain secretive, and may act as market makers generating automatic buy and sell orders continuously throughout the day. Broker-dealer proprietary desks are part of traditional broker-dealer firms but are not related to their client business, and are operated by the largest investment banks. Thirdly hedge funds focus on complex statistical arbitrage, taking advantage of pricing inefficiencies between asset classes and securities.

Today strategies using algorithmic trading and High Frequency Trading play a central role on financial exchanges, alternative markets, and banks‘ internalized (over-the-counter) dealings:

High frequency traders typically act in a proprietary capacity, making use of a number of strategies and generating a very large number of trades every single day. They leverage technology and algorithms from end-to-end of the investment chain – from market data analysis and the operation of a specific trading strategy to the generation, routing, and execution of orders and trades. What differentiates HFT from algorithmic trading is the high frequency turnover of positions as well as its implicit reliance on ultra-low latency connection and speed of the system.

The use of algorithms in computerised exchange trading has experienced a long evolution with the increasing digitalisation of exchanges:

Over time, algorithms have continuously evolved: while initial first-generation algorithms – fairly simple in their goals and logic – were pure trade execution algos, second-generation algorithms – strategy implementation algos – have become much more sophisticated and are typically used to produce own trading signals which are then executed by trade execution algos. Third-generation algorithms include intelligent logic that learns from market activity and adjusts the trading strategy of the order based on what the algorithm perceives is happening in the market. HFT is not a strategy per se, but rather a technologically more advanced method of implementing particular trading strategies. The objective of HFT strategies is to seek to benefit from market liquidity imbalances or other short-term pricing inefficiencies.

1. HFT do not have an affirmative market making obligation, that is they are not obliged to provide liquidity by constantly displaying two sides quotes, which may translate into a lack of liquidity during volatile conditions.
2. HFT contribute little market depth due to the marginal size of their quotes, which may result in larger orders having to transact with many small orders, and this may impact on overall transaction costs.
3. HFT quotes are barely accessible due to the extremely short duration for which the liquidity is available when orders are cancelled within milliseconds.

Besides the shallowness of the HFT contribution to liquidity, are the real fears of how HFT can compound and magnify risk by the rapidity of its actions:

There is evidence that high-frequency algorithmic trading also has some positive benefits for investors by narrowing spreads – the difference between the price at which a buyer is willing to purchase a financial instrument and the price at which a seller is willing to sell it – and by increasing liquidity at each decimal point. However, a major issue for regulators and policymakers is the extent to which high-frequency trading, unfiltered sponsored access, and co-location amplify risks, including systemic risk, by increasing the speed at which trading errors or fraudulent trades can occur.

Although there have always been occasional trading errors and episodic volatility spikes in markets, the speed, automation and interconnectedness of today‘s markets create a different scale of risk. These risks demand that exchanges and market participants employ effective quality management systems and sophisticated risk mitigation controls adapted to these new dynamics in order to protect against potential threats to market stability arising from technology malfunctions or episodic illiquidity. However, there are more deliberate aspects of HFT strategies which may present serious problems for market structure and functioning, and where conduct may be illegal, for example in order anticipation seeks to ascertain the existence of large buyers or sellers in the marketplace and then to trade ahead of those buyers and sellers in anticipation that their large orders will move market prices. A momentum strategy involves initiating a series of orders and trades in an attempt to ignite a rapid price move. HFT strategies can resemble traditional forms of market manipulation that violate the Exchange Act:

1. Spoofing and layering occurs when traders create a false appearance of market activity by entering multiple non-bona fide orders on one side of the market at increasing or decreasing prices in order to induce others to buy or sell the stock at a price altered by the bogus orders.
2. Painting the tape involves placing successive small amount of buy orders at increasing prices in order to stimulate increased demand.

3. Quote Stuffing and price fade are additional HFT dubious practices: quote stuffing is a practice that floods the market with huge numbers of orders and cancellations in rapid succession which may generate buying or selling interest, or compromise the trading position of other market participants. Order or price fade involves the rapid cancellation of orders in response to other trades.

The World Federation of Exchanges insists: ― Exchanges are committed to protecting market stability and promoting orderly markets, and understand that a robust and resilient risk control framework adapted to today‘s high speed markets, is a cornerstone of enhancing investor confidence. However this robust and resilient risk control framework‘ seems lacking, including in the dark pools now established for trading that were initially proposed as safer than the open market.

# Speculatively Accelerated Capital

Is high frequency trading good or bad? A reasonable answer must differentiate. Various strategies can be classified as high frequency; each needs to be considered separately before issuing a general verdict.

First, one should distinguish passive and active high frequency strategies. Passive strategies engage in non-designated market making by submitting resting orders. Profits come from earning the bid-ask spread and liquidity rebates offered by exchanges. Active strategies involve the submission of marketable orders. Their profit often directly translates into somebody else’s loss. Consequently, they have raised more (and eloquent) suspicion (including FLASH BOYS by Michael Lewis). Active strategies typically exploit short-term predictability of asset prices. This is particularly evident in order anticipation strategies, which

ascertain the existence of large buyers or sellers in the marketplace and then trade ahead of these buyers or sellers in anticipation that their large orders will move market prices (Securities and Exchange Commission, 2014, p. 8).

Hirschey demonstrates that high frequency traders indeed anticipate large orders with the help of complex algorithms. Large orders are submitted by institutional investors for various reasons. New information (or misinformation) on the fundamental asset value is one of them. Others include inventory management, margin calls, or the activation of stop-loss limits.

Even in the absence of order anticipation strategies, large orders are subject to execution shortfall, i.e. the liquidation value falls short of the mark-to-market value. Execution shortfall is explained in the literature as a consequence of information asymmetry (Glosten and Milgrom) and risk aversion among market makers (Ho and Stoll).

Institutional investors seek to achieve optimal execution (i.e. minimize execution shortfall and trading costs) with the help of execution algorithms. These algorithms, e.g. the popular VWAP (volume weighted average price), are typically based on the observation that price impact depends on the relative volume of an order: Price impact is lower when markets are busy. When high frequency traders detect such an execution algorithm, they obtain information on future trades and can earn significant profits with an order anticipation strategy.

That such order anticipation strategies have been described as aggressive, predatory  and “algo-sniffing” (MacKenzie) suggests that the Securities and Exchange Commission is not alone in suspecting that they “may present serious problems in today’s market structure”. But which problems exactly? There is little doubt that order anticipation strategies increase the execution shortfall of large orders. This is bad news for institutional investors. But, to put it bluntly, “the money isn’t gone, it’s just somewhere else”. The important question is whether order anticipation strategies decrease market quality.

Papers on the relationship between high frequency trading and market quality have identified two issues where the influence of high frequency trading remains inconclusive:

• How do high frequency traders influence market efficiency under normal market conditions?

An important determinant of market efficiency is volatility. Zhang and Riordan finds that high frequency traders increase volatility, Hasbrouck and Saar finds the opposite. Benos and Sagade point out that intraday volatility is “good” when it is the result of price discovery, but “excessive” noise otherwise. They study high frequency trading in four British stocks, finding that high frequency traders participate in 27% of all trading volume and that active high frequency traders in particular “can significantly amplify both price discovery and noise”, but “have higher ratios of information-to-noise contribution than all other traders”.

• Do high frequency traders increase the risk of financial breakdowns? Bershova and Rakhlin echo concerns that liquidity provided by (passive) high frequency traders could be

fictitious; although such liquidity is plentiful during ‘normal’ market conditions, it disappears at the first sign of trouble

has increasingly shifted market liquidity toward a smaller subset of the investable universe […]. Ultimately, this […] contributes to higher short-term correlations across the entire market.

Thus, high frequency trading may be beneficial most of the time, but dangerous when markets are under pressure. The sociologist Donald MacKenzie agrees, arguing that high frequency trading leaves no time to react appropriately when something goes wrong. This became apparent during the 2010 Flash Crash. When high frequency traders trade ahead of large orders in their model of price impact, they cause price overshooting. This can lead to a domino effect by activating stop-loss limits of other traders, resulting in new large orders that cause even greater price overshooting, etc. Empirically, however, the frequency of market breakdowns was significantly lower during 2007-2013 than during 1993-2006, when high frequency trading was less prevalent.

Even with high-quality data, empirical studies cannot fully entangle different strategies employed by high frequency traders, but what is required instead is an integration of high frequency trading into a mathematical model of optimal execution. It features transient price impact, heterogeneous transaction costs and strategic interaction between an arbitrary number of traders. High frequency traders may decrease the price deviation caused by a large order, and thus reduce the risk of domino effects in the wake of large institutional trades….

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# High Frequency Traders: A Case in Point.

Events on 6th May 2010:

In the ordinary course of business, HFTs use their technological advantage to profit from aggressively removing the last few contracts at the best bid and ask levels and then establishing new best bids and asks at adjacent price levels ahead of an immediacy-demanding customer. As an illustration of this “immediacy absorption” activity, consider the following stylized example, presented in Figure and described below.

Suppose that we observe the central limit order book for a stock index futures contract. The notional value of one stock index futures contract is \$50. The market is very liquid – on average there are hundreds of resting limit orders to buy or sell multiple contracts at either the best bid or the best offer. At some point during the day, due to temporary selling pressure, there is a total of just 100 contracts left at the best bid price of 1000.00. Recognizing that the queue at the best bid is about to be depleted, HFTs submit executable limit orders to aggressively sell a total of 100 contracts, thus completely depleting the queue at the best bid, and very quickly submit sequences of new limit orders to buy a total of 100 contracts at the new best bid price of 999.75, as well as to sell 100 contracts at the new best offer of 1000.00. If the selling pressure continues, then HFTs are able to buy 100 contracts at 999.75 and make a profit of \$1,250 dollars among them. If, however, the selling pressure stops and the new best offer price of 1000.00 attracts buyers, then HFTs would very quickly sell 100 contracts (which are at the very front of the new best offer queue), “scratching” the trade at the same price as they bought, and getting rid of the risky inventory in a few milliseconds.

This type of trading activity reduces, albeit for only a few milliseconds, the latency of a price move. Under normal market conditions, this trading activity somewhat accelerates price changes and adds to the trading volume, but does not result in a significant directional price move. In effect, this activity imparts a small “immediacy absorption” cost on all traders, including the market makers, who are not fast enough to cancel the last remaining orders before an imminent price move.

This activity, however, makes it both costlier and riskier for the slower market makers to maintain continuous market presence. In response to the additional cost and risk, market makers lower their acceptable inventory bounds to levels that are too small to offset temporary liquidity imbalances of any significant size. When the diminished liquidity buffer of the market makers is pierced by a sudden order flow imbalance, they begin to demand a progressively greater compensation for maintaining continuous market presence, and prices start to move directionally. Just as the prices are moving directionally and volatility is elevated, immediacy absorption activity of HFTs can exacerbate a directional price move and amplify volatility. Higher volatility further increases the speed at which the best bid and offer queues are being depleted, inducing HFT algorithms to demand immediacy even more, fueling a spike in trading volume, and making it more costly for the market makers to maintain continuous market presence. This forces more risk averse market makers to withdraw from the market, which results in a full-blown market crash.

Empirically, immediacy absorption activity of the HFTs should manifest itself in the data very differently from the liquidity provision activity of the Market Makers. To establish the presence of these differences in the data, we test the following hypotheses:

Hypothesis H1: HFTs are more likely than Market Makers to aggressively execute the last 100 contracts before a price move in the direction of the trade. Market Makers are more likely than HFTs to have the last 100 resting contracts against which aggressive orders are executed.

Hypothesis H2: HFTs trade aggressively in the direction of the price move. Market Makers get run over by a price move.

Hypothesis H3: Both HFTs and Market Makers scratch trades, but HFTs scratch more.

To statistically test our “immediacy absorption” hypotheses against the “liquidity provision” hypotheses, we divide all of the trades during the 405 minute trading day into two subsets: Aggressive Buy trades and Aggressive Sell trades. Within each subset, we further aggregate multiple aggressive buy or sell transactions resulting from the execution of the same order into Aggressive Buy or Aggressive Sell sequences. The intuition is as follows. Often a specific trade is not a stand alone event, but a part of a sequence of transactions associated with the execution of the same order. For example, an order to aggressively sell 10 contracts may result in four Aggressive Sell transactions: for 2 contracts, 1 contract, 4 contracts, and 3 contracts, respectively, due to the specific sequence of resting bids against which this aggressive sell order was be executed. Using the order ID number, we are able to aggregate these four transactions into one Aggressive Sell sequence for 10 contracts.

Testing Hypothesis H1. Aggressive removal of the last 100 contracts by HFTs; passive provision of the last 100 resting contracts by the Market Makers. Using the Aggressive Buy sequences, we label as a “price increase event” all occurrences of trading sequences in which at least 100 contracts consecutively executed at the same price are followed by some number of contracts at a higher price. To examine indications of low latency, we focus on the the last 100 contracts traded before the price increase and the first 100 contracts at the next higher price (or fewer if the price changes again before 100 contracts are executed). Although we do not look directly at the limit order book data, price increase events are defined to capture occasions where traders use executable buy orders to lift the last remaining offers in the limit order book. Using Aggressive sell trades, we define “price decrease events” symmetrically as occurrences of sequences of trades in which 100 contracts executed at the same price are followed by executions at lower prices. These events are intended to capture occasions where traders use executable sell orders to hit the last few best bids in the limit order book. The results are presented in Table below

For price increase and price decrease events, we calculate each of the six trader categories’ shares of Aggressive and Passive trading volume for the last 100 contracts traded at the “old” price level before the price increase or decrease and the first 100 contracts traded at the “new” price level (or fewer if the number of contracts is less than 100) after the price increase or decrease event.

Table above presents, for the six trader categories, volume shares for the last 100 contracts at the old price and the first 100 contracts at the new price. For comparison, the unconditional shares of aggressive and passive trading volume of each trader category are also reported. Table has four panels covering (A) price increase events on May 3-5, (B) price decrease events on May 3-5, (C) price increase events on May 6, and (D) price decrease events on May 6. In each panel there are six rows of data, one row for each trader category. Relative to panels A and C, the rows for Fundamental Buyers (BUYER) and Fundamental Sellers (SELLER) are reversed in panels B and D to emphasize the symmetry between buying during price increase events and selling during price decrease events. The first two columns report the shares of Aggressive and Passive contract volume for the last 100 contracts before the price change; the next two columns report the shares of Aggressive and Passive volume for up to the next 100 contracts after the price change; and the last two columns report the “unconditional” market shares of Aggressive and Passive sides of all Aggressive buy volume or sell volume. For May 3-5, the data are based on volume pooled across the three days.

Consider panel A, which describes price increase events associated with Aggressive buy trades on May 3-5, 2010. High Frequency Traders participated on the Aggressive side of 34.04% of all aggressive buy volume. Strongly consistent with immediacy absorption hypothesis, the participation rate rises to 57.70% of the Aggressive side of trades on the last 100 contracts of Aggressive buy volume before price increase events and falls to 14.84% of the Aggressive side of trades on the first 100 contracts of Aggressive buy volume after price increase events.

High Frequency Traders participated on the Passive side of 34.33% of all aggressive buy volume. Consistent with hypothesis, the participation rate on the Passive side of Aggressive buy volume falls to 28.72% of the last 100 contracts before a price increase event. It rises to 37.93% of the first 100 contracts after a price increase event.

These results are inconsistent with the idea that high frequency traders behave like textbook market makers, suffering adverse selection losses associated with being picked off by informed traders. Instead, when the price is about to move to a new level, high frequency traders tend to avoid being run over and take the price to the new level with Aggressive trades of their own.

Market Makers follow a noticeably more passive trading strategy than High Frequency Traders. According to panel A, Market Makers are 13.48% of the Passive side of all Aggressive trades, but they are only 7.27% of the Aggressive side of all Aggressive trades. On the last 100 contracts at the old price, Market Makers’ share of volume increases only modestly, from 7.27% to 8.78% of trades. Their share of Passive volume at the old price increases, from 13.48% to 15.80%. These facts are consistent with the interpretation that Market Makers, unlike High Frequency Traders, do engage in a strategy similar to traditional passive market making, buying at the bid price, selling at the offer price, and suffering losses when the price moves against them. These facts are also consistent with the hypothesis that High Frequency Traders have lower latency than Market Makers.

Intuition might suggest that Fundamental Buyers would tend to place the Aggressive trades which move prices up from one tick level to the next. This intuition does not seem to be corroborated by the data. According to panel A, Fundamental Buyers are 21.53% of all Aggressive trades but only 11.61% of the last 100 Aggressive contracts traded at the old price. Instead, Fundamental Buyers increase their share of Aggressive buy volume to 26.17% of the first 100 contracts at the new price.

Taking into account symmetry between buying and selling, panel B shows the results for Aggressive sell trades during May 3-5, 2010, are almost the same as the results for Aggressive buy trades. High Frequency Traders are 34.17% of all Aggressive sell volume, increase their share to 55.20% of the last 100 Aggressive sell contracts at the old price, and decrease their share to 15.04% of the last 100 Aggressive sell contracts at the new price. Market Makers are 7.45% of all Aggressive sell contracts, increase their share to only 8.57% of the last 100 Aggressive sell trades at the old price, and decrease their share to 6.58% of the last 100 Aggressive sell contracts at the new price. Fundamental Sellers’ shares of Aggressive sell trades behave similarly to Fundamental Buyers’ shares of Aggressive Buy trades. Fundamental Sellers are 20.91% of all Aggressive sell contracts, decrease their share to 11.96% of the last 100 Aggressive sell contracts at the old price, and increase their share to 24.87% of the first 100 Aggressive sell contracts at the new price.

The number of price increase and price decrease events increased dramatically on May 6, consistent with the increased volatility of the market on that day. On May 3-5, there were 4100 price increase events and 4062 price decrease events. On May 6 alone, there were 4101 price increase events and 4377 price decrease events. There were therefore approximately three times as many price increase events per day on May 6 as on the three preceding days.

Testing Hypothesis H2. HFTs trade aggressively in the direction of the price move; Market Makers get run over by a price move. To examine this hypothesis, we analyze whether High Frequency Traders use Aggressive trades to trade in the direction of contemporaneous price changes, while Market Makers use Passive trades to trade in the opposite direction from price changes. To this end, we estimate the regression equation

Δyt = α + Φ . Δyt-1 + δ . yt-1 + Σi=120i . Δpt-1 /0.25] + εt

(where yt and Δyt denote inventories and change in inventories of High Frequency Traders for each second of a trading day; t = 0 corresponds to the opening of stock trading on the NYSE at 8:30:00 a.m. CT (9:30:00 ET) and t = 24, 300 denotes the close of Globex at 15:15:00 CT (4:15 p.m. ET); Δpt denotes the price change in index point units between the high-low midpoint of second t-1 and the high-low midpoint of second t. Regressing second-by-second changes in inventory levels of High Frequency Traders on the level of their inventories the previous second, the change in their inventory levels the previous second, the change in prices during the current second, and lagged price changes for each of the previous 20 previous seconds.)

for Passive and Aggressive inventory changes separately.

Table above presents the regression results of the two components of change in holdings on lagged inventory, lagged change in holdings and lagged price changes over one second intervals. Panel A and Panel B report the results for May 3-5 and May 6, respectively. Each panel has four columns, reporting estimated coefficients where the dependent variables are net Aggressive volume (Aggressive buys minus Aggressive sells) by High Frequency Traders (∆AHFT), net Passive volume by High Frequency Traders (∆PHFT), net Aggressive volume by Market Makers (∆AMM), and net Passive volume by Market Makers (∆PMM).

We observe that for lagged inventories (NPHFTt−1), the estimated coefficients for Aggressive and Passive trades by High Frequency Traders are δAHFT = −0.005 (t = −9.55) and δPHFT = −0.001 (t = −3.13), respectively. These coefficient estimates have the interpretation that High Frequency Traders use Aggressive trades to liquidate inventories more intensively than passive trades. In contrast, the results for Market Makers are very different. For lagged inventories (NPMMt−1), the estimated coefficients for Aggressive and Passive volume by Market Makers are δAMM = −0.002 (t = −6.73) and δPMM = −0.002 (t = −5.26), respectively. The similarity of these coefficients estimates has the interpretation that Market Makers favor neither Aggressive trades nor Passive trades when liquidating inventories.

For lagged price changes, coefficient estimates for Aggressive trades by High Frequency Traders and Market Makers are positive and statistically significant at lags 1-4 and lags 1-10, respectively. These results have the interpretation that both High Frequency Traders’ and Market Makers’ trade on recent price momentum, but the trading is compressed into a shorter time frame for High Frequency Traders than for Market Makers.

When High Frequency Traders are net buyers on May 3-5, prices rise by 17% of a tick in the next second. When HFTs execute Aggressively or Passively, prices rise by 20% and 2% of a tick in the next second, respectively. In subsequent seconds, prices in all cases trend downward by about 5% of a tick over the subsequent 19 seconds. For May 3-5, the results are almost symmetric for selling.