Trading on the Floor after Sweeping the Book
TTRADING ON THE FLOOR AFTERSWEEPING THE BOOK
Vassilis Polimenis*
Informed traders need to trade fast in order to profit from their private informationbefore it becomes public. Fast electronic markets provide such liquidity. Slowmarkets provide execution in an auction based trading floor. Hybrid marketscombine both execution venues. In its main result, the paper shows that tocompensate for their slow and risky executions, trading floors need to be at leasttwice as deep as the sweeping facility. Furthermore, when a stand-alone tradingfloor is enhanced with the addition of a sweeping facility, overall informed tradingwill decline because it is easier for informed traders to extract the full value oftheir private info.
The NYSE Hybrid Market will marry the best of electronic trading and theauction market in a way no other can match. — John A. Thain, NYSE CEO A dvances in communication and computing technologies are dramaticallyaltering the trading landscape. The traditional trading paradigm, that of an“open-outcry” auction based trading floor, is giving its place to a morecomplicated and diverse order matching environment. As the introduction of newtechnologies brings promise for faster, less risky and more accurate trading execu-tions, it is becoming increasingly difficult for traditional exchanges to compete fororder flow. This difficulty is evident in the recent merger activity; during April2005, all four major U.S. equity markets were seeking to merge. Mergers betweentraditional and electronic exchanges will produce hybrid markets that combinemanual and automatic execution elements.Such transformations have, at the very best, been received with skepticism bythe floor-trading communities of brokers and specialists — the intermediaries whooversee the matching of buy and sell orders — who are at the heart of the auction-based price-discovery process. In a decimalized world, complete automation that
Vassilis Polimenis is an assistant professor in the A. Gary Anderson Graduate School ofManagement at the University of California. Contact information: University of California, A.Gary Anderson Graduate School of Management, 900 University Ave., Riverside, CA 92521-0203. E-mail: [email protected]
Keywords : hybrid markets, trading floor, limit order book, liquidity premia
JEL Classification:
G121. On April 20, NYSE and Archipelago aanounced an agreement to merge and, furthermore, tobecome a public company. Two days later, NASDAQ announced an agreement to buy the INETECN. eview of Futures Markets eliminates latency in order execution, and creates perfect order timing sequencing,will make it dramatically more difficult, or even impossible, for dealers andexchange professionals to trade profitably. As Peterffy and Battan noted in theSEC’s Market Structure Hearings held in New York on November 2002,“[d]esignated liquidity providers, therefore, have had to rely on their inherent timeand place advantage in the manual market place — specifically, that they can seeorders before others can see them and can take their time (sometimes up to 90seconds) to decide whether to interact with those orders or not — in order to reapa reward for the services they provide.” (See also the recent Peterffy and Battan2004 piece in
Financial Analysts Journal .)The key difference between traditional “slow” floor-based exchanges relativeto their “fast” electronic competitors for liquidity is the firmness of quotes postedin slow and fast markets. A fast market quote is a true quote that an order canundoubtedly get filled against, while a slow market quote is more of an indicationof a price. In fast-changing market conditions, a slow market may advantageouslyuse its “slowness” in updating prices, thus posting artificially attractive quotes,and block away electronic exchanges that cannot do so.To modernize the National Market System [NMS] to reflect advances inelectronic stock markets and order routing, in March 2004 the SEC established thenew Regulation NMS, which among other things differentiates between automatic(fast) and manual (slow) markets. According to the new ruling, fast markets arenot allowed to execute an order at a price that is inferior to an electronic market’sbest price but may trade through better but non-immediately executable prices onslow markets. The rule takes away the biggest competitive advantage of traditionaltrading floors in the battle for liquidity. As a response, many exchanges are proposingenhancements to their electronic-trading platforms.One of the highly contested issues for these newly emerging hybrid exchangesis whether to allow for so-called “sweeping of the book.” Sweeping refers to theability to electronically execute transactions not only at the best price, as for exampleNYSE’s electronic platform Direct+ now does, but also at quotes above or belowthe best price. It is argued that the electronic component creates in-house competitionfor the trading floor, since electronic access to the full limit order book wouldmake it much more challenging for the floor community to co-exist within thenewly emerging hybrid markets.In the face of these significant developments, the academic community hasbeen slow in offering definite answers to the questions surrounding these issues.Are hybrid markets good for price discovery? Do trading floors enhance theefficiency of the market? What changes in trading should would we expect when atraditional exchange becomes hybrid by offering electronic access to its entirelimit order book through sweeping? What are the parameters that will determinethe inter-market competitiveness?The main reason for the lack of definite academic answers is the inherentcomplexity of the trading process and the difficulty of modeling liquidity discovery.An important growing branch of literature endogenously explains liquidity rading on the Floor after Sweeping the Book constraints and their effect on trading. In the two canonical models of marketmicrostructure described in Glosten and Milgrom (1985) and Kyle (1985), liquiditydiscounts are related to the size of the order because large orders are moreinformative. Recently, Back and Baruch (2004) provide some joint analysis of theKyle and Glosten and Milgrom papers with a single informed trader and risk neutralmarket makers.Despite their indisputable significance, the above equilibrium analyses havebeen criticized in that they do not lend themselves to real market calibrations, andtheir conclusions cannot be generalized to more realistic models of trading (Black1995). For the discussion of slow versus fast markets, these models are silentbecause, in order to endogenously derive market depth, they posit efficient marketsthat reflect all publicly available information immediately.In this paper we take some initial steps in the direction of modeling fast andslow markets and discussing their basic competitive characteristics. Slow markets,as their name implies, take some time to execute orders; fast markets are the limitof slow markets with infinite speed of adjustment. There is another importantdifference: In electronic markets limit orders are pre-posted and cannot be removeddepending on the incoming order size; the electronic market does not have time todifferentiate for small versus large traders. On the contrary, in trading floors, dueto the slow speed of execution, market makers have the necessary time to adjusttheir quotes when a block is coming and may thus avoid to be price discriminatedagainst (by not providing liquidity at the early stages of a block’s execution).The first contribution of the paper is to model market dynamics as a functionof trading volume rather than the usual calendar time, t. When the market evolvesstochastically as a function of volume, execution time of a size q order becomes astochastic process with respect to q; for a particular execution scenario wÎW, atrade will be completed at time t = t(q, w). Price dynamics are then subordinated toexecution time dynamics in a way that will formally be defined in the text. Thevalue of trading volume in explaining returns is being discussed in Clark (1973),and Ané and Geman (2000).The paper argues that hybrid markets do not generate more informed tradingthan stand-alone electronic markets. That is, under the assumptions of the modelhere, a slow market only competes for the same liquidity with the fast market. It isshown that, when competing for liquidity with an electronic (Kyle-type) market,which is l -1F shares per dollar deep, a trading floor will be able to divert trades awayfrom its electronic competitor only if it is at least twice as deep.Actually, at its concluding section the paper shows that when a stand-alonefloor is transformed to a hybrid market, with the addition of a sweeping facility, wecould expect the overall amount of informed trading to decline. The fundamentalreason for such a decline is that information mainly flows from the fast to the slowmarket; thus, it is the lower depth of the fast market that determines overall trade.The first task of the paper is to present a formal model of the liquidity andprice discovery mechanisms on both the trading floor as well as the hybrid marketsthat combine a floor with an electronic sweeping facility. The model presented eview of Futures Markets here, using only mild assumptions, builds upon the basic notions of liquidity —order size that can be absorbed in a given time — and depth — units of liquidityfor a given cost — as market participants truly understand them. For a short period,before her proprietary information becomes public, an informed trader is a liquiditymonopsonist. In return for releasing information in the market, through her trades,the informed trader is compensated with liquidity supplied by uninformed traders.Unlike the instantaneous and riskless execution offered by electronic markets,auction-based trading floors are characterized by “slow” executions. In a relatedstudy, Polimenis (2005) investigates the optimal trading of a trader who may onlytrade in a slow market. The central idea in that paper, which we borrow here, is thatin a slow market price adjustment is subordinated to liquidity discovery; prices areupdated while we search for liquidity. Thus, a slow order execution is undesirablefor an informed trader, not due to lost interest in the money “tied” to the order but,rather, due to informational dimension of time. The motivation behind Polimenis(2005) is to capture the fact that a slow execution is risky because it exposes theblock order to the market for prolonged periods of time and thus carries informationalcosts. As is discussed there, execution time provides the medium for incorporatingnew information in prices.As a side note, it may be argued that such an approach is not only motivatedby purely practical reasons but in a fundamental way relates to the complexity ofcomputing the efficient price by processing in real time trading patterns and otherinformation available to market makers. By prolonging executions, illiquid marketsallow more time to the market to absorb and reflect private information, thus limitingthe informed agent’s profits.The model of the slow market here is related to Polimenis (2005), but here thetrader’s options are extended to allow for the choice to participate, up to anendogenously chosen degree, in a fast market. Another difference between the twomodels of slow markets is, in Polimenis (2005), noise trades arrive continually asa Brownian motion, while here, noise orders arrive at discrete but totally randomsizes and times.Notably, the applicability of the results is limited in that the paper does notendogenize the entire liquidity markets but rather only addresses in an endogenousfashion inter-market competition. The paper thus builds upon the groundbreakingwork by Kyle and others, who provide a purely endogenous lambda, by enhancingthe available trading venues.In Sections I and II, the fast and slow markets are introduced. In Sections IIIand IV, we model liquidity discovery and optimal trading on the stand-alone floor.In Section V hybrid markets are discussed. Finally, in Section VI, the nature of themarket for liquidity is discussed.
I. FAST MARKETS AND KYLE’S LAMDA
Modeling the operation of fast markets is relatively straightforward. Fastmarkets (typically electronic order books) are characterized by certain andimmediate executions; that is, when sweeping a transparent book, the agent knows rading on the Floor after Sweeping the Book all dimensions of her order execution quality. The order will be immediatelyexecuted at a price away enough (from the current price) to activate the requiredamount of standing limit orders.The most important parameter that determines execution quality in a fast marketis the density of standing limit orders. Such markets are characterized by theirbreadth, λ F (F stands for Fast market), measured in dollars of expected impact pertransacted share . That is, λ F is the sensitivity of the fast market to trading. Kyle(1985) endogenously calculates the λ of the market as a signal-to-noise ratio. Thetrader who trades q shares in a fast market with λ F will experience a deterministicadverse impactThe inverse quantity λ F-1 is the familiar market depth, measured in shares perdollar, with the natural interpretation of capturing the density of shares, placed bylimit orders, per dollar in the order book. Clearly, breadth λ F is a negative attributesince it measures the cost per unit of liquidity. Depth measures the liquidity thatone dollar buys in the market.The value of private information is measured by the implied misvaluation from their fundamental value at which securities currently trade. According to herprivate information, the trader observes that, at time zero, the stock trades at aprice that deviates from its intrinsic value by ∆ P dollars. If the trader were a pricetaker, the existence of a non-zero ∆ P would be an arbitrage opportunity, and thusan unacceptable condition for an infinitely deep market.In reality, even though the condition ∆ P>0 will certainly lead to trading, itdoes not amount to an arbitrage opportunity. To remain profitable, the trader has tobalance the amount of trading with the impact it generates. In this environment,the profitability of her strategy is determined by the value of the private information, ∆ P, which captures the gain due to the current misvaluation of the security, and theprice impact of the released information (liquidity cost), I.
A. Sweeping the Book
When sweeping q shares from the book, the agent releases information thatimmediately impacts the price by λ F q. It should be emphasized here that limitorders on the book are already standing and the instantaneous nature of orderexecution in fast markets implies that sweeping the book is an inherently anonymousoperation. That is, the fast market here differs from the typical one-shot model(e.g., Kyle) where a unique price that clears the market applies for the entire order.When the insider chooses to sweep the book, her orders are being executed atprogressively deteriorating price points. Thus, the total cost of buying the requiredliquidity from the book equals , and her profit is λ F q (5) ydy λ F ∫ q0
2. This will be contrasted later with trading floors that do not provide anonymous trading. eview of Futures Markets
The trader will trade q F shares, whereAfter q F shares the price reflects the full private information and trading stops.Thus, a fast market with instantaneous executions will immediately return to itsefficient state even in the presence of a single informed trader. II. TRADING IN SLOW MARKETS
Slow markets, as their name implies, are characterized by slow and thusuncertain executions; that is, when sending her order to a trading floor, the agentdoes not know when her order will be executed and at what price. Observe that if,as in Kyle, the trader is risk neutral, risk and thus execution speed don’t matter.Since trading floors are risky, in order to attract traders, they have to provide,in a sense that will become clear, “cheaper” liquidity than books. In liquid tradingfloors, the large number of uninformed traders allows an informed trader to releaseless information per share traded. In this sense, the model of a slow market hereresembles Kyle (1985), but with an important difference: In Kyle market makersdecide an efficient price by completely “distilling” all public information. Here,market makers still correct prices in the right direction (i.e., towards efficiency),but not in a completely efficient way. Instead, they mechanically tend to correctprices upwards for a buy, and downwards for a sell. Even though this implies thatprices are to some degree inefficient, it resembles more with realistic markets,where market makers may not want to impose a strong structure on the missinginside information (i.e., on what they don’t know). Since liquid markets are characterized by the large number of shares that canbe traded before private information becomes public, it is natural to define floorliquidity, a , as the expected rate at which marketable orders arrive; and thus measureit in shares per second.In a way analogous to the lambda of a book, λ F , a floor is also characterized byits sensitivity to new trades, λ S (S for Slow market). The fundamental differencebetween the sensitivities of a fast and slow market, is that the latter refers to a riskyexecution and thus needs to be corrected for risk.Unlike a fast Kyle-type market that adjusts prices immediately to reflect newinformation, the λ S of a slow market is only an expected rate of adjustment. In aslow market, prices are expected to adjust at a rate ydy λ Pq ∫ − q0 F ∆ (2) ÄP = ë F q F (3)
3. In Kyle, the endogenous λ depends on knowledge of the variance of the inside signal, and inthe more recent version in Back and Baruch (2004), it depends in restricting the two possibleinside values (zero or one). rading on the Floor after Sweeping the Book dollars per second. A proper model of slow markets should include the fast market as a specialcase for which a = .When liquidity is infinite, there is effectively no search forliquidity, and prices adjust immediately, µ = . Furthermore, if two markets arecharacterized by the same λ , the one with the higher a is superior and will bestrictly preferred by the agent. This is because, as we will see later, a higher liquidity, a, implies faster execution and thus smaller execution risk.The agent who observed that the illiquid security was last traded at a priceaway from its actual value by ∆ P dollars knows that if she chooses to trade she willnot enjoy the entire ∆ P, since information will affect the market in an adverse way.For example, a large buy order (for an undervalued security) will tend to get executedat a higher transaction price. Thus, the trader will pocket a per share profit equal to ∆ P-I(q), where I(q) is the total dollar impact of information released during thetime, τ (q), it takes for the q-order to get executed. III. LIQUIDITY DISCOVERY ON THE FLOOR SATISFIES
The increasing process t(q) is clearly of bounded variation with no negativejumps or diffusive component. The rate of arrivals of execution-time jumps withsize between z and z+dz, the Lévy measure L(dz), in this case satisfies ì = ë S a (4) ∞ ∞
4. We are used to processes with respect to time; here time (and price impact) is a process withrespect to q.5.For more on subordinator Lévy processes and their generating functions see ch.III in Bertoin(1996).
In order to model a slow market, we have to first model the process of liquidity discovery on the floor. Generally speaking, execution on the floor is stochastic, and determined as a particular realization ω from a universe Ω of possible execution paths. Under a particular execution scenario ω ∈ Ω , a trade will be completed at time τ (q, ω ) when enough traders willing to take the opposite side have submitted their marketable orders. If we make the natural assumption that, for every execution ω , larger orders will take more time to get executed, we may think of execution time, τ , when plotted against size, q, as an increasing process taking values in (0, ∞ ). If we further assume that the extra time it takes to execute another q’ shares does not depend on how many shares q the trader has executed so far — formally τ (q+q’)- τ (q) is distributed as τ (q’) —we are left with τ (q) being an increasing Lévy process in q. In the language of Lévy processes, τ (q), when viewed as a function of size q, is a subordinator. An important feature of Lévy subordinators is that we may work with their cumulant kernel K τ (), defined for any s through Ee s τ = exp(qK τ (s)) (5) eview of Futures Markets In the most general case, a subordinator includes a locally deterministic continuousand a pure jump part. In the case of the τ (q) process, given that order arrivals fromnoise traders in reality are totally random in size and unpredictable, it makes littlesense to include a deterministic continuous component. Thus, we will assume herethat τ (q) includes only positive jumps ; in this case the Lévy-Kintchine representa-tion of the kernel is given byThen, the extra time it takes for an extra dq shares to get executed followswhere, the law of N is that of a Poisson random measure on + x + with Lévymeasure L(dz) that models the pure jump liquidity arrival delay from the totallyrandom market orders submitted by noise traders. That is, N(dq,dz) is one when ittakes between z and z+dz seconds for an extra dq shares to get executed. Thismeans that an incremental infinitesimal quantity may take a large amount of time toget executed.Since liquidity is formally measured as the expected rate, a , at which marketableorders arrive, and is measured in transacted shares per second, the trader expectsthat her order will take E τ = q/a seconds to get executed on the floor. On the otherhand, from the Lévy property we have thatwhich implies that liquidity has to satisfy Example.
In this example, provided to make the discussion specific, we specify the min(1, ) ( ) z L dz +∞ < ∞ ∫ (6) sz τ K (s) (e 1)L(dz) +∞ = − ∫ (7)(8) d τ zN(dq,dz) +∞ = ∫ (9) E τ =qK τ ’(0) (10) a -1 = K τ ’(0)= ∫ +∞ )( dzzL rading on the Floor after Sweeping the Book exact dynamics of the execution times τ as a process of order size q. Exampleequations retain the original label with an e-suffix. As we discussed already, τ (q) isa subordinator with respect to q. The Gamma process provides an important familyof subordinatorsIn this case the distribution of the execution time τ for an order of size q isThe Gamma process evolves purely through totally unpredictable jumps, and inthis case (7) specializes toThe idea of subordinating price impact to the liquidity discovery is introducedfor the first time in Polimenis (2005), and our slow market here strongly parallelsthe trading floor in that paper. One difference between the two models, besides thefact that here the slow market will compete with a fast market for liquidity, is thatin Polimenis (2005) liquidity discovery is purely continuous, and the executionpoint in (11) is defined as the stopping time when a Brownian motion meets thelevel q. As a consequence in Polimenis (2005), the execution time follows an InverseGaussian law.Given (8), the uncertainty in execution delays is given by Example, continued.
Given (11) and (13), the variance of execution delays for the Gamma executionfollowsUncertainty grows with size and declines with the liquidity of the market.Having modeled liquidity discovery, the next issue is that of the evolution ofprices during execution, that is, price discovery.
The uncertain execution time andthe uncertain price evolution during execution introduce execution risk in slowmarkets. If the price discovery process was deterministic, as in a fast market, theinformed trader would know that when her order execution would be completed, τ = (a -1 ,q) (11)(12) P( τ (q) ∈ d τ ) = ae -a τ (a τ ) q-1 / Γ (q)d τ for τ >0 K τ (s) = -log(1-s/a) (7e) Var( τ (q))= q ∫ +∞ L(dz)z =qK τ ”(0) (13) Var( τ ) = q/a (13e) eview of Futures Markets her per share profit would be exactly equal to ∆ P- λ S q. In reality, µ in (4) provides only an expected rate at which the price drifts“against” the trader. To capture the stochastic nature of real markets, impact ismodeled as a Brownian motion with drift µ >0.where τ (q) is the stopping time defined above. Furthermore, the impact Brownianmotion W is independent of noise trade arrivals. Even though price impact evolvesas a Brownian motion with respect to execution time, when impact is studied as afunction of size, which is the true control variable for the informed agent, it doesnot obey a Gaussian law anymore.In real markets price discovery continues while the floor searches for liquidity,and this is captured here by the mathematical subordination of the price discoveryto the liquidity discovery process in (14). Taking iterated expectations, byconditioning on the delay τ and using the moment generating function for τ , werecoverso that the cumulant generator of the impact process is related to the cumulantkernal of the delay through Example, continued.
In the case of the delay process in (11), given (7e), the impact follows a
VarianceGamma (VG) process. The cumulant function K(s) in this case equalsWe already know that the agent expects that the impact of her order will beFrom this we find the first derivative of the impact cumulant function at zero
I(q) = µ τ (q) + σ W( τ (q)) (14)
7. And this would trivialize the issue since the trader would just choose to trade the entirequantity on the low λ market.8. The VG process (with respect to time) is introduced in Madan and Milne (1991), Madan andSeneta (1990), and Madan, Carr, and Chang (1998). EI S = λ S q = K’(0) q (17) K’(0) = λ S (18) E q e sI(q) = E q E τ e sI( τ (q)) = exp(qK τ (µs+.5 σ s ))=exp(qK(s)) (15)(16) K(s) = K τ (µs+.5 σ s ) K(s)=log(a/(a-µs-.5 σ s )) (16e) rading on the Floor after Sweeping the Book The expected impact may also be recovered from (16) by noting thatwhich, using (10), leads towhich is a proof of the originally desired relation (4). Execution price risk equalsGiven (16), we haveor more intuitivelyCumulant functions are always convex, K”(0)>0, so that a large order will increasethe price impact and its variance. Liquidity lowers risk because it affects the execu-tion delay.
Example, continued.
When execution delays follow a Gamma process, Var( τ ) follows (13e), and(21) becomesNotice that, keeping λ S fixed, liquidity does not affect expected price impact. Nevertheless, a high- a security will be preferred because, by accelerating execution, a lowers price risk. IV. TRADING ON THE FLOOR
The trader exhibits a constant absolute risk aversion η and knows the liquidityparameters of the market; i.e. knows the liquidity offered by the floor, a , and the λ of the floor, so that she can use (4) to calculate the impact drift µ for her orders. Shealso knows the volatility parameter for the security, σ. By offering slow executions, a trading floor has another important disadvantagefor the insider; knowing it will impact the price, market makers who observe alarge incoming order will refuse to provide the crucial early liquidity and be pricediscriminated against. Unlike fast markets, where the instantaneous execution allowsthe trader to “hit” limit orders at their pre-committed levels, in slow markets, marketmakers may change their quotes after having seen the order.
As was discussed in
K’(s) = (µ+ σ s) K τ ’(µs+.5 σ s ) λ S = K’(0) = µ K τ ’(0) = µ a -1 (21) Var(I) = Var( τ ) µ + E( τ ) σ Var(I) = K”(0) q (19)(20)
K”(0) = µ K τ ”(0) + σ K τ ’(0) Var(I) = ( λ S2 + σ /a)q (21e) eview of Futures Markets the introduction, this is really the most crucial point in the entire debate about slowand fast markets. In other words, here, the slow market is a one-shot model in thesense that the entire order gets executed at a single price.When the order is submitted to the trading floor, the block will be transactedat a single price that fully reflects the entire impact of the information releasedduring the order execution. The utility — a trader with initial wealth W gets —from issuing a block order of size q equalsThe q-superscript in the expectation operator explicitly shows that the trader is nota price taker; that is, the order size will determine not only the trader’s position butthe price impact as well. That is, the expectation is taken conditionally on the cho-sen order size . Observe that, for the analysis of the optimal order, the direction ofthe mispricing does not matter, since it will only determine the direction of thetrade (buy for underpriced and sell for overpriced securities). Without loss of gen-erality, we may assume that ∆ P and q are positive.We find that the utility gain from block trading in a slow market equalswith the price of floor liquidity being given by Example, continued.
Given the Gamma dynamics, the price of liquidity in (24) specializes toObserve that, with this interpretation of P qS as the price of liquidity, and since K(0) = 0,we get the proper result that for small traders (price takers) the price of liquidity P iszero; a small investor is a price taker since she does not have to wait for liquidity.When trading on the floor, as the sole owner of her “aging” proprietaryinformation, the trader is faced with a dilemma: trade a small quantity (quick) thuscapturing the maximum per share benefit, or trade a large quantity (slow) at smallerper share gain. The informed trader behaves as a time-limited liquidity monopsonist confronted with what essentially amounts to an increasing supply curve forliquidity (22) U S = E q –exp(- η (W + q( ∆ P – I S (q)))) G S = log (U /U S ) = η q ∆ P – qP qS (23) P qS = K( η q) = K τ (µ η q + .5 σ η q ) (24)(24e) P qS = log(a/(a – µ η q – .5 σ η q ))
9. Exponential utility is negative, and thus the agent is better off by actually lowering his utilityin absolute terms.10. Clearly, since K’(0) = λ S >0, and given the convexity of cumulant functions K(s), dP/dq>0. rading on the Floor after Sweeping the Book Given Gamma dynamics, the price derivative specializes to A. The Optimal Order on the Standalone Floor
In slow markets, the informed agent’s problem is to decide for the optimalblock order size. Given (23), a large trader will choose a size q that satisfiesAs a liquidity monopsony, the informed trader trades to the point that equatesher constant marginal revenueto her increasing marginal liquidity costThe optimal trade point q s is shown in Figure 1. V. TRADING ON THE FLOOR AFTER SWEEPING THE BOOK
In a hybrid market, where a trader has access to a trading floor as well as theentire limit order book, she has one more control variable. Namely, she can choosehow many shares to sweep from the book, q F , and send her remaining order, q S , tothe floor for execution. Notice that we do not assume here any ex ante preferencefor one market over the other; the order split choice and submission to both marketshappens simultaneously. Nevertheless, since sweeping the book is deterministicand instantaneous, while trading on the floor is slow, the book trade will materializebefore the floor trade has even started.
This important observation is central in theentire debate around slow versus fast markets; due to their ability to instantaneouslyreflect it, information flows easier from fast to slow markets rather than the otherway around. Some who, probably unfairly, argue that trading floors profit on theexpense of electronic markets imply this kind of consideration.As we saw previously, when a block order is submitted to a floor, the blockwill be transacted at a single price that fully reflects the entire impact of theinformation released during the order execution. If q H = q HF + q HS is the entire trade(25) dP/dq = K τ ’(µ η q+ .5 σ η q )(µ η + σ η q)>0 (25e) dP/dq = (µ η + σ η q)/(a- µ η q- .5 σ η q )>0 q S = argmax q η q ∆ P – qP qS (26)
11. Clearly, the trade quantity q will always be such that a– µη q– .5 ó η q > 0; otherwise, thetrader will incur an unbounded liquidity price (24e). MR = η∆ P = AR (27)(28)
MC = P qS + qdP/dq > AC = P qS eview of Futures Markets Figure 1. size in a hybrid market, then the ratio q HF /q H , denotes the fraction of her order thetrader chooses to sweep the book for and thus captures the immediacy of the order.Equivalently, the immediacy factor q HF /q H determines how deeply (i.e., how farfrom the current market) she sweeps the book.The total impact of a hybrid trade is decomposed into two different sources:(1) a deterministic component because of sweeping and (2) a stochastic due toprivate information leaking and becoming public during execution on the floor.Thus, the total utility gain from trading in a fast electronic market, and a slowand transparent trading floor equalsEssentially, the trader buys the liquidity to trade in two different places. Sweepingthe book happens instantaneously, before the floor has had any time to incorporateinformation, and thus the liquidity cost of the fast component in a hybrid marketremains unaffected as if the fast market where to operate alone (not as part of ahybrid market), P qF = ηλ F q.On the other hand, since sweeping the book releases information immediately,this information will also be immediately reflected in trading ensuing on the floor.Essentially, market makers on the floor observe the electronic component and adjust ∫ −−−∆= qF qSSFSFHH PqIqydyPqG ηληη (29) rading on the Floor after Sweeping the Book prices accordingly. Thus, the liquidity price in the slow market (trading floor) willalso depend on the amount traded on the fast market (limit order book). This explainsthe term η q S I F = η q S λ F q F in (29) above.Taking the derivative of (29) with respect to the amount of total trading, wefind that the amounts traded in the fast and slow markets have to combine as followsBut from (3), we see that a stand-alone fast market would provide the same result, Lemma 1.
Hybrid markets do not generate more informed trading than stand-alone fast markets.In other words, the total trading in the hybrid market is determined from the λ of the fast component. When it co-exists with a fast market, a slow market “steals”liquidity without offering any informational benefits.To some degree, lemma 1 is a negative statement for the existence of the so-called hybrid markets that combine electronic and auction based execution facilities.But we have to be careful in that lemma 1 is only valid under the restricted tradingpolicies considered here, that is, one-shot policies. That is here, we have notconsidered the fully dynamic problem where the trader trades small quantities, andas the impact of her previous sub-orders gets realized, she decides how to tradenext.Finally, lemma 1 predicts something many market participants have assertedall along: The two types of markets are inherently competitive since they will haveto share the same liquidity. Even though the sensitivity of the slow market will notaffect the overall informed trading, it will play a central role in determining thedegree to which the trading floor will be able to compete in “stealing” liquidityaway from the book.Having solved for the optimal q H , rewrite (29) as a function only of the tradingon the hybrid floor, q HS and take the derivative of (31) with respect to q HS The optimal trade point q HS is clearly shown in Figure 1 as the point where theline with slope çë F meets the MC = P qS + qdP/dq curve.Since this partial at zero is zero, the trader will only trade at the floor when thesecond derivative at zero is positive. Taking the second derivative of G H , withrespect to floor trading, at zero, and observing from (25) that (30) ∆ P = λ F q H = λ F (q HF + q HS ) (31) ∫ − −−−−∆= qHSqH qHSHSHSHFHSFHH PqqqqydyPqG )( ληληη dqdPqPq ηλ qG qHSHSqHSHSFHSH −−=∂∂ (32) eview of Futures Markets we find that Lemma 2.
In a hybrid market, trading will be diverted from the fast to thetransparent slow market only if the lambda of the floor satisfiesLemma 2 formalizes and quantifies the previous informal reasoning that slowmarkets, being risky, will have to somehow provide cheaper trading to attractliquidity. Indeed, criterion (34) points to cheaper trading, since the trader expectsless than half the impact per share. Lemma 2 shows that the proper dimension atwhich competing markets are measured is depth, ë -1 , rather than liquidity alone. VI. THE NATURE OF THE MARKET FOR LIQUIDITY
It is easy to see that as a liquidity monopsonist, the informed trader buys lessliquidity thus revealing less information in the markets. But how much informedtrading, q max , can we expect at the best? What is the maximum profitable amount oftrading from a social point of view? The value of the private information has beencompletely collected when there are no more utility gains from re-distributing theprivate information.Specifically, this is the point at which a new trader, endowed with the insider’sinformation, cannot profitably trade.Let’s start with a market where informed traders have already submitted tradesfor q shares. Since a new trade will have to be executed in a floor that alreadyworks on executing market orders for q shares, the post-trade utility for a newtrader j who has been endowed with the information equalsThe critical observation is that, as a subordinated Brownian motion, pricediscovery in (14) is a Lévy process with respect to trading q, not time; that is, Ee sI(q) = e qK(s) . A Lévy process is characterized by increments that are independent andidentically distributed. Thus, given the trade q, the utility gain equalsEquation (36) shows that our model captures the following fundamental, and,at first, counter-intuitive characteristic of the market for liquidity. In commoditymarkets, the price of the commodity is determined by the total demand, and thetotal cost for the j th agent equals the product of her demand times the market-wideclearing price, q j P q+qj . In the market for liquidity, each agent pays an individual (33) [dP/dq] = K τ ’(0)µ η = ηλ S (34) λ S < λ F /2 U(q j ) = U o E qj exp(- η q j ( ∆ P – I(q+q j ))) (35)(36) qjjjjoj PqqPq η qUU G )()(log +−== ∆ rading on the Floor after Sweeping the Book price totally determined only by their trade P qj , but their total cost depends on theentire liquidity demand, q+q j . This happens because, liquidity markets differentiateagents by the amount they trade. Unlike “small” traders, who pay little as P =0,agents who trade large quantities are informed and they pay a large price. From (36) the marginal gain of the new trader isSince, at q max , no trader can benefit if given the information, the maximumamount of trading is such that the initial marginal gain for a new trader has to bezerowhere, we used the fact that P = 0. From (33) we haveand we recoverWhen contrasted with (3) and (30), we observe that eventually a stand-alonetrading floor will reach the competitive point (38), but that will only happen whenthe private information leaks to an increasing number of traders.Equation (30) points to another result. When a stand-alone floor (type S) isenhanced with a fast execution component to become a hybrid market (type H),overall informed trading is expected to decline. This decline happens because,from lemma 2, the ë of the fast market is quite higher. Since the overall trading inthe hybrid market is determined by the ë F , and not the ë s anymore, total tradingdeclines. Essentially, it will be easier for many informed traders to extract themaximum value of their private information and they will trade less. Of course, ifthere is only a single informed trader, the introduction of the book sweeping facilitywill lower trading on the floor but increase overall informed trading as is clearlyshown in Figure 1. CONCLUDING REMARKS
The paper introduces a formal model of the liquidity and price discoverymechanisms in a hybrid market, which combines a trading floor with an electronicsweeping facility. We show that, despite some common beliefs, hybrid markets do
12. Clearly, anonymous markets, which we don’t study here, to the benefit of informed traderscannot differentiate them by their size. Such markets are discussied in Polimenis (2005).13. Since traders here are risk averse, the maximum trade will happen only by a new trader whois not already exposed to risk. ?Gj/?qj = η∆ P – Pqj – (q+qj) dPqj/dqj η∆ P – q max [dP/dq] = 0 (37) [dP/dq] = ηλ S q max = ∆ P/ λ S (38) eview of Futures Markets not generate more informed trading than stand-alone electronic markets. Underthe assumptions of the model here, a slow market may only divert away liquidityfrom a fast market when it is at least twice as deep. Hybrid markets improve stand-alone trading floors, when the informed trader is the sole information owner. Onthe contrary, when there are multiple informed traders, transforming a trading floorto a hybrid market may lead to a decline of the overall amount of informed trading. References
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