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- The High-Frequency Trading Arms Race: Frequent Batch Auctions as a Market Design Response

mml mi mml mi l mml mi mml mi o mml mi mml mi w mml mi mml mrow mml msub mml mrow mml math τ δslow and mml math display inline mml mrow mml mi τ mml mi mml mo mml mo mml msub mml mi δ mml mi mml mrow mml mi f mml mi mml mi a mml mi mml mi s mml mi mml mi t mml mi mml mrow mml msub mml mrow mml math τ δfast create an asymmetry between slow and fast traders because fast traders observe them in time for the next batch auction but slow traders do not This critical interval constitutes proportion mml math display inline mml mrow mml mfrac mml mi δ mml mi mml mi τ mml mi mml mfrac mml mrow mml math δτ of the trading day where mml math display inline mml mrow mml mi δ mml mi mml mo mml mo mml msub mml mi δ mml mi mml mrow mml mi s mml mi mml mi l mml mi mml mi o mml mi mml mi w mml mi mml mrow mml msub mml mo mml mo mml msub mml mi δ mml mi mml mrow mml mi f mml mi mml mi a mml mi mml mi s mml mi mml mi t mml mi mml mrow mml msub mml mrow mml math δ δslow δfast For more details see the text of Section VII B This Article First published online July 23 2015 doi 10 1093 qje qjv027 Quarterly Journal of Economics November 1 2015 vol 130 no 4 1547 1621 Abstract Free Full Text HTML Free Full Text PDF Supplementary Data Search this journal Advanced Current Issue November 2015 130 4 Alert me to new issues The Journal About the

Original URL path: https://qje.oxfordjournals.org/content/130/4/1547/F7.expansion.html (2016-02-18)

Open archived version from archive - The High-Frequency Trading Arms Race: Frequent Batch Auctions as a Market Design Response

costs of speed mml math display block mml mrow mml msub mml mi λ mml mi mml mrow mml mi i mml mi mml mi n mml mi mml mi v mml mi mml mi e mml mi mml mi s mml mi mml mi t mml mi mml mrow mml msub mml mo mml mo mml mfrac mml mi s mml mi mml mn 2 mml mn mml mfrac mml mo mml mo mml msub mml mi λ mml mi mml mrow mml mi j mml mi mml mi u mml mi mml mi m mml mi mml mi p mml mi mml mrow mml msub mml mo mml mo mml mtext Pr mml mtext mml mo mml mo mml mo stretchy true mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mi s mml mi mml mn 2 mml mn mml mfrac mml mo stretchy true mml mo mml mo mml mo mml mi mathvariant double struck E mml mi mml mo stretchy true mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mi s mml mi mml mn 2 mml mn mml mfrac mml mo mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mi s mml mi mml mn 2 mml mn mml mfrac mml mo stretchy true mml mo mml mo mml mo mml mfrac mml mrow mml mi N mml mi mml mo mml mo mml mn 1 mml mn mml mrow mml mi N mml mi mml mfrac mml mo mml mo mml msub mml mi c mml mi mml mrow mml mi s mml mi mml mi p mml mi mml mi e mml mi mml mi e mml mi mml mi d mml mi mml mrow mml msub mml mo mml mo mml mrow mml math λinvest s2 λjump Pr J s2 E J s2 J s2 N 1N cspeed 5 Second is the zero profit condition for stale quote snipers which says that the rents from sniping as written in equation 2 equal the costs of speed mml math display block mml mrow mml msub mml mi λ mml mi mml mrow mml mi j mml mi mml mi u mml mi mml mi m mml mi mml mi p mml mi mml mrow mml msub mml mo mml mo mml mtext Pr mml mtext mml mo mml mo mml mo stretchy true mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mi s mml mi mml mn 2 mml mn mml mfrac mml mo stretchy true mml mo mml mo mml mo mml mi mathvariant double struck E mml mi mml mo stretchy true mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mi s mml mi mml mn 2 mml mn mml mfrac mml mo mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mi s mml mi mml mn 2 mml mn mml mfrac mml mo stretchy true mml mo mml mo mml mo mml mfrac mml mn 1 mml mn mml mi N mml mi mml mfrac mml mo mml mo mml msub mml mi c mml mi mml mrow mml mi s mml mi mml mi p mml mi mml mi e mml mi mml mi e mml mi mml mi d mml mi mml mrow mml msub mml mo mml mo mml mrow mml math λjump Pr J s2 E J s2 J s2 1N cspeed 6 Together equations 5 and 6 characterize the equilibrium bid ask spread mml math display inline mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml math s and the equilibrium quantity of entry mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mrow mml math N Notice that subtracting equation 6 from equation 5 yields exactly equation 3 hence the equilibrium bid ask spread is the same as in the exogenous entry case We can then solve for the equilibrium entry quantity by adding equation 5 and N 1 times equation 6 to obtain mml math display block mml mrow mml msub mml mi λ mml mi mml mrow mml mi i mml mi mml mi n mml mi mml mi v mml mi mml mi e mml mi mml mi s mml mi mml mi t mml mi mml mrow mml msub mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo mml mo mml msup mml mi N mml mi mml mo mml mo mml msup mml mo mml mo mml msub mml mi c mml mi mml mrow mml mi s mml mi mml mi p mml mi mml mi e mml mi mml mi e mml mi mml mi d mml mi mml mrow mml msub mml mo mml mo mml mrow mml math λinvest s 2 N cspeed 7 The economic interpretation of equation 7 is that all of the expenditure by trading firms on speed technology RHS is ultimately borne by investors via the cost of liquidity LHS Examining equation 3 as well we have an equivalence between the total prize in the arms race the total expenditures on speed in the arms race and the cost to investors 26 Hence the rents created by the continuous limit order book are dissipated by the speed race 27 Proposition 3 There is an equilibrium of the continuous limit order book market design with endogenous entry with play as described above The equilibrium number of fast trading firms mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mrow mml math N and the equilibrium bid ask spread mml math display inline mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml math s are uniquely determined by the zero profit conditions equations 5 and 6 The structure of play in this equilibrium is identical to that in the exogenous entry case as characterized by Proposition 1 but replacing the exogenous N trading firms with the endogenous mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mrow mml math N fast trading firms Slow trading firms play no role in equilibrium The following three quantities are equivalent in any equilibrium The total rents to trading firms mml math display inline mml mrow mml msub mml mi λ mml mi mml mrow mml mi j mml mi mml mi u mml mi mml mi m mml mi mml mi p mml mi mml mrow mml msub mml mo mml mo mml mtext Pr mml mtext mml mo mml mo mml mo stretchy false mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo stretchy false mml mo mml mo mml mo mml mi mathvariant double struck E mml mi mml mo stretchy false mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo stretchy false mml mo mml mrow mml math λjump Pr J s 2 E J s 2 J s 2 That is the sum of the value of all arbitrage opportunities that the snipers are racing to capture The total revenue liquidity providers earn from investors via the positive bid ask spread mml math display inline mml mrow mml msub mml mi λ mml mi mml mrow mml mi i mml mi mml mi n mml mi mml mi v mml mi mml mi e mml mi mml mi s mml mi mml mi t mml mi mml mrow mml msub mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mrow mml math λinvest s 2 The total equilibrium expenditure by trading firms on speed technology mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mo mml mo mml msub mml mi c mml mi mml mrow mml mi s mml mi mml mi p mml mi mml mi e mml mi mml mi e mml mi mml mi d mml mi mml mrow mml msub mml mrow mml math N cspeed VI E Discussion of the Equilibrium 1 Welfare Costs of the Arms Race A Prisoner s Dilemma among Trading Firms The equilibrium derived above can be interpreted as the outcome of a prisoner s dilemma among trading firms To see this compare the equilibrium outcome with endogenous entry to the equilibrium outcome with exogenous entry if the exogenous number of trading firms is mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mrow mml math N and their latency is δ slow In both cases the mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mrow mml math N trading firms sort themselves into 1 liquidity provider and mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mo mml mo mml mn 1 mml mn mml mrow mml math N 1 stale quote snipers and in both cases the bid ask spread mml math display inline mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml math s is characterized by trading firms indifference between liquidity provision and stale quote sniping The only difference is that now all trading firms both the liquidity provider and the snipers respond to changes in y with a delay of δ slow instead of δ fast Investors still get to trade immediately and still pay the same bid ask spread cost of mml math display inline mml mrow mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mrow mml math s 2 so their welfare is unchanged The welfare of the mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mrow mml math N trading firms is strictly greater though since they no longer pay the cost of speed Proposition 4 Prisoner s Dilemma Consider the model of Section VI D modified so that the number of trading firms is mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mrow mml math N Social welfare would be higher by mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mo mml mo mml msub mml mi c mml mi mml mrow mml mi s mml mi mml mi p mml mi mml mi e mml mi mml mi e mml mi mml mi d mml mi mml mrow mml msub mml mrow mml math N cspeed per unit time if the mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mrow mml math N trading firms could commit not to invest in the speed technology with the gains shared equally among the mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mrow mml math N trading firms But each individual trading firm has a dominant strategy incentive to deviate and invest in speed so this is not an equilibrium The situation constitutes a prisoner s dilemma with social costs equal to the total expenditure on speed As we will see below frequent batch auctions resolve this prisoner s dilemma and in a manner that allocates the welfare savings to investors instead of trading firms 2 Connection to the Empirics The Arms Race Is a Constant Proposition 5 Comparative Statics of the Arms Race Prize The size of the prize in the arms race mml math display inline mml mrow mml msub mml mi λ mml mi mml mrow mml mi j mml mi mml mi u mml mi mml mi m mml mi mml mi p mml mi mml mrow mml msub mml mo mml mo mml mtext Pr mml mtext mml mo mml mo mml mo stretchy false mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo stretchy false mml mo mml mo mml mo mml mi mathvariant double struck E mml mi mml mo stretchy false mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo stretchy false mml mo mml mrow mml math λjump Pr J s 2 E J s 2 J s 2 has the following comparative statics The size of the prize is increasing in the frequency of jumps λ jump If jump distribution mml math display inline mml mrow mml mi F mml mi mml msub mml mo mml mo mml mrow mml mi j mml mi mml mi u mml mi mml mi m mml mi mml mi p mml mi mml mrow mml msub mml mrow mml math F jump is a mean preserving spread of F jump then the size of the prize is strictly larger under mml math display inline mml mrow mml mi F mml mi mml msub mml mo mml mo mml mrow mml mi j mml mi mml mi u mml mi mml mi m mml mi mml mi p mml mi mml mrow mml msub mml mrow mml math F jump than mml math display inline mml mrow mml msub mml mi F mml mi mml mrow mml mi j mml mi mml mi u mml mi mml mi m mml mi mml mi p mml mi mml mrow mml msub mml mrow mml math Fjump The size of the prize is invariant to the cost of speed c speed The size of the prize is invariant to the speed of fast trading firms δ fast The size of the prize is invariant to the difference in speed between fast and slow trading firms δ Proposition 5 suggests that the HFT arms race is best understood as an equilibrium constant of the continuous limit order book and thus helps make sense of our empirical results Specifically suppose that speed technology improves each year and reinterpret the model so that c speed is the cost of being at the cutting edge of speed technology in the current year δ fast is the speed at the cutting edge and δ is the speed differential between the cutting edge and other trading firms Under this interpretation in equilibrium of our model the speed with which information y is incorporated into prices x grows faster and faster each year as consistent with our findings in the correlation breakdown analysis Figure III And arbitrage durations decline each year as consistent with our findings on the duration of ES SPY opportunities Figure IV However the arms race prize itself is unaffected by these advances in speed which is consistent with Figures V and VI because the total size of the prize can be decomposed as per arbitrage profitability mml math display inline mml mrow mml mi mathvariant double struck E mml mi mml mo stretchy false mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo stretchy false mml mo mml mrow mml math E J s 2 J s 2 times arbitrage frequency mml math display inline mml mrow mml msub mml mi λ mml mi mml mrow mml mi j mml mi mml mi u mml mi mml mi m mml mi mml mi p mml mi mml mrow mml msub mml mo mml mo mml mtext Pr mml mtext mml mo mml mo mml mo stretchy false mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo stretchy false mml mo mml mrow mml math λjump Pr J s 2 What does affect the size of the prize are the market volatility parameters again consistent with our findings in the arbitrage analysis 3 Relationship to the Efficient Markets Hypothesis It is interesting to interpret the equilibrium derived above as it relates to the efficient markets hypothesis On the one hand the market is highly efficient in the sense of instantaneously incorporating news about y into the price of x Formally the midpoint of the bid ask spread for x is equal to fast trading firms information about x s fundamental value mml math display inline mml mrow mml msub mml mi y mml mi mml mrow mml mi t mml mi mml mo mml mo mml msub mml mi δ mml mi mml mrow mml mi f mml mi mml mi a mml mi mml mi s mml mi mml mi t mml mi mml mrow mml msub mml mrow mml msub mml mrow mml math yt δfast for proportion one of the trading day On the other hand a strictly positive volume of trade is conducted at prices known by all trading firms to be stale Formally the proportion of trade that is conducted at quotes that do not contain mml math display inline mml mrow mml msub mml mi y mml mi mml mrow mml mi t mml mi mml mo mml mo mml msub mml mi δ mml mi mml mrow mml mi f mml mi mml mi a mml mi mml mi s mml mi mml mi t mml mi mml mrow mml msub mml mrow mml msub mml mrow mml math yt δfast is mml math display block mml mrow mml mfrac mml mrow mml msub mml mi λ mml mi mml mrow mml mi j mml mi mml mi u mml mi mml mi m mml mi mml mi p mml mi mml mrow mml msub mml mo mml mo mml mtext Pr mml mtext mml mo mml mo mml mo stretchy false mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo stretchy false mml mo mml mo mml mo mml mfrac mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mo mml mo mml mn 1 mml mn mml mrow mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mrow mml mfrac mml mrow mml mrow mml msub mml mi λ mml mi mml mrow mml mi j mml mi mml mi u mml mi mml mi m mml mi mml mi p mml mi mml mrow mml msub mml mo mml mo mml mtext Pr mml mtext mml mo mml mo mml mo stretchy false mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo stretchy false mml mo mml mo mml mo mml mfrac mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mo mml mo mml mn 1 mml mn mml mrow mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mrow mml mfrac mml mo mml mo mml msub mml mi λ mml mi mml mrow mml mi i mml mi mml mi n mml mi mml mi v mml mi mml mi e mml mi mml mi s mml mi mml mi t mml mi mml mrow mml msub mml mrow mml mfrac mml mo mml mo mml mrow mml math λjump Pr J s 2 N 1N λjump Pr J s 2 N 1N λinvest Hence the market is extremely efficient in time space but not in volume space a lot of volume gets transacted at incorrect prices This volume is in turn associated with rents from symmetrically observed public information about securities prices which is in violation of the weak form efficient markets hypothesis see Fama 1970 28 4 Role of HFTs In equilibrium of our model fast trading firms endogenously serve two roles liquidity provision and stale quote sniping The liquidity provision role is useful to investors the stale quote sniping role is detrimental to investors because it increases the costs of liquidity provision 29 This distinction between roles is important to keep in mind when interpreting the historical evidence on the effect of HFT on liquidity The rise of HFT over the past 15 years or so conflates two distinct phenomena the increased role of information technology IT in financial markets and the speed race The empirical record is unambiguous that overall IT has improved liquidity see especially Hendershott Jones and Menkveld 2011 which uses a natural experiment to show that the transition from human based liquidity provision to computer based liquidity provision enhanced liquidity This makes intuitive economic sense as IT has lowered costs in numerous sectors throughout the economy However there is little support for the proposition that the speed race per se has improved liquidity Moreover in the time series of both bid ask spreads over time Virtu 2014 p 103 and the cost of executing large trades over time Angel Harris and Spatt 2015 p 23 Frazzini Israel and Moskowitz 2012 table IV it appears that most of the improvements in liquidity associated with the rise of IT were realized in the late 1990s and early to mid 2000s well before the millisecond and microsecond level speed race We emphasize that our results do not imply that on net HFT has been negative for liquidity or social welfare Our results say that sniping is negative for liquidity and that the speed race is socially wasteful Frequent batch auctions preserve in a sense enhance the useful function served by HFTs liquidity provision and price discovery while eliminating sniping and the speed race Previous Section Next Section VII Frequent Batch Auctions as a Market Design Response In this section we define the frequent batch auction market design and show that it directly addresses the problems we have identified with the continuous limit order book market design VII A Frequent Batch Auctions Definition Informally frequent batch auctions are just like the continuous limit order book but with two departures i time is treated as discrete not continuous and ii orders are processed in batch using a uniform price auction instead of serially in order of receipt The remainder of this subsection defines frequent batch auctions formally 30 The trading day is divided into equal length discrete time intervals each of length mml math display inline mml mrow mml mi τ mml mi mml mo mml mo mml mn 0 mml mn mml mrow mml math τ 0 We refer to the parameter τ as the batch length and to the intervals as batch intervals We refer to a generic batch interval either using the interval generically mml math display inline mml mrow mml mo stretchy false mml mo mml mn 0 mml mn mml mo mml mo mml mi τ mml mi mml mo stretchy false mml mo mml mrow mml math 0 τ or using the ending time generically t At any moment in time during a batch interval traders i e investors or trading firms may submit offers to buy and sell shares of stock in the form of limit orders and market orders Just as in the continuous market a limit order is a price quantity pair expressing an offer to buy or sell a specific quantity at a specific price and a market order specifies a quantity but not a price 31 A single trader may submit multiple orders which can be interpreted as submitting a demand function or a supply function or both Just as in the continuous market traders may freely modify or cancel their orders at any moment in time Also just as in the continuous market orders remain outstanding until either executed or canceled that is if an order is not executed in the batch at time t it automatically carries over for mml math display inline mml mrow mml mi t mml mi mml mo mml mo mml mi τ mml mi mml mo mml mo mml mo mml mo mml mi t mml mi mml mo mml mo mml mn 2 mml mn mml mi τ mml mi mml mo mml mo mml mo mml mo mml mi t mml mi mml mo mml mo mml mn 3 mml mn mml mi τ mml mi mml mrow mml math t τ t 2τ t 3τ etc At the end of each batch interval the exchange batches all outstanding orders both new orders received during this interval and orders outstanding from previous intervals and computes the aggregate demand and supply functions out of all bids and asks respectively If demand and supply do not intersect then there is no trade and all orders remain outstanding for the next batch auction If demand and supply do intersect then the market clears where supply equals demand with all transactions occurring at the same price that is at a uniform price There are two cases to consider If demand and supply intersect horizontally or at a point this pins down a unique market clearing price mml math display inline mml mrow mml msup mml mi p mml mi mml mo mml mo mml msup mml mrow mml math p and a unique maximum possible quantity mml math display inline mml mrow mml msup mml mi q mml mi mml mo mml mo mml msup mml mrow mml math q In this case offers to buy with bids strictly greater than mml math display inline mml mrow mml msup mml mi p mml mi mml mo mml mo mml msup mml mrow mml math p and offers to sell with asks strictly less than mml math display inline mml mrow mml msup mml mi p mml mi mml mo mml mo mml msup mml mrow mml math p transact their full quantity at price mml math display inline mml mrow mml msup mml mi p mml mi mml mo mml mo mml msup mml mrow mml math p whereas for bids and asks of exactly mml math display inline mml mrow mml msup mml mi p mml mi mml mo mml mo mml msup mml mrow mml math p it may be necessary to ration one side of the market to enable market clearing 32 For this rationing we adopt a time priority rule analogous to current practice under the continuous market but treating time as discrete orders that have been left outstanding for a larger integer number of batch intervals have higher priority whereas if two orders were submitted in the same batch interval they have the same priority irrespective of the precise time they were submitted within that batch interval If necessary to break ties between orders submitted during the same batch interval the rationing is random pro rata If demand and supply intersect vertically this pins down a unique quantity mml math display inline mml mrow mml msup mml mi q mml mi mml mo mml mo mml msup mml mrow mml math q and an interval of market clearing prices mml math display inline mml mrow mml mo stretchy false mml mo mml msubsup mml mi p mml mi mml mi L mml mi mml mo mml mo mml msubsup mml mo mml mo mml msubsup mml mi p mml mi mml mi H mml mi mml mo mml mo mml msubsup mml mo stretchy false mml mo mml mrow mml math pL pH In this case all offers to buy with bids weakly greater than mml math display inline mml mrow mml msubsup mml mi p mml mi mml mi H mml mi mml mo mml mo mml msubsup mml mrow mml math pH and all offers to sell with asks weakly lower than mml math display inline mml mrow mml msubsup mml mi p mml mi mml mi L mml mi mml mo mml mo mml msubsup mml mrow mml math pL transact their full quantity and the price is mml math display inline mml mrow mml mfrac mml mrow mml msubsup mml mi p mml mi mml mi L mml mi mml mo mml mo mml msubsup mml mo mml mo mml msubsup mml mi p mml mi mml mi H mml mi mml mo mml mo mml msubsup mml mrow mml mn 2 mml mn mml mfrac mml mrow mml math pL pH 2 Information policy details are as follows After each auction is computed all of the orders that were entered into the batch auction both outstanding orders from previous batch intervals and new orders entered during the just completed batch interval are displayed publicly Also displayed are details of the auction outcome the supply and demand functions and the market clearing price and quantity or no trade Activity during the batch interval is not displayed publicly during the batch interval that is information is disseminated in discrete time So for the time t auction participants see all of the orders and auction information from the auctions at time mml math display inline mml mrow mml mi t mml mi mml mo mml mo mml mi τ mml mi mml mo mml mo mml mi t mml mi mml mo mml mo mml mn 2 mml mn mml mi τ mml mi mml mo mml mo mml mi t mml mi mml mo mml mo mml mn 3 mml mn mml mi τ mml mi mml mo mml mo mml mo mml mo mml mo mml mo mml mrow mml math t τ t 2τ t 3τ but they do not see new activity for the time t auction until after the auction is completed This information policy may sound different from current practice but it is in fact closely analogous In the continuous market new order book activity is first economically processed by the exchange e g a new order is entered in the book or a new order trades against the book and only then is the order announced publicly along with the updated state of the book Similarly here new order book activity is first economically processed by the exchange and only then announced publicly the only difference is that the economic processing occurs in discrete time and hence the information dissemination occurs in discrete time as well 33 To further clarify the relationship to the continuous limit order book market design it is helpful to discuss three scenarios A first scenario is that there is no new activity during the batch interval this case would be quite common if the batch interval is short In this case all outstanding orders simply carry over to the next batch interval analogous to displayed liquidity in a continuous limit order book A second scenario is that a single investor arrives during the batch interval and submits an order to buy analogously to sell at the best outstanding offer from the previous batch interval that is the frequent batch auction version of the ask analogously bid This scenario is also closely analogous to the continuous market The investor trades at the bid or ask and which order or orders get filled is based on our version of time priority A third scenario is that there is a large amount of new activity in the batch interval for example there is a news event and many trading algorithms are reacting at once In this scenario frequent batch auctions and the continuous limit order book are importantly different frequent batch auctions process all of the new activity together in a uniform price auction at the end of the interval whereas the continuous market processes the new activity serially in order of arrival VII B Why and How Frequent Batch Auctions Address the Problems with Continuous Trading Frequent batch auctions directly address the problems we identified in Section VI with the continuous limit order book for two reasons First and most obviously discrete time reduces the value of tiny speed advantages To see this consider a situation with two trading firms one who pays the cost c speed and hence has latency δ fast and one who does not pay the cost and hence has latency δ slow In the continuous market whenever there is a jump in y the fast trading firm gets to act on it first In the frequent batch auction market the fast trading firm s speed advantage is only relevant if the jump in y occurs at a very specific time in the batch interval Any jumps in y that occur during the window mml math display inline mml mrow mml mo stretchy false mml mo mml mn 0 mml mn mml mo mml mo mml mi τ mml mi mml mo mml mo mml msub mml mi δ mml mi mml mrow mml mi s mml mi mml mi l mml mi mml mi o mml mi mml mi w mml mi mml mrow mml msub mml mo stretchy false mml mo mml mrow mml math 0 τ δslow are observed by both the slow and fast trading firm in time to react for the batch auction at τ Similarly any jumps in y that occur during the window mml math display inline mml mrow mml mo stretchy false mml mo mml mi τ mml mi mml mo mml mo mml msub mml mi δ mml mi mml mrow mml mi f mml mi mml mi a mml mi mml mi s mml mi mml mi t mml mi mml mrow mml msub mml mo mml mo mml mi τ mml mi mml mo stretchy false mml mo mml mrow mml math τ δfast τ are observed by neither the fast nor

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1 equal the costs of speed mml math display block mml mrow mml msub mml mi λ mml mi mml mrow mml mi i mml mi mml mi n mml mi mml mi v mml mi mml mi e mml mi mml mi s mml mi mml mi t mml mi mml mrow mml msub mml mo mml mo mml mfrac mml mi s mml mi mml mn 2 mml mn mml mfrac mml mo mml mo mml msub mml mi λ mml mi mml mrow mml mi j mml mi mml mi u mml mi mml mi m mml mi mml mi p mml mi mml mrow mml msub mml mo mml mo mml mtext Pr mml mtext mml mo mml mo mml mo stretchy true mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mi s mml mi mml mn 2 mml mn mml mfrac mml mo stretchy true mml mo mml mo mml mo mml mi mathvariant double struck E mml mi mml mo stretchy true mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mi s mml mi mml mn 2 mml mn mml mfrac mml mo mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mi s mml mi mml mn 2 mml mn mml mfrac mml mo stretchy true mml mo mml mo mml mo mml mfrac mml mrow mml mi N mml mi mml mo mml mo mml mn 1 mml mn mml mrow mml mi N mml mi mml mfrac mml mo mml mo mml msub mml mi c mml mi mml mrow mml mi s mml mi mml mi p mml mi mml mi e mml mi mml mi e mml mi mml mi d mml mi mml mrow mml msub mml mo mml mo mml mrow mml math λinvest s2 λjump Pr J s2 E J s2 J s2 N 1N cspeed 5 Second is the zero profit condition for stale quote snipers which says that the rents from sniping as written in equation 2 equal the costs of speed mml math display block mml mrow mml msub mml mi λ mml mi mml mrow mml mi j mml mi mml mi u mml mi mml mi m mml mi mml mi p mml mi mml mrow mml msub mml mo mml mo mml mtext Pr mml mtext mml mo mml mo mml mo stretchy true mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mi s mml mi mml mn 2 mml mn mml mfrac mml mo stretchy true mml mo mml mo mml mo mml mi mathvariant double struck E mml mi mml mo stretchy true mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mi s mml mi mml mn 2 mml mn mml mfrac mml mo mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mi s mml mi mml mn 2 mml mn mml mfrac mml mo stretchy true mml mo mml mo mml mo mml mfrac mml mn 1 mml mn mml mi N mml mi mml mfrac mml mo mml mo mml msub mml mi c mml mi mml mrow mml mi s mml mi mml mi p mml mi mml mi e mml mi mml mi e mml mi mml mi d mml mi mml mrow mml msub mml mo mml mo mml mrow mml math λjump Pr J s2 E J s2 J s2 1N cspeed 6 Together equations 5 and 6 characterize the equilibrium bid ask spread mml math display inline mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml math s and the equilibrium quantity of entry mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mrow mml math N Notice that subtracting equation 6 from equation 5 yields exactly equation 3 hence the equilibrium bid ask spread is the same as in the exogenous entry case We can then solve for the equilibrium entry quantity by adding equation 5 and N 1 times equation 6 to obtain mml math display block mml mrow mml msub mml mi λ mml mi mml mrow mml mi i mml mi mml mi n mml mi mml mi v mml mi mml mi e mml mi mml mi s mml mi mml mi t mml mi mml mrow mml msub mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo mml mo mml msup mml mi N mml mi mml mo mml mo mml msup mml mo mml mo mml msub mml mi c mml mi mml mrow mml mi s mml mi mml mi p mml mi mml mi e mml mi mml mi e mml mi mml mi d mml mi mml mrow mml msub mml mo mml mo mml mrow mml math λinvest s 2 N cspeed 7 The economic interpretation of equation 7 is that all of the expenditure by trading firms on speed technology RHS is ultimately borne by investors via the cost of liquidity LHS Examining equation 3 as well we have an equivalence between the total prize in the arms race the total expenditures on speed in the arms race and the cost to investors 26 Hence the rents created by the continuous limit order book are dissipated by the speed race 27 Proposition 3 There is an equilibrium of the continuous limit order book market design with endogenous entry with play as described above The equilibrium number of fast trading firms mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mrow mml math N and the equilibrium bid ask spread mml math display inline mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml math s are uniquely determined by the zero profit conditions equations 5 and 6 The structure of play in this equilibrium is identical to that in the exogenous entry case as characterized by Proposition 1 but replacing the exogenous N trading firms with the endogenous mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mrow mml math N fast trading firms Slow trading firms play no role in equilibrium The following three quantities are equivalent in any equilibrium The total rents to trading firms mml math display inline mml mrow mml msub mml mi λ mml mi mml mrow mml mi j mml mi mml mi u mml mi mml mi m mml mi mml mi p mml mi mml mrow mml msub mml mo mml mo mml mtext Pr mml mtext mml mo mml mo mml mo stretchy false mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo stretchy false mml mo mml mo mml mo mml mi mathvariant double struck E mml mi mml mo stretchy false mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo stretchy false mml mo mml mrow mml math λjump Pr J s 2 E J s 2 J s 2 That is the sum of the value of all arbitrage opportunities that the snipers are racing to capture The total revenue liquidity providers earn from investors via the positive bid ask spread mml math display inline mml mrow mml msub mml mi λ mml mi mml mrow mml mi i mml mi mml mi n mml mi mml mi v mml mi mml mi e mml mi mml mi s mml mi mml mi t mml mi mml mrow mml msub mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mrow mml math λinvest s 2 The total equilibrium expenditure by trading firms on speed technology mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mo mml mo mml msub mml mi c mml mi mml mrow mml mi s mml mi mml mi p mml mi mml mi e mml mi mml mi e mml mi mml mi d mml mi mml mrow mml msub mml mrow mml math N cspeed VI E Discussion of the Equilibrium 1 Welfare Costs of the Arms Race A Prisoner s Dilemma among Trading Firms The equilibrium derived above can be interpreted as the outcome of a prisoner s dilemma among trading firms To see this compare the equilibrium outcome with endogenous entry to the equilibrium outcome with exogenous entry if the exogenous number of trading firms is mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mrow mml math N and their latency is δ slow In both cases the mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mrow mml math N trading firms sort themselves into 1 liquidity provider and mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mo mml mo mml mn 1 mml mn mml mrow mml math N 1 stale quote snipers and in both cases the bid ask spread mml math display inline mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml math s is characterized by trading firms indifference between liquidity provision and stale quote sniping The only difference is that now all trading firms both the liquidity provider and the snipers respond to changes in y with a delay of δ slow instead of δ fast Investors still get to trade immediately and still pay the same bid ask spread cost of mml math display inline mml mrow mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mrow mml math s 2 so their welfare is unchanged The welfare of the mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mrow mml math N trading firms is strictly greater though since they no longer pay the cost of speed Proposition 4 Prisoner s Dilemma Consider the model of Section VI D modified so that the number of trading firms is mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mrow mml math N Social welfare would be higher by mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mo mml mo mml msub mml mi c mml mi mml mrow mml mi s mml mi mml mi p mml mi mml mi e mml mi mml mi e mml mi mml mi d mml mi mml mrow mml msub mml mrow mml math N cspeed per unit time if the mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mrow mml math N trading firms could commit not to invest in the speed technology with the gains shared equally among the mml math display inline mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mrow mml math N trading firms But each individual trading firm has a dominant strategy incentive to deviate and invest in speed so this is not an equilibrium The situation constitutes a prisoner s dilemma with social costs equal to the total expenditure on speed As we will see below frequent batch auctions resolve this prisoner s dilemma and in a manner that allocates the welfare savings to investors instead of trading firms 2 Connection to the Empirics The Arms Race Is a Constant Proposition 5 Comparative Statics of the Arms Race Prize The size of the prize in the arms race mml math display inline mml mrow mml msub mml mi λ mml mi mml mrow mml mi j mml mi mml mi u mml mi mml mi m mml mi mml mi p mml mi mml mrow mml msub mml mo mml mo mml mtext Pr mml mtext mml mo mml mo mml mo stretchy false mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo stretchy false mml mo mml mo mml mo mml mi mathvariant double struck E mml mi mml mo stretchy false mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo stretchy false mml mo mml mrow mml math λjump Pr J s 2 E J s 2 J s 2 has the following comparative statics The size of the prize is increasing in the frequency of jumps λ jump If jump distribution mml math display inline mml mrow mml mi F mml mi mml msub mml mo mml mo mml mrow mml mi j mml mi mml mi u mml mi mml mi m mml mi mml mi p mml mi mml mrow mml msub mml mrow mml math F jump is a mean preserving spread of F jump then the size of the prize is strictly larger under mml math display inline mml mrow mml mi F mml mi mml msub mml mo mml mo mml mrow mml mi j mml mi mml mi u mml mi mml mi m mml mi mml mi p mml mi mml mrow mml msub mml mrow mml math F jump than mml math display inline mml mrow mml msub mml mi F mml mi mml mrow mml mi j mml mi mml mi u mml mi mml mi m mml mi mml mi p mml mi mml mrow mml msub mml mrow mml math Fjump The size of the prize is invariant to the cost of speed c speed The size of the prize is invariant to the speed of fast trading firms δ fast The size of the prize is invariant to the difference in speed between fast and slow trading firms δ Proposition 5 suggests that the HFT arms race is best understood as an equilibrium constant of the continuous limit order book and thus helps make sense of our empirical results Specifically suppose that speed technology improves each year and reinterpret the model so that c speed is the cost of being at the cutting edge of speed technology in the current year δ fast is the speed at the cutting edge and δ is the speed differential between the cutting edge and other trading firms Under this interpretation in equilibrium of our model the speed with which information y is incorporated into prices x grows faster and faster each year as consistent with our findings in the correlation breakdown analysis Figure III And arbitrage durations decline each year as consistent with our findings on the duration of ES SPY opportunities Figure IV However the arms race prize itself is unaffected by these advances in speed which is consistent with Figures V and VI because the total size of the prize can be decomposed as per arbitrage profitability mml math display inline mml mrow mml mi mathvariant double struck E mml mi mml mo stretchy false mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo stretchy false mml mo mml mrow mml math E J s 2 J s 2 times arbitrage frequency mml math display inline mml mrow mml msub mml mi λ mml mi mml mrow mml mi j mml mi mml mi u mml mi mml mi m mml mi mml mi p mml mi mml mrow mml msub mml mo mml mo mml mtext Pr mml mtext mml mo mml mo mml mo stretchy false mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo stretchy false mml mo mml mrow mml math λjump Pr J s 2 What does affect the size of the prize are the market volatility parameters again consistent with our findings in the arbitrage analysis 3 Relationship to the Efficient Markets Hypothesis It is interesting to interpret the equilibrium derived above as it relates to the efficient markets hypothesis On the one hand the market is highly efficient in the sense of instantaneously incorporating news about y into the price of x Formally the midpoint of the bid ask spread for x is equal to fast trading firms information about x s fundamental value mml math display inline mml mrow mml msub mml mi y mml mi mml mrow mml mi t mml mi mml mo mml mo mml msub mml mi δ mml mi mml mrow mml mi f mml mi mml mi a mml mi mml mi s mml mi mml mi t mml mi mml mrow mml msub mml mrow mml msub mml mrow mml math yt δfast for proportion one of the trading day On the other hand a strictly positive volume of trade is conducted at prices known by all trading firms to be stale Formally the proportion of trade that is conducted at quotes that do not contain mml math display inline mml mrow mml msub mml mi y mml mi mml mrow mml mi t mml mi mml mo mml mo mml msub mml mi δ mml mi mml mrow mml mi f mml mi mml mi a mml mi mml mi s mml mi mml mi t mml mi mml mrow mml msub mml mrow mml msub mml mrow mml math yt δfast is mml math display block mml mrow mml mfrac mml mrow mml msub mml mi λ mml mi mml mrow mml mi j mml mi mml mi u mml mi mml mi m mml mi mml mi p mml mi mml mrow mml msub mml mo mml mo mml mtext Pr mml mtext mml mo mml mo mml mo stretchy false mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo stretchy false mml mo mml mo mml mo mml mfrac mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mo mml mo mml mn 1 mml mn mml mrow mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mrow mml mfrac mml mrow mml mrow mml msub mml mi λ mml mi mml mrow mml mi j mml mi mml mi u mml mi mml mi m mml mi mml mi p mml mi mml mrow mml msub mml mo mml mo mml mtext Pr mml mtext mml mo mml mo mml mo stretchy false mml mo mml mi J mml mi mml mo mml mo mml mfrac mml mrow mml msup mml mi s mml mi mml mo mml mo mml msup mml mrow mml mn 2 mml mn mml mfrac mml mo stretchy false mml mo mml mo mml mo mml mfrac mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mo mml mo mml mn 1 mml mn mml mrow mml mrow mml msup mml mi N mml mi mml mo mml mo mml msup mml mrow mml mfrac mml mo mml mo mml msub mml mi λ mml mi mml mrow mml mi i mml mi mml mi n mml mi mml mi v mml mi mml mi e mml mi mml mi s mml mi mml mi t mml mi mml mrow mml msub mml mrow mml mfrac mml mo mml mo mml mrow mml math λjump Pr J s 2 N 1N λjump Pr J s 2 N 1N λinvest Hence the market is extremely efficient in time space but not in volume space a lot of volume gets transacted at incorrect prices This volume is in turn associated with rents from symmetrically observed public information about securities prices which is in violation of the weak form efficient markets hypothesis see Fama 1970 28 4 Role of HFTs In equilibrium of our model fast trading firms endogenously serve two roles liquidity provision and stale quote sniping The liquidity provision role is useful to investors the stale quote sniping role is detrimental to investors because it increases the costs of liquidity provision 29 This distinction between roles is important to keep in mind when interpreting the historical evidence on the effect of HFT on liquidity The rise of HFT over the past 15 years or so conflates two distinct phenomena the increased role of information technology IT in financial markets and the speed race The empirical record is unambiguous that overall IT has improved liquidity see especially Hendershott Jones and Menkveld 2011 which uses a natural experiment to show that the transition from human based liquidity provision to computer based liquidity provision enhanced liquidity This makes intuitive economic sense as IT has lowered costs in numerous sectors throughout the economy However there is little support for the proposition that the speed race per se has improved liquidity Moreover in the time series of both bid ask spreads over time Virtu 2014 p 103 and the cost of executing large trades over time Angel Harris and Spatt 2015 p 23 Frazzini Israel and Moskowitz 2012 table IV it appears that most of the improvements in liquidity associated with the rise of IT were realized in the late 1990s and early to mid 2000s well before the millisecond and microsecond level speed race We emphasize that our results do not imply that on net HFT has been negative for liquidity or social welfare Our results say that sniping is negative for liquidity and that the speed race is socially wasteful Frequent batch auctions preserve in a sense enhance the useful function served by HFTs liquidity provision and price discovery while eliminating sniping and the speed race Previous Section Next Section VII Frequent Batch Auctions as a Market Design Response In this section we define the frequent batch auction market design and show that it directly addresses the problems we have identified with the continuous limit order book market design VII A Frequent Batch Auctions Definition Informally frequent batch auctions are just like the continuous limit order book but with two departures i time is treated as discrete not continuous and ii orders are processed in batch using a uniform price auction instead of serially in order of receipt The remainder of this subsection defines frequent batch auctions formally 30 The trading day is divided into equal length discrete time intervals each of length mml math display inline mml mrow mml mi τ mml mi mml mo mml mo mml mn 0 mml mn mml mrow mml math τ 0 We refer to the parameter τ as the batch length and to the intervals as batch intervals We refer to a generic batch interval either using the interval generically mml math display inline mml mrow mml mo stretchy false mml mo mml mn 0 mml mn mml mo mml mo mml mi τ mml mi mml mo stretchy false mml mo mml mrow mml math 0 τ or using the ending time generically t At any moment in time during a batch interval traders i e investors or trading firms may submit offers to buy and sell shares of stock in the form of limit orders and market orders Just as in the continuous market a limit order is a price quantity pair expressing an offer to buy or sell a specific quantity at a specific price and a market order specifies a quantity but not a price 31 A single trader may submit multiple orders which can be interpreted as submitting a demand function or a supply function or both Just as in the continuous market traders may freely modify or cancel their orders at any moment in time Also just as in the continuous market orders remain outstanding until either executed or canceled that is if an order is not executed in the batch at time t it automatically carries over for mml math display inline mml mrow mml mi t mml mi mml mo mml mo mml mi τ mml mi mml mo mml mo mml mo mml mo mml mi t mml mi mml mo mml mo mml mn 2 mml mn mml mi τ mml mi mml mo mml mo mml mo mml mo mml mi t mml mi mml mo mml mo mml mn 3 mml mn mml mi τ mml mi mml mrow mml math t τ t 2τ t 3τ etc At the end of each batch interval the exchange batches all outstanding orders both new orders received during this interval and orders outstanding from previous intervals and computes the aggregate demand and supply functions out of all bids and asks respectively If demand and supply do not intersect then there is no trade and all orders remain outstanding for the next batch auction If demand and supply do intersect then the market clears where supply equals demand with all transactions occurring at the same price that is at a uniform price There are two cases to consider If demand and supply intersect horizontally or at a point this pins down a unique market clearing price mml math display inline mml mrow mml msup mml mi p mml mi mml mo mml mo mml msup mml mrow mml math p and a unique maximum possible quantity mml math display inline mml mrow mml msup mml mi q mml mi mml mo mml mo mml msup mml mrow mml math q In this case offers to buy with bids strictly greater than mml math display inline mml mrow mml msup mml mi p mml mi mml mo mml mo mml msup mml mrow mml math p and offers to sell with asks strictly less than mml math display inline mml mrow mml msup mml mi p mml mi mml mo mml mo mml msup mml mrow mml math p transact their full quantity at price mml math display inline mml mrow mml msup mml mi p mml mi mml mo mml mo mml msup mml mrow mml math p whereas for bids and asks of exactly mml math display inline mml mrow mml msup mml mi p mml mi mml mo mml mo mml msup mml mrow mml math p it may be necessary to ration one side of the market to enable market clearing 32 For this rationing we adopt a time priority rule analogous to current practice under the continuous market but treating time as discrete orders that have been left outstanding for a larger integer number of batch intervals have higher priority whereas if two orders were submitted in the same batch interval they have the same priority irrespective of the precise time they were submitted within that batch interval If necessary to break ties between orders submitted during the same batch interval the rationing is random pro rata If demand and supply intersect vertically this pins down a unique quantity mml math display inline mml mrow mml msup mml mi q mml mi mml mo mml mo mml msup mml mrow mml math q and an interval of market clearing prices mml math display inline mml mrow mml mo stretchy false mml mo mml msubsup mml mi p mml mi mml mi L mml mi mml mo mml mo mml msubsup mml mo mml mo mml msubsup mml mi p mml mi mml mi H mml mi mml mo mml mo mml msubsup mml mo stretchy false mml mo mml mrow mml math pL pH In this case all offers to buy with bids weakly greater than mml math display inline mml mrow mml msubsup mml mi p mml mi mml mi H mml mi mml mo mml mo mml msubsup mml mrow mml math pH and all offers to sell with asks weakly lower than mml math display inline mml mrow mml msubsup mml mi p mml mi mml mi L mml mi mml mo mml mo mml msubsup mml mrow mml math pL transact their full quantity and the price is mml math display inline mml mrow mml mfrac mml mrow mml msubsup mml mi p mml mi mml mi L mml mi mml mo mml mo mml msubsup mml mo mml mo mml msubsup mml mi p mml mi mml mi H mml mi mml mo mml mo mml msubsup mml mrow mml mn 2 mml mn mml mfrac mml mrow mml math pL pH 2 Information policy details are as follows After each auction is computed all of the orders that were entered into the batch auction both outstanding orders from previous batch intervals and new orders entered during the just completed batch interval are displayed publicly Also displayed are details of the auction outcome the supply and demand functions and the market clearing price and quantity or no trade Activity during the batch interval is not displayed publicly during the batch interval that is information is disseminated in discrete time So for the time t auction participants see all of the orders and auction information from the auctions at time mml math display inline mml mrow mml mi t mml mi mml mo mml mo mml mi τ mml mi mml mo mml mo mml mi t mml mi mml mo mml mo mml mn 2 mml mn mml mi τ mml mi mml mo mml mo mml mi t mml mi mml mo mml mo mml mn 3 mml mn mml mi τ mml mi mml mo mml mo mml mo mml mo mml mo mml mo mml mrow mml math t τ t 2τ t 3τ but they do not see new activity for the time t auction until after the auction is completed This information policy may sound different from current practice but it is in fact closely analogous In the continuous market new order book activity is first economically processed by the exchange e g a new order is entered in the book or a new order trades against the book and only then is the order announced publicly along with the updated state of the book Similarly here new order book activity is first economically processed by the exchange and only then announced publicly the only difference is that the economic processing occurs in discrete time and hence the information dissemination occurs in discrete time as well 33 To further clarify the relationship to the continuous limit order book market design it is helpful to discuss three scenarios A first scenario is that there is no new activity during the batch interval this case would be quite common if the batch interval is short In this case all outstanding orders simply carry over to the next batch interval analogous to displayed liquidity in a continuous limit order book A second scenario is that a single investor arrives during the batch interval and submits an order to buy analogously to sell at the best outstanding offer from the previous batch interval that is the frequent batch auction version of the ask analogously bid This scenario is also closely analogous to the continuous market The investor trades at the bid or ask and which order or orders get filled is based on our version of time priority A third scenario is that there is a large amount of new activity in the batch interval for example there is a news event and many trading algorithms are reacting at once In this scenario frequent batch auctions and the continuous limit order book are importantly different frequent batch auctions process all of the new activity together in a uniform price auction at the end of the interval whereas the continuous market processes the new activity serially in order of arrival VII B Why and How Frequent Batch Auctions Address the Problems with Continuous Trading Frequent batch auctions directly address the problems we identified in Section VI with the continuous limit order book for two reasons First and most obviously discrete time reduces the value of tiny speed advantages To see this consider a situation with two trading firms one who pays the cost c speed and hence has latency δ fast and one who does not pay the cost and hence has latency δ slow In the continuous market whenever there is a jump in y the fast trading firm gets to act on it first In the frequent batch auction market the fast trading firm s speed advantage is only relevant if the jump in y occurs at a very specific time in the batch interval Any jumps in y that occur during the window mml math display inline mml mrow mml mo stretchy false mml mo mml mn 0 mml mn mml mo mml mo mml mi τ mml mi mml mo mml mo mml msub mml mi δ mml mi mml mrow mml mi s mml mi mml mi l mml mi mml mi o mml mi mml mi w mml mi mml mrow mml msub mml mo stretchy false mml mo mml mrow mml math 0 τ δslow are observed by both the slow and fast trading firm in time to react for the batch auction at τ Similarly any jumps in y that occur during the window mml math display inline mml mrow mml mo stretchy false mml mo mml mi τ mml mi mml mo mml mo mml msub mml mi δ mml mi mml mrow mml mi f mml mi mml mi a mml mi mml mi s mml mi mml mi t mml mi mml mrow mml msub mml mo mml mo mml mi τ mml mi mml mo stretchy false mml mo mml mrow mml math τ δfast τ are observed by neither

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