Informed trading, limit order book and implementation shortfall: equilibrium and asymptotics
aa r X i v : . [ q -f i n . T R ] M a r INFORMED TRADING, LIMIT ORDER BOOK ANDIMPLEMENTATION SHORTFALL: EQUILIBRIUM ANDASYMPTOTICS
UMUT C¸ ET˙IN AND HENRI WAELBROECK
Abstract.
We propose a static equilibrium model for limit order book where N ≥ last slice traded against the limit order book is priced at the liquidation value of the asset.However, competition among the insiders leads to aggressive trading causing theaggregate profit to vanish in the limiting case N → ∞ . The numerical resultsalso show that the spread increases with the number of insiders keeping the otherparameters fixed. Finally, an equilibrium may not exist if the liquidation value isunbounded. We conjecture that existence of equilibrium requires a sufficient amountof competition among insiders if the signal distribution exhibit fat tails. Introduction
Kyle [16] studies in a simple but remarkably powerful framework a single riskneutral informed trader and a number of non-strategic uninformed liquidity traderssubmitting orders to a market maker, who aggregates all the orders and clear themarket at a single price. Consequently, Kyle’s batch trading model does not producea bid-ask spread. However, his model allows for an explicit characterisation of equi-librium parameters - including the optimal strategy of the informed trader as well as
Date : March 11, 2020. the equilibrium pricing rule. This in turn allows us to analyse how the private infor-mation is disseminated to the market and gets incorporated into the prices over time.In particular,
Kyle’s lambda yields an explicit measurement of market’s liquidity andprice impact of trades.While the role of the market makers and the price setting mechanism of Kyle’sbatch trading model were in line with the practices of specialists and floor tradersof the main exchanges in the the 80s, the role of the designated market makers hasdiminished in recent years. Nowadays most of the equity and derivative exchangeshave moved to the electronic limit order book format. What distinguishes limit ordermarkets from markets with a uniform market-clearing price as in Kyle [16] is thateach limit order is executed at its respective limit price leading to discriminatorypricing .The case of liquidity suppliers moving first and submitting limit orders to be laterhit by potentially informed market order was first studied by Rock [23] and Glosten[12]. In equilibrium even infinitesimal trades have significant impact, which resultsin a positive bid-ask spread unlike the uniform price auction models. Following theapproach initiated by Rock and Glosten, Seppi [24] studies the liquidity provisionby a specialist competing against a competitive limit order book. His model alsoallows a discussion on market design issues such as the effect of ‘tick size.’ Based onthe model assumptions of [24] Parlour and Seppi [18] build a model of competitionbetween exchanges. In another static model Foucault and Menkveld [10] consider anequilibrium among limit order traders and a broker than can be one of two types anduse this model to study the effects of market fragmentation in the context of rivalrybetween Euronext and the London Stock Exchange in the Dutch stock market.While useful for policy purposes, the above static models have also some undesir-able features. Probably the most unrealistic feature of these models is that, quotingParlour and Seppi [19], “investors either have an inelastic motive to trade, and arewilling to pay for immediate execution via market orders, or they are entirely disin-terested liquidity providers with no reason to trade other than to be compensated forsupplying liquidity via limit orders.” In particular, unlike the Kyle’s model, their fo-cus is more concentrated on the supply side of the liquidity and the investors’ tradingstrategies are not completely endogenous. As a result, the shape of the limit orderbook, which in general can only be obtained numerically, that arises in these modelsdoes not truly capture the impact of trades by a rational investor with price elasticmotives.In this paper we present a static microstructure model for the limit order book,where the adverse selection occurs due to the existence of informed traders (henceforthcalled insider s). Following Kyle [16], the insiders know the liquidation value V ofthe asset in advance and place a market order to maximise their expected profit.They submit their order to a competitive dealer who already has a position thatis independent from the liquidation value. The dealer trades the aggregate amount NFORMED TRADING AND THE LIMIT ORDER BOOK 3 against a competitive limit order book populated by liquidity suppliers that arrivedto the market before the dealer and the insiders.Our assumption that the limit order book is competitive requires a justificationin view of the results of Biais et al. [5] and Back and Baruch [3]. First of all, thecompetitive offer curve should be viewed as a limiting book when the number oflimit order traders increases to infinity. However, as shown in an example in [3] thecompetitive book is not always obtained as a limit of Nash equilibiria among finitelymany liquidity providers if the level of adverse selection faced by the limit ordertraders is not high enough. In our model there is always sufficient adverse selectionsince we assume that the dealer’s own demand for the asset is normally distributed.Even though we model the interactions in a single period model, our framework alsohas the flavour of a continuous-time Kyle model considered by Back [2]. Indeed, incase of a monopolistic insider, we show that conditional on V = v the last infinitesimalslice of the aggregate order traded by the dealer against the order book is priced at v on average, reminiscing the convergence of the equilibrium price to the liquidationvalue by the end of the trading horizon in the continuous-time Kyle model. Thus,all private information gets incorporated into the order book once the last slice hastraded. This is in contrast with the corresponding Kyle model in one period , whichis not surprising given the uniform pricing in the Kyle model.The above ‘convergence’ result motivates the analysis of the case of multiple in-siders. In the same vein as Holden and Subrahmanyam [14] we also model and solvethe imperfect competition among N ≥ V perfectly. As in [14] this competition leads to more aggressive trading. We findthat (conditional on V = v ) the average price for the last slice traded by the dealerexceeds v if the insiders are buying in equilibrium, which can be attributed to moreaggressive buying due to the competition. An analogous observation holds when theoptimal strategy in equilibrium is to sell. We obtain an explicit formula for the ag-gregate profit of the insiders that shows that the aggregate profit converges to 0 as N → ∞ .We characterise the equilibrium strategy of insiders and the corresponding equilib-rium order book as a solution of an integral equation, which is equivalently given bythe fixed point of an integral operator. Although this equation admits an analyticsolution only in very restrictive settings, its numerical implementation is straight-forward. We establish the existence of equilibrium when the fundamental value isbounded and obtain numerical solutions for a number of unbounded distributionscommonly found in the literature and used in practice.Despite the fact that the exact form of the order book can only be solved numeri-cally, we are nevertheless able to obtain the exact analytic asymptotics when the ordersize is large. Moreover, due to the scaling property of the normal distribution, theseasymptotics still provide a good approximation for small orders if the variance of the In the corresponding one-period Kyle model, where V is normally distributed with mean 0, theaverage price for the aggregate order conditional on V = v equals v . UMUT C¸ ET˙IN AND HENRI WAELBROECK dealer’s demand is sufficiently small. The error in such an approximation appears tobe still low even in the case of higher variance as confirmed visually by our numericalstudies.Due to the discriminatory pricing in the order book, our model produces a non-zerobid-ask spread in equilibrium. We find that the bid-ask spread imposed by the limitorder traders is the same regardless of the variance of the initial inventory of the dealer.The shape of the order book, however, is not invariant to the changes in this variance;on the contrary, the order book flattens as it increases. That is, the limit order tradersdo not need to extract significantly higher rents for larger trades if the informationcontent of the order is very small. In the limit the equilibrium converges to one wherethe transaction cost is proportional to the trade size. Moreover, similar to the modelconsidered by Foucault [9], our model also predicts that a larger price volatility leadsto a bigger bid-ask spread - a phenomenon that is empirically documented by Ranaldo[22]. Indeed, we show that the spread gets wider as the variance of the private signalof the insiders increases, which corresponds to the case of a higher informationaladvantage.That we can compute the price impact of large trades has profound implicationsfor practitioners. Portfolio managers endeavour to find mispriced assets in order tobeat their benchmarks. When they decide to change their holdings to take advantageof a mispricing, they create orders in an order management system; the trading deskroutes these orders to broker-dealers for execution. When they do so, on average, theprice to buy is greater than at the time of the decision, and vice-versa, the sellingprice is lower, because the trade’s effect on supply and demand causes market impact.The implementation shortfall is thereby the value lost by not being able to execute atthe decision price [20]. Understanding how this implementation shortfall scales withtrade size is essential to optimize the investment management process. A marketimpact model is needed in order to maximize net trading profits. It is also importantto optimize position sizes, limit liquidity risk and estimate a portfolio’s capacity.Empirical data provides some clues as to the shape and scale of market impact. Inparticular, it has been known for some time that impact is a concave function of tradesize (see Torre [26]). But biases in the data, a low signal-to-noise ratio and other issuesprevent practitioners from determining the functional form of market impact precisely,particularly for large trades where data is sparse. Models used by practitioners includesquare root and logarithm (e.g. Torre [26], Potters and Bouchaud [21], Almgren etal. [1], Bershova and Rakhlin [4], and Zarinelli et al. [29]). When calibrated to data,these models yield similar results for small to moderate trade sizes, but they yield verydifferent predictions for the impact of for very large trades. This is unfortunate giventhat the largest trades typically dominate the aggregate implementation shortfall formost portfolios. The absence of a consensus on the shape of market impact for verylarge trades motivates the search for a theoretical framework that would capture theessential features of trading and yet be sufficiently parsimonious to be testable.
NFORMED TRADING AND THE LIMIT ORDER BOOK 5
We show that the market impact follows a power law or a logarithmic law dependingon the distribution of the liquidation value. The price impact, and equivalently theimplementation shortfall, has a power law if the liquidation value has fat tails whilelighter tails lead to a logarithmic behaviour.Our framework also allows us to compute the tails of the probability distributionassociated with aggregate volume. For a large class of distributions that can be usedto model the liquidation value we show that the tail probability distribution for thetrade volume obeys a power law. The power-law behavior of order sizes has beendocumented previously in various contexts: Gopikrishnan et al. [13] showed that themarket volume in a time interval ∆ t has a power-law tail with exponent 1.7; Lillo et al.consider off-book trades [17] and find an exponent ranging from 1.59 to 1.74. Vaglicaet al. reconstructed metaorders in Spanish stock exchange using data with brokeragecodes and found that the metaorder transaction size is distributed has a power lawtail with exponent 1.7 [28]. And in a study performed directly on institutional tradedata from Alliance Bernstein, Bershova and Rakhlin found a tail exponent 1.56 formetaorder sizes [4].Our proof of the existence of equilibrium assumes that the liquidation value ofthe asset is bounded. However, our numerical experiments suggests that equilibriumexists for a large class of unbounded signals. Moreover, the asymptotics of thesesolutions agree with the analytical forms that our theory predicts. However, unlikethe bounded case, an equilibrium may not exist for all unbounded distributions.Given the premise of numerical results and our formal calculations we conjecture thatexistence of equilibrium requires a sufficient amount of competition among insiders ifthe signal distribution exhibit fat tails. For instance, when the private signal is givenby a Student’s t-distribution our numerical iterations diverge in case of a monopolisticinsider.We also consider an alternative model where instead of sending the order to a dealerwith an existing inventory, the informed investors send the order to an institutionaltrading desk that also receives orders from noise traders. The aggregate is liquidatedagainst a limit order book; however, in the case of the institutional trading desk theinsiders and noise traders all receive allocations at the same average price. While wewere not able to find an existence proof for this model, numerical solutions are similarto those of the dealer inventory model.Structure of the paper is as follows: We present the dealer inventory model in thenext section. In Section 3, we show the existence of an equilibrium and characterizesome of its properties. Section 4 considers the asymptotic behaviour of market impact.Numerical solutions are provided in Section 5 for the dealer inventory model. Section6 explores the trading desk model numerically and compares results with the dealerinventory model. UMUT C¸ ET˙IN AND HENRI WAELBROECK The market structure and equilibrium
Our model is built upon Glosten [12] and the trading takes place in one-period:There are three class of investors that are all risk-neutral: 1) competitive liquiditysuppliers who post limit orders, 2) a competitive dealer who clears the market, and3) N ≥ V of the asset. All randomvariables in this section are assumed to be define on a complete probability space(Ω , F , P ) and E is the expectation operator associated to P .Liquidity suppliers move first and place limit orders that give rise to an order book.That is, if the limit order book is defined by some function h : R → R , the marketorder moving up (or down) the book faces a cost of h ( y ) dy as soon as hitting thelimit order at level y . The dealer already has Z ∼ N (0 , σ ) number of shares of theasset, which is assumed to be independent of V . The dealer’s initial inventory in theasset is unknown by other market participants. The insider chooses a trade size X tomaximise her expected profit conditional on her private information.Let us first consider the case N = 1. We assume that the cost of a market order of X units by the insider is Z X h ( Z + y ) dy. (2.1)The above cost can be justified as a result of Bertrand competition among dealerswhose initial mean-zero inventories are normally distributed with identical variance σ . Indeed, first observe that when the insider trades X units via the dealer, thedealer’s inventory changes from Z to Z + X . If the cost to the insider is given by(2.1), the total profit of the dealer after trading via the liquidity suppliers is given by Z X h ( Z + y ) dy − Z X + Z h ( y ) dy = Z X + ZZ h ( y ) dy − Z X + Z h ( y ) dy = − Z Z h ( y ) dy, which is the cost of liquidating his initial inventory directly via the limit order book.Therefore, no dealer will be willing to charge less than (2.1). The insider knows thedistribution of Z but do not know the inventories of individual dealers. Dealers firstannounce that they will price according to 2.1 plus a premium (the dealer’s profit).Insider then chooses a dealer based on this information, before knowing Z . Bertrandcompetition will then drive the dealer’s premium to zero. That is, an individual dealerknows that if he charges higher than what is proposed by (2.1), he will be undercutby another dealer. Thus, a Bertrand competition will lead to (2.1) for the cost of thetrades of the insider. Remark 1.
Note that the quote given by the competitive dealer that leads to (2.1)does not violate the limit order protection rule that is mandated by the SEC in theUS. To see this suppose the insider wants to buy
X > many units. She is charged p ( X ) := X R X + ZZ h ( y ) dy on average for this transaction. If p ( X ) is bigger than thebest ask, i.e. h (0+) , the price priority implies that the limit sell orders at price p ( X ) and less must be filled first. Thus, the dealer will first buy h − ( p ( X )) many shares NFORMED TRADING AND THE LIMIT ORDER BOOK 7 at a cost of R h − ( p ( X ))0 h ( y ) dy and the remaining X + Z − h − ( p ( X )) shares in hisinventory at R X + Zh − ( p ( X )) h ( y ) dy . This makes his cumulative cost R X + Z h ( y ) dy and finalprofit − R Z h ( y ) dy coinciding with above calculations. Thus, in view of (2.1) the expected profit of the insider from a market order of size x is given by E v (cid:20) V x − Z x h ( Z + y ) dy (cid:21) , where E v is the expectation operator for the insider with the private information V = v . Since h is assumed nondecreasing, the first order condition characterises theunique X ∗ achieving the maximum expected profit via V = F ( X ∗ ), where F ( x ) := Z ∞−∞ h ( x + z ) q ( σ, z ) dz (2.2)and q ( σ, · ) is the probability density function of a mean-zero Gaussian random variablewith variance σ . Note that since h ∗ is non-decreasing and not constant, F is strictlyincreasing and one-to-one. Thus, X ∗ = F − ( V ).We shall also consider the case of multiple insiders trading via the same competitivedealer and knowing the value of V . Assuming that the dealer charges each insider anamount proportional to their order size, the expected profit of an individual insiderplacing an order of size x is given by E v (cid:20) V x − xU + x Z U + x h ( Z + y ) dy (cid:21) , where U denotes the aggregate demand of the other insiders. The number of insiderswill be denoted by N .The first order condition associated with the above optimisation problem of anindividual insider is again given by V = E v (cid:20) xU + x h ( Z + U + x ) + U (( U + x ) Z U + x h ( Z + u ) du (cid:21) . As every insider has symmetric information and is risk-neutral, the equilibrium de-mand x ∗ for each insider must be the same and satisfy V = E v (cid:20) h ( Z + N x ∗ ) N + N − N x ∗ Z Nx ∗ h ( Z + u ) du (cid:21) . Denoting the total informed demand by X ∗ , the above can be rewritten as V = F ( X ∗ ), where F ( x ) := E v (cid:20) h ( Z + x ) N + N − N x Z x h ( Z + u ) du (cid:21) , (2.3) UMUT C¸ ET˙IN AND HENRI WAELBROECK and F (0) is interpreted by continuity to be E v (cid:20) h ( Z ) N + ( N − h ( Z ) N (cid:21) = E v [ h ( Z )] . Moreover, it follows from the monotonicity of h that F defined via (2.3) is strictlyincreasing. Also note that (2.3) reduces to (2.2) when N = 1. Thus, we shall alwaysrefer to (2.3) when discussing the optimal strategies of the insiders regardless of thevalue of N .In what follows the total informed demand will be denoted by X and we assumefollowing Glosten [12] that limit prices are given by ‘tail expectations.’ That is,denoting the total demand X + Z by Y , h ( y ) = (cid:26) E [ V | Y ≥ y ] , if y > E [ V | Y ≤ y ] , if y < . (cid:27) (2.4)The value h (0) is not relevant for the subsequent computations and can be freelychosen to be any value between the best ask h (0+) := lim y ↓ h ( y ) and the best bid h (0 − ) := lim y ↑ h ( y ).The definition (2.4) entails that liquidity suppliers earn zero aggregate profit onaverage. Indeed, the expected profit is given by E (cid:20)Z Y ( h ( y ) − V ) dy (cid:21) = E (cid:20) [ Y > Z Y ( h ( y ) − V ) dy + [ Y < Z Y ( V − h ( y )) dy (cid:21) = Z ∞ E [( h ( y ) − V ) [ Y ≥ y ] ] dy + Z −∞ E [( V − h ( y )) [ Y ≤ y ] ] dy = 0 , where the last equality is due to the definition of the conditional expectation. Definition 2.1.
The pair ( h ∗ , X ∗ ) is said to be a Glosten equilibrium if h ∗ is non-decreasing and non-constant, X ∗ ∈ R and i) h ∗ satisfies (2.4) with Y = X ∗ + Z ; ii) X ∗ is the profit maximising order size for the insider(s) given h ∗ . That is, V = F ( X ∗ ) , where F is given by (2.3). The strict monotonicity of F leads to the following result that in particular yieldsan explicit formula for the aggregate profit of the insiders. Proposition 2.1.
Let ( X ∗ , h ) be an equilibrium and F defined by (2.3). Then, thefollowing hold: (1) For any x ∈ R we have E v [ h ( x + Z )] = F ( x ) + N ( N − x N Z x ( F ( x ) − F ( y )) y N − dy = F ( x ) + N ( N − Z ( F ( x ) − F ( xy )) y N − dy. (2.5) NFORMED TRADING AND THE LIMIT ORDER BOOK 9 (2) E v [ h ( X ∗ + Z )] = v when N = 1 and, for N > , E v [ h ( X ∗ + Z )] > v (resp. E v [ h ( X ∗ + Z )] < v ) if X ∗ > (resp. X ∗ < ). More precisely, E v [ h ( X ∗ + Z )] = v + N ( N − Z ( v − F ( yX ∗ )) y N − dy. (2.6)(3) The aggregate expected liquidation profit of the insiders conditional on V = v is given by π ∗ ( v ) := E v (cid:20)Z X ∗ ( v − h ( y + Z )) dy (cid:21) = N Z F − ( v )0 ( v − F ( y )) (cid:18) yF − ( v ) (cid:19) N − dy = F − ( v ) N Z ( v − F ( yF − ( v ))) y N − dy. (2.7)The expression (2.6) reveals an interesting feature of our model akin to Kyle’s modelin continuous time. Observe that h ( X ∗ + Z ) can be viewed as the last slice that istraded with the limit order traders. Thus, when there is a monopolistic insider, (2.6)shows that the final slice is priced at the actual value of V similar to the convergenceof the equilibrium price to the liquidation value of the asset in the continuous timeversion of the Kyle model studied in Back [2] for general payoffs. As expected, this‘convergence’ disappears when the liquidation value of the asses is observed by morethan one insider. Such a competition leads to a more aggressive trading as in Holdenand Subrahmanyam [14] and results in higher (resp. lower) market valuation if theoptimal strategy is to buy (resp. sell).In practice an important trading benchmark for traders is the implementation short-fall . Perold [20] defines it as the difference between a ‘paper trading’ benchmark andthe actual trading costs. Assuming that the benchmark is given by the ex-ante valu-ation E [ V ], the associated shortfall in our context can be defined as follows: Definition 2.2.
Let h be a function that defines the limit order book in a Glostenequilibrium. Then, the implementation shortfall associated with trading x units isgiven by IS ( x ) := 1 x Z x E [ h ( Z + y )] dy. Observe that IS ( x ) is simply the expected average cost of trading x units. As thefollowing result shows, it is smaller than the marginal cost F ( x ), which is given bythe first order condition (2.3). Proposition 2.2.
Let h be a function that defines the limit order book in a Glostenequilibrium and F ( x ) be given by the first order condition (2.3). Then, IS ( x ) = N Z F ( xy ) y N − dy. In particular, IS ( x ) < F ( x ) for x > and IS ( x ) > F ( x ) for x < . Characterisation of Glosten equilibrium
Suppose that ( X ∗ , h ∗ ) is an equilibrium. Writing h instead of h ∗ to ease the expo-sition and using the definition of h when y >
0, we get h ( y ) = E [ V | X ∗ + Z ≥ y ] = E [ V | F − ( V ) ≥ y − Z ] = E [ V | V ≥ F ( y − Z )] . Similarly, h ( y ) = E [ V | V ≤ F ( y − Z )] for y < right-continuous functions Φ ± and Π ± viaΦ + ( y ) := E [ V [ V >y ] ] , Π + ( y ) := P ( V > y )Φ − ( y ) := E [ V [ V ≤ y ] ] , Π − ( y ) := P ( V ≤ y ) = 1 − Π + ( y ) . Note that Φ + ( y ) + Φ − ( y ) = E [ V ] for all y ∈ R . Moreover, Φ + ( y − ) = E [ V [ V ≥ y ] ] andΠ + ( y − ) = P ( V ≥ y ).Now let us compute h ( y ) for y > h ( y ) = E [ V | V ≥ F ( y − Z )] = E [ V [ V ≥ F ( y − Z )] ] P ( V ≥ F ( y − Z ))= R ∞−∞ Φ + ( F ( y − z ) − ) q ( σ, z ) dz R ∞−∞ Π + ( F ( y − z ) − ) q ( σ, z ) dz . On the other hand, Π + ( x ) = Π + ( x − ) at most for countably many x . Thus, Φ + ( x ) =Φ + ( x − ) for almost all x and we have for all y > h ( y ) = R ∞−∞ Φ + ( F ( y − z )) q ( σ, z ) dz R ∞−∞ Π + ( F ( y − z )) q ( σ, z ) dz . (3.8)Similarly, for y < h ( y ) = R ∞−∞ Φ − ( F ( y − z )) q ( σ, z ) dz R ∞−∞ Π − ( F ( y − z )) q ( σ, z ) dz . (3.9)In order to obtain an equation for F it will be convenient to define, for any continuous g , the mappings φ + g ( x ) := R ∞−∞ Φ + ( g ( z )) q ( σ, x − z ) dz R ∞−∞ Π + ( g ( z )) q ( σ, x − z ) dz and φ − g ( x ) := R ∞−∞ Φ − ( g ( z ) q ( σ, x − z ) dz R ∞−∞ Π − ( g ( z )) q ( σ, x − z ) dz . Let us also set φ g ( x ) := φ + g ( x ) x ≥ + φ − g ( x ) x< . (3.10)Now, combining (2.3), (3.8) and (3.9) yields an equation for F : F ( x ) = 1 N Z ∞−∞ q ( σ, x − z ) φ F ( z ) dz + N − N x Z x dy Z ∞−∞ q ( σ, y − z ) φ F ( z ) dz, (3.11) NFORMED TRADING AND THE LIMIT ORDER BOOK 11
If one can change the order of integration in above , then (3.11) can be rewritten as F ( x ) = Z ∞−∞ (cid:26) N q ( σ, x − z ) + N − N ¯ q ( σ, x, z ) (cid:27) φ F ( z ) dz, (3.12)with ¯ q ( σ, x, z ) := x =0 x Z x q ( σ, y − z ) dy + x =0 q ( σ, z ) . (3.13)The preceding calculations show that the existence of a solution for the above integralequation is a necessary condition for equilibrium. We in fact have the converse, too. Theorem 3.1.
A Glosten equilibrium exists if and only if there exists a function F : R → R that satisfies (3.11). Given such a solution F , ( X ∗ , h ∗ ) constitutes anequilibrium, where X ∗ = F − ( V ) and h ∗ is defined via (3.8) and (3.9). In view of the above theorem finding an equilibrium boils down to finding a solutionof (3.11). Moreover, equilibrium will be uniquely defined if there exists a uniquesolution of (3.11). As usual, solutions of (3.11) will be identified as fixed-points ofa mapping. Before analysing in depth this fixed-point problem, we shal observe fewproperties of equilibrium. The following scaling property of F is inherited from theanalogous property of the Gaussian distribution. Proposition 3.1.
Let F (1; · ) be a solution of (3.11) with σ = 1 . Then the function x F (1; xσ ) solves (3.11).Proof. F (cid:16) xσ (cid:17) = 1 N Z ∞−∞ q (cid:16) , xσ − z (cid:17) φ F ( z ) dz + σ N − N x Z xσ dy Z ∞−∞ q (1 , y − z ) φ F ( z ) dz = 1 N Z ∞−∞ q (cid:16) σ, x − z (cid:17) φ F (cid:16) zσ (cid:17) dz + N − N x Z x dy Z ∞−∞ q (cid:16) , yσ − z (cid:17) φ F ( z ) dz = 1 N Z ∞−∞ q (cid:16) σ, x − z (cid:17) φ F (cid:16) zσ (cid:17) dz + N − N x Z x dy Z ∞−∞ q ( σ, y − z ) φ F (cid:16) zy (cid:17) dz. Similar manipulations yield φ ± F (cid:16) zy (cid:17) = R ∞−∞ Φ ± (cid:16) F (cid:16) uy (cid:17)(cid:17) q (cid:16) σ, u − z (cid:17) du R ∞−∞ Π ± (cid:16) F (cid:16) uy (cid:17)(cid:17) q ( σ, u − z ) du , which establishes the claim. (cid:3) A straightforward corollary to the above is the following.
Corollary 3.1.
Consider the solutions of (3.11) for any σ . (1) If (3.11) has a unique solution for some σ , it has a unique solution for all σ s. (2) If (3.11) has a unique solution for some σ , F (0) does not depend on σ . We shall see later that this is justified when V is bounded from below (3) Let h ( σ ; · ) be the function defined via (3.8) and (3.9), where F is the uniquesolution of (3.11) for the given σ . Then, h ( σ ; x ) = h (1; xσ ) for all x = 0 . (4) Suppose (3.11) has a unique solution for some σ and let X ∗ ( σ ) be the optimalorder size for the given σ . Then, X ∗ ( σ ) = σX ∗ (1) . In particular, h inherits the same scaling property of F . A simple but strikingconsequence of this property concerns the equilibrium bid-ask spread . Definition 3.1.
Let h be a function with right and left limits defining an order book.The bid-ask spread of the associated order book is given by h (0+) − h (0 − ) . Corollary 3.2.
Suppose that uniqueness holds for the solutions of (3.11). Then, h (0+) − h (0 − ) does not depend on σ . Moreover, for fixed x , h ( x ) is decreasing in σ for x > and increasing in σ for x < . Thus, the liquidity suppliers charge a bid-ask spread that does not vanish even ifthe amount of ‘noise’ trading is excessively large. Moreover, the dependency of thelimit price on σ is not monotone. In particular, h ( x ) approaches to the supremum(resp. infimum) of the set of possible values of V for x > x <
0) when σ →
0. We shall see these phenomena occurring explicitly in Examples 3.1 and 3.2.Furthermore, if we consider the limiting behaviour in the other direction as σ → ∞ ,we observe that the order book gets flatten and converges to the one that yieldstransaction costs that are proportional to the order size.A similar scaling property will hold when the signal distribution possesses a formof self-similarity, which can be proved by similar methods. Corollary 3.3.
Suppose that uniqueness holds for the solutions of (3.11) and that V = V ( t ) for some t > such that the random variables ( V ( t )) t> are self-similar inthe sense that V ( t ) d = t α V (1) for some α > for all t > . Let F t be the solution of(3.11) for V = V ( t ) . Then, F t = t α F . Moreover, the bid-ask spread is increasing in t . A typical example of a self-similar random variable is mean-zero Normal randomvariable. In this case V ( t ) d = tV (1), where t is the parameter corresponding to stan-dard deviation. Similarly, V ( t ) d = tV (1), when V ( t ) corresponds to an exponentialrandom variable with mean (hence, standard deviation) t . The above corollary there-fore shows that the bid ask spread gets bigger as the variance of the information signalgets higher, indicating that the liquidity suppliers charge a bigger bid-ask spread whenthe informational asymmetry gets bigger.Another consequence of the uniqueness of solutions of (3.11) is that the aggregateexpected profit of the insiders vanishes as N → ∞ . Proposition 3.2.
Suppose that −∞ < m < M < ∞ , there exists a unique solution F N of (3.11) for each N ≥ , and uniqueness holds for the solutions of F ( x ) = 1 x Z x dy Z ∞−∞ q ( σ, y − z ) φ F ( z ) dz. (3.14) NFORMED TRADING AND THE LIMIT ORDER BOOK 13
Assume further that Π + is continuous. Then F ∞ = lim N →∞ F N exists and solves(3.14). Moreover, lim N →∞ π ∗ ( v ) = 0 , where π ∗ is the aggregate expected profit asdefined in (2.7). Definition 3.2. V is said to be symmetric if V and − V have the same distribution.That is, Π + ( y − ) = Π − ( − y ) for all y . Proposition 3.3.
Suppose that V is symmetric and there exists a unique solution F to (3.11). Then F is symmetric, i.e. F ( x ) = − F ( − x ) for all x . Moreover, anysymmetric solution of (3.11) is also a solution of F ( x ) = 1 N Z ∞ q ( σ, x − z ) φ + F ( z ) dz + N − N x Z x dy Z ∞ q ( σ, y − z ) φ + F ( z ) dz, (3.15) where q ( σ, x, z ) := q ( σ, x − z ) − q ( σ, x + z ) .Proof. Observe that − Φ + ( y − ) = E [ − V [ V ≥ y ] ] = E [ − V [ − V ≤− y ] ] = E [ V [ V ≤− y ] ] =Φ − ( − y ). Thus, utilising the fact that Φ + and Π + differ from their left limits at mostat countably many points and q is symmetric around zero, we obtain Z ∞−∞ q ( σ, − x − z ) φ F ( z ) dz = Z −∞ dz q ( σ, x − z ) R ∞−∞ Φ + ( F ( − u )) q ( σ, z − u ) du R ∞−∞ Π + ( F ( − u )) q ( σ, z − u ) du + Z ∞ dz q ( σ, x − z ) R ∞−∞ Φ − ( F ( − u )) q ( σ, z − u ) du R ∞−∞ Π − ( F ( − u )) q ( σ, z − u ) du = − Z −∞ dz q ( σ, x − z ) R ∞−∞ Φ − ( − F ( − u )) q ( σ, z − u ) du R ∞−∞ Π − ( − F ( − u )) q ( σ, z − u ) du − Z ∞ dz q ( σ, x − z ) R ∞−∞ Φ + ( − F ( − u )) q ( σ, z − u ) du R ∞−∞ Π + ( − F ( − u )) q ( σ, z − u ) du = − Z ∞−∞ q ( σ, − x − z ) φ G ( z ) d, where G ( x ) := − F ( − x ). Moreover, − x Z − x dy Z ∞−∞ q ( σ, y − z ) φ F ( z ) dz = 1 x Z x dy Z ∞−∞ q ( σ, − y − z ) φ F ( z ) dz = − x Z x dy Z ∞−∞ q ( σ, y − z ) φ G ( z ) dz. Thus, − G is also a solution of (3.11), which establishes the first assertion. The secondassertion follows from a change of variable in (3.11) as above and using the assumedsymmetry of F . (cid:3) Example 3.1.
Suppose that P ( V = 1) = P ( V = −
1) = . Then, the uniquesymmetric solution of (3.11) is defined by F ( x ) = 1 N Z ∞ q ( σ, x, z ) dz + N − N x Z x dy Z ∞ dzq ( σ, y, z ) , x ≥ . In this case it is easily seen that in equilibrium X ∗ = ∞ (resp. x ∗ = −∞ ) if V = 1 (resp. V = − ). Moreover, h ∗ ( y ) = [ y> − [ y< .Although the insiders’ optimal market order is to buy or sell an infinite amount,their profit remains finite. Indeed, when V = 1 , the aggregate expected profit is givenby Z ∞ E (1 − h ( Z + y )) dy = 2 E (cid:18)Z ∞ [ Z< − y ] dy (cid:19) = 2 Z ∞ P ( Z > y ) dy = 2 E [ Z + ] = E [ | Z | ] = σ r π , where Z + = max { Z, } and the second line is due to the fact that P ( Z > y ) = P ( Z + > y ) . Note that the total profit is independent of N . Example 3.2.
Consider the case P ( V = −
1) = P ( V = 0) = P ( V = 1) = . Then,similar considerations as above should yield F ( ∞ ) = 1 and F ( −∞ ) = − . Moreover,symmetry considerations must lead to F (0) = 0 , which suggests that the insider doesnot trade when V = 0 . Indeed, the unique solution of (3.11) is given by F ( x ) = 1 N Z ∞ q ( σ, x, z ) 11 + P ( Z ≥ z ) dz + N − N x Z x dy Z ∞ dzq ( σ, y, z ) 11 + P ( Z ≥ z ) . Differently from Example 3.1, the order book will not be flat since the insider does nottrade when V = 0 . One can obtain h ∗ via (3.9) and (3.8). Alternatively, for y > h ( y ) = E [ V [ X ∗ ( V )+ Z ≥ y ] ] P ( X ∗ ( V ) + Z ≥ y ) = P ( V = 1) P ( V = 1) + P ( V = 0 , Z ≥ y ) = P ( V = 1) P ( V = 1) + P ( V = 0) P ( Z ≥ y )= 11 + P ( Z ≥ y ) , where the first equality follows from the fact that X ∗ ( V ) is infinite when V = − or , and the third follows from the independence of V and Z . Similarly, for y < , h ( y ) = −
11 + P ( Z ≤ y ) . In particular, the bid-ask spread is given by h (0+) − h (0 − ) = , independent of thenoise volume and of N . NFORMED TRADING AND THE LIMIT ORDER BOOK 15
Existence of Glosten equilibrium.
We shall denote the interior of the sup-port of the random variable V by ( m, M ), where −∞ ≤ m < M ≤ ∞ , and define onthe support of V Ψ ± ( y ) := Φ ± ( y )Π ± ( y ) (3.16)so that Ψ + ( y ) = E [ V | V > y ] and Ψ − ( y ) = E [ V | V ≤ y ].We impose the following condition on the function F to ensure that the integralequation (3.11) is well-defined and changing the order of integration in (3.12) is jus-tified. Assumption 3.1. Z −∞ φ − F ( z ) q ( σ, z ) dz > −∞ . Observe that the above is automatically satisfied if V is bounded from below inview of (A.38). Moreover, if F is a continuous function satisfying (3.11) and V is symmetric, it satisfies the above assumption in view of Proposition 3.3 and thefiniteness of F .One of the useful consequences of the above assumption is the strict monotonicityof solutions of (3.11). Lemma 3.1.
Let F be a continuous non-decreasing solution of (3.11) or (3.14) sat-isfying Assumption 3.1. Then, lim x →∞ F ( x ) = M and lim x →−∞ F ( x ) = m . Conse-quently, F is strictly increasing. Theorem 3.2.
Suppose −∞ < m < M < ∞ . Then, there exists a Glosten equilib-rium. Under the hypotheses of Theorem 3.2 it is clear that m < F ( x ) < M for x ∈ R .However, in view of the bounds given by (A.37) and (A.38) it is possible to obtainsharper bounds on any solution of (3.11). Theorem 3.3.
Suppose −∞ < m < M < ∞ and let Ψ ± be as in (3.16). Then thefollowing statements are valid. (1) There exists a maximal nondecreasing solution to R ( x ) = Z ∞ dz (cid:26) N q ( σ, x − z ) + N − N ¯ q ( σ, x, z ) (cid:27) Z ∞−∞ Ψ + ( R ( y )) q ( σ, z − y ) dy + E [ V ] Z −∞ dz (cid:26) N q ( σ, x − z ) + N − N ¯ q ( σ, x, z ) (cid:27) . (3.17) Moreover, this maximal solution is not constant and any solution of (3.11) isbounded from above by this maximal solution. R is a maximal nondecreasing solution if R ( x ) ≥ r ( x ) for all x ∈ R , where r is any other nonde-creasing solution. (2) There exists a minimal nondecreasing solution to l ( x ) = E [ V ] Z ∞ dz (cid:26) N q ( σ, x − z ) + N − N ¯ q ( σ, x, z ) (cid:27) (3.18)+ Z −∞ dz (cid:26) N q ( σ, x − z ) + N − N ¯ q ( σ, x, z ) Z ∞−∞ Ψ − ( l ( y )) q ( σ, z − y ) dy (cid:27) . Moreover, this minimal solution is not constant and any solution of (3.11) isbounded from below by this minimal solution.
We shall see in the next section that F behaves like R (resp. l ) as x → ∞ (resp. x → −∞ ). 4. Market impact asymptotics
In this section we are interested in the market impact associated with large orders.More precisely, we will be computing the asymptotics of the marginal cost of trades,which is given by the function F . As we shall see later, the asymptotic form of F willcoincide (up to a scaling factor) with that of implementation shortfall IS . We willalso be able to compute the tail asymptotics of the distribution of the total demandin equilibrium. Definition 4.1.
A function g : (0 , ∞ ) → (0 , ∞ ) is said to be regularly varying ofindex ρ at ∞ if lim λ →∞ g ( λx ) g ( λ ) = x ρ , ∀ x > . Analogously, a function g : ( −∞ , → (0 , ∞ ) is said to be regularly varying of index ρ at −∞ if g ( − x ) is regularly varying of index ρ at ∞ . We shall first start with the asymptotics of solutions of (3.17) and (3.18) which willlater allow us to compute the asymptotics of interest.
Theorem 4.1.
Suppose that
N > , −∞ < m < M < ∞ , and Π + has a continuousderivative. Then, the following statements are valid: (1) Define Ψ + x ( M ) := lim x → M ddx Ψ + ( x ) and set G := M − R , where R is anynondecreasing solution of (3.17). Then, G is regularly varying of index ρ + at ∞ , where ρ + = Ψ + x ( M ) − − Ψ + x ( M ) N . (4.19) l is a minimal nondecreasing solution if l ( x ) ≤ L ( x ) for all x ∈ R , where L is any other nondecreasingsolution. NFORMED TRADING AND THE LIMIT ORDER BOOK 17 (2)
Define Ψ − x ( m ) := lim x → m ddx Ψ − ( x ) and set G := l − m , where l is any nonde-creasing solution of (3.18). Then, G is regularly varying of index ρ − at −∞ ,where ρ − = Ψ − x ( m ) − − Ψ − x ( m ) N . (4.20) Remark 2.
Observe that when M is finite, Ψ + x ( M ) = lim x → M M − Ψ + ( x ) M − x ≤ lim x → M M − xM − x = 1 . Similarly, Ψ − x ( m ) ≤ if m is finite. The above shows that R (resp. l ) is slowly varying at ∞ (resp. −∞ ) if Ψ + x ( M ) = 1(resp. Ψ − x ( M ) = 1). We can obtain a better estimate of how slow its variation isunder a further assumption on Ψ ± . Theorem 4.2.
Assume that
N > , −∞ < m < M < ∞ , Π + has a continuous deriv-ative and Ψ + x ( M ) and Ψ − x ( m ) are as in Theorem 4.1. Then, the following statementsare valid: (1) Suppose that Ψ + x ( M ) = 1 and there exist an integer n ≥ and a real constant k ∈ (0 , ∞ ) such that lim x → M Ψ + ( x ) − x ( M − x ) n +1 = 1 k . Then, the following asymptotics hold: M − R ( x ) ∼ (cid:18) NN − nk (cid:19) − n (log x ) − n , x → ∞ , (4.21) where R is any solution of (3.17). (2) Suppose that Ψ − x ( m ) = 1 and there exist an integer n ≥ and a real constant k ∈ (0 , ∞ ) such that lim x → m x − Ψ − ( x )( x − m ) n +1 = 1 k . Then l ( x ) − m ∼ (cid:18) NN − nk (cid:19) − n (log | x | ) − n , x → −∞ , (4.22) where l is any solution of (3.18). Corollary 4.1.
Assume that
N > , −∞ < m < M < ∞ , Π + has a continu-ous derivative and Ψ + x ( M ) and Ψ − x ( m ) are as in Theorem 4.1. Then, the followingstatements are valid: We write f ( x ) ∼ g ( x ), x → ±∞ if lim x →±∞ f ( x ) g ( x ) = 1. (1) Suppose that there exist an integer n ≥ and a real constant k ∈ (0 , ∞ ) suchthat lim x → M Ψ + ( x ) − x ( M − x ) n +1 = 1 k . Then, Π + ( R ) is regularly varying at ∞ of index − Ψ + x ( M )1 − Ψ + x ( M ) N , where R is any solution of (3.17). (2) Suppose that there exist an integer n ≥ and a real constant k ∈ (0 , ∞ ) suchthat lim x → M x − Ψ − ( x )( x − m ) n +1 = 1 k . Then, Π − ( l ) is regularly varying at −∞ of index − Ψ − x ( m )1 − Ψ − x ( m ) N , where l is any solution of (3.18). Remark 3.
Note that if Ψ + x ( M ) = 1 , we have lim x → M Ψ + ( x ) − xM − x = 1 − Ψ + x ( M ) , implying n = 0 and k − = 1 − Ψ + ( M ) in Part 1) of Corollary 4.1. An analogousconsideration applies to the second part. The results above now allow us to compute the asymptotics of solutions of (3.11).
Theorem 4.3.
Assume that
N > , −∞ < m < M < ∞ , Π + has a continuousderivative and Ψ + x ( M ) and Ψ − x ( m ) are as in Theorem 4.1. Let F be any solution of(3.11). Then, the following statements are valid: (1) M − F is regularly varying of index ρ + at ∞ , where ρ + is given by (4.19).Moreover, if Ψ + x ( M ) = 1 and there exist an integer n ≥ and a real constant k ∈ (0 , ∞ ) such that lim x → M Ψ + ( x ) − x ( M − x ) n +1 = 1 k , the following asymptotics hold: M − F ( x ) ∼ (cid:18) NN − nk (cid:19) − n (log x ) − n , x → ∞ . (4.23) NFORMED TRADING AND THE LIMIT ORDER BOOK 19 (2) F − m is regularly varying of index ρ − at −∞ , where ρ − is given by (4.20).Moreover, if Ψ − x ( m ) = 1 , and there exist an integer n ≥ and a realconstant k ∈ (0 , ∞ ) such that lim x → m x − Ψ − ( x )( x − m ) n +1 = 1 k , then F ( x ) − m ∼ (cid:18) NN − nk (cid:19) − n (log | x | ) − n , x → −∞ . (4.24) Remark 4.
Upon integrating by parts we arrive at Ψ + ( x ) − x ( M − x ) n +1 = R Mx Π + ( y ) dy Π + ( x )( M − x ) n +1 . Now suppose that Π + ( x ) = R Mx p ( y ) dy for some differentiable p such that − p ( x ) p ′ ( x ) ∼ k ( M − x ) n +1 as x → M, for some k > and n ≥ . A straightforward application of L’Hospital rule showsthat lim x → M Π + ( x ) p ( x )( M − x ) n +1 = 1 k , which in turn implies lim x →∞ R Mx Π + ( y ) dy Π + ( x )( M − x ) n +1 = 1 k . For instance, if p ( x ) ∝ ( M − x ) β exp (cid:18) − Σ( M − x ) n (cid:19) , for some β ∈ R , Σ > , and n ≥ , we deduce that lim x → M Ψ + ( x ) − x ( M − x ) n +1 = 1 n Σ . We also have the exact analogue of Corollary 4.1 that can be proven by exactly thesame arguments.
Corollary 4.2.
Assume that
N > , −∞ < m < M < ∞ , Π + has a continuousderivative and Ψ + x ( M ) and Ψ − x ( m ) are as in Theorem 4.1. Let F be any solution of(3.11). Then, the following statements are valid: (1) Suppose that there exist an integer n ≥ and a real constant k ∈ (0 , ∞ ) suchthat lim x → M Ψ + ( x ) − x ( M − x ) n +1 = 1 k . Then, Π + ( F ) is regularly varying at ∞ of index − Ψ + x ( M )1 − Ψ + x ( M ) N . (2) Suppose that there exist an integer n ≥ and a real constant k ∈ (0 , ∞ ) suchthat lim x → M x − Ψ − ( x )( x − m ) n +1 = 1 k . Then, Π − ( F ) is regularly varying at −∞ of index − Ψ − x ( m )1 − Ψ − x ( m ) N . Remark 5.
Although Corollary 4.2 appears rather technical, it uncovers the distri-bution of the total volume traded in equilibrium. Indeed, for x > P ( X ∗ > x ) = P ( F − ( V ) > x ) = P ( V > F ( x )) = Π + ( F ( x )) . Thus, under the hypothesis of Corollary 4.2, the tail distribution of equilibrium X ∗ isregularly varying at infinity. That is, P ( X ∗ > x ) = x − ζ + s ( x ) , where s is a slowly varying function and ζ + := Ψ + x ( M )1 − Ψ + x ( M ) N Moreover, since the aggregate order is given by Y ∗ = X ∗ + Z and Z and V areindependent, we have for y > P ( Y ∗ > y ) = Z ∞−∞ dzP ( X ∗ > y − z ) q ( σ, z ) = Z ∞−∞ dzP ( X ∗ > z ) q ( σ, y − z ) , which can easily be shown to be regularly varying at infinity with the same index.Thus, P ( Y ∗ > y ) = y − ζ + s ( y ) , y > , (4.25) for some regularly varying s . In particular, if V has light tails, i.e. Ψ + x ( M ) = 1 , P ( Y ∗ > y ) is regularly varying of index − NN − .Analogous computations yield for the sell side P ( Y ∗ < − y ) = y − ζ − s ( y ) , y > , (4.26) where ζ − := Ψ − x ( m )1 − Ψ − x ( m ) N . We have seen in Proposition 2.2 that the implementation shortfall is smaller than F . In view of Theorem 4.3 we have a more precise relationship for large x . NFORMED TRADING AND THE LIMIT ORDER BOOK 21
Corollary 4.3.
Assume that
N > , −∞ < m < M < ∞ , Π + has a continuousderivative, and let ( h ∗ , X ∗ ) be an equilibrium. Suppose that F ∗ is given by (2.3), with h being replaced by h ∗ . Then, M − IS ∗ ( x ) ∼ NN + ρ + ( M − F ∗ ( x )) , x → ∞ , (4.27) IS ∗ ( x ) − m ∼ NN + ρ − ( F ∗ ( x ) − m ) , x → −∞ , (4.28) where ρ + and ρ − are as in Theorem 4.1. Corllary 4.3 shows that for large N implementation shortfall and the marginal costof trading are almost indistinguishable. This is even more pronounced when M − F (resp. F − m ) is slowly varying, i.e. ρ + = 0 (resp. ρ − = 0). Observe that IS canbe estimated from market data using the open limit order book governed by h , whilethe computation of F depends also on the estimate of N . Remark 6.
We have focused in this section the asymptotics of F and IS . However,one also naturally wonders the asymptotic shape of the limit order book. Using ourframework it is easy to show that F and h behave similarly for large values. Indeed,recalling the measure ν defined in (B.50), we formally obtain lim x →∞ M − h ( x ) M − F ( x ) = lim x →∞ Z ∞−∞ ν ( x, , dy ) M − Ψ + ( F ( xy )) M − F ( x )= Ψ + x ( M )1 ρ + = Ψ + x ( M ) , using the mean value theorem and the continuity of the derivative of Ψ + together withthe fact that the F is regularly varying with index ρ + since the measure ν ( x, , dy ) converges to the point mass at as x → ∞ . In particular, the order book h is alsoregularly varying with the same index ρ + . Numerical studies
This section is devoted to description of results obtained in previous version via anaive numerical search for a fixed point: Starting with an F , we compute F n +1 = T F n until the distance between successive iterations become negligibly small, where T F corresponds to the right side of (3.11). Although the proofs of the statementsconcerning the existence of equilibrium and its asymptotics relied on a boundednessassumption, we shall also presents the solutions of the fixed point problems associatedwith equilibria with unbounded signals.Since σ can be absorbed into the units for measuring equity in view of (3.1), wewill set σ = 1 in all numerical tests with no loss of generality in this section.The iteration converges exponentially for all cases considered below. We illustratethe convergence in Figure 1 by showing the uniform distance d ( g n , g n +1 ) betweensuccessive iterations for log-normal signals with N = 25. −6 −5 −4 −3 −2 d ( g n , g n + ) Uniform convergence: log-normal, N=25
Figure 1.
The uniform distance between successive iterations falls offexponentially with the iteration numberIn practice, the signals distribution is at least partly known to practitioners sinceit can be inferred from the options montage and the event calendar. For example, aninsider may have advance knowledge of the result of clinical trials of a new treatmentfrom a biopharmaceutical company, or the insider may have been tipped about anumber to be released in a scheduled earnings call. The signals distributions in thesetwo examples are quite different. In the case of clinical trial results, the signal is drawnfrom a bounded distribution since the outcome is bounded by total success and totalfailure. On the other hand, earnings surprises can be arbitrarily large. Earnings callscan be priced in a naive jump-diffusion model by assuming the surprise is drawnfrom a log-normal distribution; the jump variance is estimated by comparing theat-the-money implied volatility(ATMIV) for the two nearest options expirations, asproposed by Dubinsky and Johannes [7].The shape of market impact as a function of trade size is important to practitionersto address capacity and position sizing. Various functional forms have been exploredin the literature (see, e.g., [4], [11], [21] and [26]), including √ X and log(1 + aX ).Rejecting either of these forms empirically is challenging due to three practical dif-ficulties: (1) the spread and market impact terms are collinear for small orders, (2)signal-to-noise ratios are weak for executions that take a small percentage of mar-ket volume, and (3) for very large trades, order sizes are often increased if liquidityis available, or reduced if liquidity is hard to find, leading to bias in the data. Inabsence of clear empirical evidence, theoretical predictions for the shape of marketimpact provide valuable insight into how trading costs scale with trade size. Onetestable conclusion from this paper is that market impact should be different ahead NFORMED TRADING AND THE LIMIT ORDER BOOK 23 of an event with an unusual distribution of signals, if market makers believe that aninformed trader may have advance knowledge of the signal.We consider first the case of bounded signals that was the focus of the previoussection.5.1.
Bounded signals.
Truncated Gaussian distribution.
If signals are drawn from the truncated Gaussian distribution with density p ( v ) = erf ( M √ ) √ π Σ e − v for v ∈ [ − M, M ], the numerical solution for equilibrium F convergesto the upper bound as M − F ( x ) ∼ /x N − N in accordance with the predictions ofTheorem 4.3. We show F and the theoretical prediction for its asymptotic behaviorin Figure 2. s u p ( V ) - F ( x ) Convergence to upper bound of signals distribution equilibrium
Figure 2.
The asymptotic behavior of F is shown for the case wheresignals are drawn from a truncated Gaussian distribution5.1.2. Logit-normal distribution.
If price is a probablility-weighted average over two possible outcomes v ± = p ± g as p = e − g , the signals distribution is the logit-normal distribution with density p ( v ) = q π Σ e − ln
2( 1 − v v )2Σ − v . This distribution has support in [ − , F , the order book h and the implementationshortfall for logit-normal signals, in Figure 3. i m p a c t Equilibrium: logit-normal, N=1 h(x)IS(x)F(x)
Figure 3.
Equilibrium impact and shortfall for logit-normal signalsfor the case of an insider ( N = 1)Note that the logit normal distribution does not satisfy the hypothesis of Theorem4.3. Thus, our theory cannot predict the asymptotics of F for this distribution. How-ever, the following formal arguments yield the asymptotics that seem to be verifiedby the numerical experiments. Observe that for x >> σ , 2.3 becomes F ( x ) ≈ N h ( x ) + N − N x Z x h ( u ) du (5.29)It follows that F ( x ) + xF ′ ( x ) ≈ h ( x ) + N xh ′ ( x ). Moreover, for large values of x ,we roughly have h ( x ) ≈ Ψ + ( F ( x )). Let us consider the large N limit and drop the1 /N term. Using the approximation erf c ( x ) ≈ e − x x √ π (1 − x ) and expanding to firstorder in 1 / log( x ) we find that asymptotically xF ′ = Σ(1 − F ) log (1 − F ) , whihc yields F → − e − k √ log( x ) The numerical solution shown in Figure 4 for N = 25 is consistent with this as-ymptotic form.5.2. Unbounded signals.
In what follows, we place emphasis on unbounded signals;that is, when the support of V is unbounded. Although we do not have a theoreticaljustification for the existence of a solution for (3.11) in the case of unbounded signals,we were able to arrive at numerical solutions via the above numerical search. NFORMED TRADING AND THE LIMIT ORDER BOOK 25 M - F ( x ) Convergence to upper bound: logit-normal, N=25 − √ equilibrium Figure 4.
Convergence to the upper bound for M = 1 when signalsare drawn from a logit-normal distribution and shared with N = 25traders.The asymptotic behavior of F ( x ) for large signals will depend on the tail behaviourof the distribution of V . Assuming the interchange of limits and integrals in the proofof Theorem 4.3, we can show that γ ( x ) = Ψ + x ( ∞ ) N γ ( x ) + N − N x Ψ + x ( ∞ ) Z x γ ( y ) dy, x > , where γ ( x ) := lim α →∞ F ( αx ) F ( α ) in case of M = ∞ . Solution of the above equationimmediately yields that, for N > F is regularly varying at ∞ of order ρ + , where ρ + = Ψ + x ( ∞ ) − − Ψ + x ( ∞ ) N . (5.30)Observe that when M = ∞ , Ψ + x ( ∞ ) ≥ + x ( M ) ≤
1. Since ρ + must be non-negative, this places the restriction on N : N ≥ Ψ + x ( ∞ ) (5.31)Thus, we conjecture that (5.31) is a necessary condition for the existence of equilib-rium when M = ∞ . Observe that for a fat tailed unbounded distribution, Ψ + x ( ∞ ) >
1. Thus, a sufficient competition among insiders is necessary for the equilibrium toexist. Such a condition is always satisfied in the bounded case since Ψ + x ( M ) ≤ M < ∞ .As in the bounded case F will be slowly varying at infinity when Ψ + x ( ∞ ) = 1. Inthis case, if we assume lim x →∞ (Ψ + ( x ) − x ) x n − = 1 k (5.32) Table 1.
Distributions with power-law impactDistribution Density ρ + Beta prime x λ − (1 + x ) − ( λ + α ) (cid:0) N − N α − (cid:1) − Fr´echet ( x − β ) − (1+ α ) exp n − (cid:0) x − βs (cid:1) − α o (cid:0) N − N α − (cid:1) − Lomax (cid:0) xλ (cid:1) − ( α +1) (cid:0) N − N α − (cid:1) − Pareto x − ( α +1) (cid:0) N − N α − (cid:1) − Student (cid:16) x α (cid:17) − ( α +1) / (cid:0) N − N α − (cid:1) − In above probability densities are given up to a scaling factor and implicit constraints are enforcedto ensure they are well defined with finite mean. Moreover,
N > αα − in all of the above. Table 2.
Distributions with logarithmic impactDistribution Density AsymptoticsExponential exp( − λx ) Nλ ( N − log x Gaussian exp( − ( x − µ ) / Σ) q NN − √ log x Inverse Gaussian x − / exp (cid:16) − λ ( x − µ ) µ x (cid:17) Nµ λ ( N − log x Normal Inverse Gaussian K ( λζ ( x )) πζ ( x ) exp( δγ + β ( x − µ ) N ( N − λ + β − log x Weibull x d − exp( − λ p x p ) (cid:16) Nλ p ( N − (cid:17) /p (log x ) /p In above probability densities are given up to a scaling factor and implicit constraints are enforcedto ensure they are well defined with finite mean. Moreover, ζ ( x ) := δ + ( x − µ ) for the NormalInverse Gaussian distributiuon. for some k > n ≥
1, formal calculations yield F ( x ) ∼ (cid:18) NN − nk (cid:19) n (log x ) n , x → ∞ . (5.33)Tables 1 and 2 summarise the predicted asymptotics for a class of distributionscommonly used in the literature and practice.5.2.1. Gaussian signals.
We assume that the mean of V equals 0. The numericalsolution is shown in Figure 5 together with the √ log x asymptotic behavior. NFORMED TRADING AND THE LIMIT ORDER BOOK 27 F ( x ) Functional form of the eq ilibri m (ga ssian N=25) √ ) − √ Figure 5.
Functional form of the equilibrium for Gaussian signals, for N = 255.2.2. Log-normal signals.
For a log-normal distribution, the mean is an arbitraryscale factor which we set to 1 and, thus, the density is p ( v ) = √ π Σ v e − ( ln ( v )+Σ / .We choose a large signal variance √ Σ = 10% in our numerical experiments below,illustrative of an earnings announcement for a high-volatility name. Moreover, wetranslate the distribution by 1 so that the mean is 0. The equilibrium solutions for h, F and IS are shown for various values of N in Figure 6.Observe that the equilibrium F is not symmetric around its mean: a market makerwho is short the stock runs the risk of unbounded losses, whereas for a long positionthe maximum loss is always bounded since price cannot fall below zero.The log-normal distribution does not satisfy the conditions of Theorem 4.3. Thus,we do not have a theoretical prediction for the asymptotic market impact. However,we find numerically that the implementation shortfall fits p log(1 + ax δ ). Asymp-totically, both the shortfall and F are consistent with p log( x ). We note that thelarge-size behavior p log ( x ) is more concave than both the square root commonlyused by practitioners and also the log (1 + x ) model that has been suggested in someempirical studies, e.g. [4].The asymmetry between positive and negative signals is clearly visible in the figureas we chose a rather large signal with a standard deviation of 10%.The aggregate profit is a decreasing function of the number of informed investorsand we show the profit as a function of trade size below for various values of N in Figure 8. Moreover, Figure 9 shows how the spread depends on the number ofinformed investors.The greater positive tail mass for the log-normal signals implies that large positivesignals yield a greater profit than for Gaussian signals with the same Σ, and vice-versa,negative signals yield a smaller profit in the log-normal case (Figure 10).5.2.3. Student signals.
We explore the effect of fat tails in the signal distributionnext. We consider the case where signals drawn from a Student t-distribution with α = 3, p ( v ) ∼ v ) . This is reminiscent of some empirical studies such as Plerou[27]. However, we note that we are assuming a Student distribution of arithmetic returns. The power-law tail of geometric returns in Plerou’s study implies an infiniteexpected price.In view of Table 1 the expected asymptotics is F ( x ) ∼ X / ( α − − αN ) . Moreover, ourconjecture predicts no equilibrium when N ≤ αα − = . Indeed, no numerical solutionfor F can be found when N = 1. The equilibrium asymptotically parabolic for N = 2(theoretical ρ + = 2), linear for N = 3 (theoretical ρ + = 1) and concave for N ≥ N = 25, the asymptotic exponentis F ( x ) ∼ x / according to our theory. The numerical solution is compared to thisprediction in Figure 12.The case of power-law tailed signal distributions was considered previously byFarmer et al. in the case of perfect competition between insiders [8]. One can viewtheir model as the limiting case of the one considered herein as N → ∞ in case of aPareto-tailed distribution with exponent 3.5.2.4. Exponential signals.
We are not aware of any situation that gives rise to anexponential distribution for an asset’s price . However, the case is of interest toillustrate the effect of extreme asymmetry. For exponential signals, the probabilitydensity function is given by p ( v ) = λ e − λv for v >
0. We take λ = 1 and translate thedistribution by 1 so that the mean equals 0 for the numerical solutions. The numericsindicate that the equilibrium is asymptotically linear for the case of monopolisticinsider and concave in case of competitive investors, as shown in Figure 13. A possible exception is Kou and Wang [15] who consider an asymmetric double exponential distri-bution for jumps in the asset price
NFORMED TRADING AND THE LIMIT ORDER BOOK 29 −20 −15 −10 −5 0 5 10 15 20−0.4−0.20.00.20.40.60.8 F ( x ) Equilibrium olution (Log-normal ignal )
F(x): N=1F(x): N=2F(x): N=25−20 −15 −10 −5 0 5 10 15 20x−0.4−0.20.00.20.40.60.8 h ( x ) h(x): N=1h(x): N=2h(x): N=25−20 −15 −10 −5 0 5 10 15 20−0.3−0.2−0.10.00.10.20.30.40.5 I S ( x ) IS(x): N=1IS(x): N=2IS(x): N=25
Figure 6.
Equilibrium solutions for log-normal signals for the casesof an insider ( N = 1), and shared signals with N = 2, N = 25 I S ( x ) Equilibrium IS: log-normal, N=25 √ ln(1 + 0.065x ) + 0.01positive branch0.079 √ ln(1 + 0.095x ) − 0.01negative branch Figure 7.
Functional form of the equilibrium for Log-normal signals,for N = 25. The log-normal distribution is not symmetric and this re-sults in a notable difference between the positive and negative branchesfor cost as a function of trade size. −0.2 −0.1 0.0 0.1 0.2v0.000.020.040.060.080.10 π Aggregate profit per share, log-normal signals
N=1N=2N=10
Figure 8.
Aggregate investor profit per share for Log-normal signals
NFORMED TRADING AND THE LIMIT ORDER BOOK 31 A s k - B i d Spread vs N : Log − normal signals, Σ = 0.1 Figure 9.
The spread is shown as a function of the number of informedinvestors, for log-normal signals with √ Σ = 10%. −0.2 −0.1 0.0 0.1 0.2v0.000.010.020.030.040.050.060.070.08 π Profit per hare: log-normal v Gau ian ignal (N=2)
F(x): N=2 log-normalF(x): N=2 Gau ian
Figure 10.
Aggregate profit per share for Gaussian signals vs. Log-normal, for N = 2 −20 −15 −10 −5 0 5 10 15 20−20−1001020 F ( x ) Equilibrium sol tions (St dent signals)
F(x): N=2F(x): N=3F(x): N=25−20 −15 −10 −5 0 5 10 15 20−30−20−100102030 h ( x ) h(x): N=2h(x): N=3h(x): N=25−20 −15 −10 −5 0 5 10 15 20x−7.5−5.0−2.50.02.55.07.5 I S ( x ) IS(x): N=2IS(x): N=3IS(x): N=25
Figure 11.
Equilibrium solutions for Student signals for the cases N = 2, N = 3 and N = 25 NFORMED TRADING AND THE LIMIT ORDER BOOK 33 F ( x ) Functional form of he equilibrium F(x): S uden , N=25 (0.10 + 0.42x) − (0.1) equilibrium
Figure 12.
Functional form of the equilibrium for Student signals for α = 3, N = 25 −20 −15 −10 −5 0 5 10 15 2005101520 F ( x ) Equilibrium sol tions (Exponential signals)
F(x): N=1F(x): N=2F(x): N=25−20 −15 −10 −5 0 5 10 15 20x05101520 h ( x ) h(x): N=1h(x): N=2h(x): N=25−20 −15 −10 −5 0 5 10 15 200246810 I S ( x ) IS(x): N=1IS(x): N=2IS(x): N=25
Figure 13.
Equilibrium solutions for Exponential signals for the cases N = 1, N = 2 and N = 25 NFORMED TRADING AND THE LIMIT ORDER BOOK 35 Same-price liquidation
Up to now, we assumed that the dealer had an initial position Z and liquidatedhis total position X + Z after trading with the insider via the limit order book andargued that a Bertrand competition leads to a price to the insider given by (2.1). Analternative framework also of interest to practitioners is one where portfolio managersand noise traders submit their orders to an aggregator (institutional trading desk),which merges the orders into a block (“metaorder”, in the literature), liquidates X + Z for some average price, and allocates shares with the same average price tonoise traders and insiders. We refer to this framework as same-price liquidation . Theexpected profit of the insiders in this case is E v (cid:2) V X − XX + Z Z X + Z h ( y ) dy (cid:3) . (6.34)In this case the corresponding first order condition for the maximisation problemis given by V = F ( X ∗ ), where F ( x ) = E v (cid:20) xZ + x h ( Z + x ) − ¯ h ( Z + x ) N + ¯ h ( Z + x ) (cid:21) , where ¯ h ( x ) = x R x h ( y ) dy and h is given by the tail expectation as above.The problem with this first order condition is that it is not clear whether it yieldsa maximum as F defined above is not necessarily increasing, i.e. we may not have aconcave function to maximise in contrast to the problem studied in previous section.However, if one assumes that F is increasing and obtains the corresponding in-tegral equation, one can still get a fixed point. Moreover, our numerical solutionsalways suggest an increasing solution yielding a ‘numerical’ proof of the existence ofequilibrium.The solutions are similar in form and share the same asymptotic behaviour forboth bounded and unbounded signal distributions. We show as an example the caseof log-normal signals in Figure 14.Figure 15 compares the same-price liquidation model to the dealer inventory modelin the case of Gaussian signals. The dotted lines represent the same-price liquidationequilibria. Market impact is somewhat greater with same-price liquidation than forthe dealer inventory model, for N >
1. For the insider case, the two are essentiallyidentical. −20 −15 −10 −5 0 5 10 15 20−0.4−0.20.00.20.40.60.8 F ( x ) Same-price e uilibrium (Log-normal signals)
F(x): N=1F(x): N=2F(x): N=25−20 −15 −10 −5 0 5 10 15 20x−0.4−0.20.00.20.40.60.8 h ( x ) h(x): N=1h(x): N=2h(x): N=25−20 −15 −10 −5 0 5 10 15 20−0.3−0.2−0.10.00.10.20.30.40.5 I S ( x ) IS(x): N=1IS(x): N=2IS(x): N=25
Figure 14.
Equilibrium solutions for same-price liquidation with log-normal signals for the cases of an insider ( N = 1), and shared signalswith N = 2, N = 25 NFORMED TRADING AND THE LIMIT ORDER BOOK 37 −20 −15 −10 −5 0 5 10 15 20x−4−3−2−101234 I S ( x ) N=1N=25N=1 s.p.N=25 s.p. −20 −15 −10 −5 0 5 10 15 20x−20−15−10−505101520 h ( x ) N=1N=25N=1 s.p.N=25 s.p.
Same-p ice vs. cash desk liquidation equilib ium: Gaussian signals
Figure 15.
Equilibrium for Gaussian signals, comparing the same-price and cash desk liquidation models ( N = 1 , , Conclusion
In this article, we explored how private information is transferred into the marketprice through a limit order book. Since empirical data on very large trades is sparseand often biased, it is important to develop a theoretical understanding of the pro-cess in order to discriminate between various proposals for the shape of the impactfunction.We proposed an asymmetric information based equilibrium model where informedinvestors draw a signal and send their orders to a dealer with an initial position whoexecutes at the net cost to liquidate the aggregate amount against a limit order book.Unlike the earlier static equilibrium models developed for limit order markets, theinformed traders’ positions are determined endogenously in equilibrium. We showedthat solutions exist in the case of bounded signals and discussed properties of theequilibrium including the asymptotic behavior of the implementation shortfall forlarge trades.Our results provide the micro-foundations for a large number of empirical findingsincluding those on price impact and volume. We found that market impact is asymp-totically a power of trade size if the signal has fat tails, whereas the impact becomesof the form (log x ) /p for some p > α = 3, there is no equilibriumin the case of a monopolistic insider, and the shape of market impact tends to a squareroot in the limit where the number of informed investors is large, N → ∞ . Althoughwe do not have an analytic proof, the bid-ask spread seems to be an increasing andbounded function of the number of informed investors.A relevant and arguably more realistic extension of our framework while still re-maining in a static setting is to consider the scenario in which the insiders receivedifferent but possibly correlated signals regarding the liquidation value. On the otherhand, due to our assumption that the insiders’ orders arrive simultaneously to thedealer, the optimisation problem of each insider requires the solution of a nonlinearfiltering problem even in the case of Gaussian signals. Our present technology is notyet able to deal with such complications and, therefore, we postpone the discussionof this extension to subsequent research.In reality limit order markets are dynamic and thus the order books change overtime reflecting the changes in market parameters. The analytic characterisation of theequilibrium in the current framework in terms of the fixed point of an integral operatormakes us optimistic regarding an extension of the current framework to a dynamicsetting in continuous time. However, continuous trading brings extra flexibilities toportfolio choice - including the option to place a market or limit order at each trade -resulting in a more complicated model. This extension, though extremely interesting,will thus be left for future research. NFORMED TRADING AND THE LIMIT ORDER BOOK 39
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Appendix A. Auxiliary results
Lemma A.1. Ψ + and Ψ − are non-decreasing on the support of V .Proof. Suppose x < y . Note that Ψ + is non-decreasing if E [ V [ V >y ] ] P ( V > x ) − E [ V [ V >x ] ] P ( V > y ) ≥ . Indeed, the left side of the above equals E [( V − y ) [ V >y ] ] P ( V > x ) − E [( V − y ) [ V >x ] ] P ( V > y )= E [( V − y ) [ V >y ] ] ( P ( V > x ) − P ( V > y )) − E [( V − y ) [ x
Let g : R → R be a continuous function and u + (resp. u − ) be theunique solution of u t + σ u xx = 0 , u (1 , x ) = Π + ( g ( z )) ( resp. u (1 , x ) = Π − ( g ( z ))) . (A.35) Then, the following hold: (1)
There exits a solution B on a filtered probability space (Ω , F , ( F t ) , Q ) to thefollowing SDE: dB t = σdW t + σ u x ( t, B t ) u ( t, B t ) dt, B = x, (A.36) where u is either u + or u − and W is a Brownian motion with W = 0 . (2) φ + g ( x ) = E Q + [Ψ + ( g ( B ))] and φ − g ( x ) = E Q − [Ψ − ( g ( B ))] , where ( B, Q + ) (resp. ( B, Q − ) ) corresponds to the solution of (A.36) if u = u + (resp u = u − ) and E Q stands for the expectation under Q . (3) φ + g (0) > φ − g (0) . (4) Suppose further that g is non-decreasing. Then, φ ± g are non-decreasing, too.Consequently, φ g is non-decreasing. Moreover, φ + g ( x ) ≤ E Q + (cid:2) Ψ + ( g ( σW + x )) (cid:3) (A.37) φ − g ( x ) ≥ E Q − (cid:2) Ψ − ( g ( σW + x )) (cid:3) . (A.38) NFORMED TRADING AND THE LIMIT ORDER BOOK 41
Proof.
We shall prove the claims for u + only, the corresponding proof for u − beinganalogous.(1) Note that u + ( t, x ) = Z ∞−∞ Π + ( g ( z )) 1 p πσ (1 − t ) exp (cid:18) − ( y − z ) σ (1 − t ) (cid:19) dz. Then, if β is a Brownian motion on a filtered probability space (Ω , F , ( F t ) , P )with β = 0, u ( t, B t ) is a bounded martingale with u (1 , B ) = Π + ( g ( B ),where B = σβ + x . Thus, we can define a new measure Q on (Ω , F ) by d Q d P = u (1 , B ) u (0 , B ) . By means of Girsanov’s theorem, under Q , B solves (A.36).(2) Observe that φ + g ( x ) = E [Ψ + ( g ( B ))Π + ( g ( B ))] E [Π + ( g ( B ))] = E [Ψ + ( g ( B )) u + (1 , B )] E [ u (1 , B )]= E Q (cid:2) Ψ + ( g ( B )) (cid:3) . (3) The claim is equivalent to Z ∞−∞ Φ + ( g ( z )) q ( σ, z ) dz Z ∞−∞ Π − ( g ( z )) q ( σ, z ) dz − Z ∞−∞ Φ − ( g ( z )) q ( σ, z ) dz Z ∞−∞ Π + ( g ( z )) q ( σ, z ) dz > . Using Φ + = E [ V ] − Φ − and Π + + Π − = 1, the above is valid if and only if0 < Z ∞−∞ Φ + ( g ( z )) q ( σ, z ) dz − E [ V ] Z ∞−∞ Π + ( g ( z )) q ( σ, z ) dz = Z ∞−∞ (cid:0) Ψ + ( g ( z )) − E [ V ] (cid:1) q ( σ, z )Π + ( g ( z )) dz, which holds since for any x we have Ψ + ( x ) ≥ Ψ + ( m ) = E [ V ] in view ofLemma A.1 and Ψ + is not constant.(4) Now, suppose g is non-decreasing, which in turn implies u + x ≤ + isnon-increasing. Therefore, as u x ( t,x ) u ( t,x ) is Lipschitz on [0 , t ] for any t < T , thestandard comparison results for SDEs applied to (A.36) in conjunction withLemma A.1 imply E Q (cid:2) Ψ + ( g ( B )) (cid:12)(cid:12) B = y (cid:3) ≥ E Q (cid:2) Ψ + ( g ( B )) (cid:12)(cid:12) B = x (cid:3) if y ≥ x since we can construct all these solutions indexed by their starting point onthe same probability space due to the local Lipschitz property of u x /u . Thisshows the desired monotonicity φ + ( g ).Similarly, the same comparison principle yields that the solution of (A.36)is bounded from above by σW t + x in case of u = u + since u + x ≤
0. Combinedwith the monotonicity property of Ψ + ( g ), we deduce φ + g ( x ) ≤ E Q [Ψ + ( g ( σW + x ))]. (cid:3) Appendix B. Proofs
Proof of Proposition 2.1. (1) Let g ( x ) := E v [ h ( x + Z )]. By direct differentiation,the expression (2.3) implies xF ( x ) = x g ( x ) N + N − N Z x g ( y ) dy, (B.39)which is equivalent to x g ′ ( x ) N + g = F ( x ) + xF ′ ( x ) . Recall that g (0) = E v [ h ( Z )] = F (0) by construction. Thus, the unique solu-tion of the above ODE with this initial condition is given by g ( x ) = N F ( x ) − N ( N − x N Z x F ( y ) y N − dy = F ( x )+ N ( N − x N Z x ( F ( x ) − F ( y )) y N − dy. This yields (2.5) after a change of variable.(2) The above also yields (2.6) due to the first order condition F ( X ∗ ) = V . Theremaining assertions are direct consequences of the strict monotonicity of F .(3) Finally, note that the total expected profit is given by Z X ∗ ( v − E v [ h ( y + Z )]) dy = Z X ∗ ( v − g ( y )) dy = vX ∗ − NN − (cid:18) X ∗ F ( X ∗ ) − X ∗ g ( X ∗ ) N (cid:19) = − vX ∗ N − X ∗ g ( X ∗ ) N − X ∗ N − g ( X ∗ ) − v )= N Z X ∗ ( v − F ( y )) (cid:16) yX ∗ (cid:17) N − dy, where the second equality follows from (B.39) and the third is due to F ( X ∗ ) = V . (cid:3) Proof of Proposition 2.2.
Note that E [ h ( y + Z )] = E v [ h ( y + Z )] for all y . We shallshow the result for N >
1, the remaining case is similar and easier.
NFORMED TRADING AND THE LIMIT ORDER BOOK 43
Using the first representation in (2.5), we obtain Z x E [ h ( y + Z )] dy = N Z x F ( y ) dy − N ( N − Z x dyy − N Z y dzF ( z ) z N − = N Z x F ( y ) dy − N ( N − Z x dzF ( z ) z N − Z xz dyy − N = N Z x F ( y ) dy + N Z x dzF ( z ) z N − (cid:0) x − N +1 − z − N +1 (cid:1) = N Z x dzF ( z ) (cid:16) zx (cid:17) N − = xN Z dzF ( xz ) z N − , which yields the first assertion once divided by x .The remaining claims follow from F ( xy ) < F ( x ) (resp. F ( xy ) > F ( x )) for x > x <
0) and for all y ∈ (0 ,
1) since F is strictly increasing. (cid:3) Proof of Proposition 3.2.
First observe that φ F is bounded since V is a bounded ran-dom variable by assumption. Thus the dominated convergence theorem in conjunctionwith the continuity of Π + , and thus that of Φ + and Φ − , show that both of lim inf F N and lim sup F N solve (3.14). As (3.14) can have at most one solution, F ∞ := lim F N exists.Also note that F ∞ is strictly increasing and continuous by Lemma 3.1. Thus,lim N →∞ F − N ( v ) = F − ∞ ( v ) ∈ R . Thus,lim N →∞ π ∗ ( v ) = lim N →∞ F − N ( v ) N Z ( v − F N ( F − N ( v ) y ) y N − dy = F − ∞ ( v ) lim N →∞ N Z ( v − F N ( F − N ( v ) y ) y N − dy Since each F N takes values in ( m, M ), lim N →∞ F N ( F − N ( v ) y ) = F ∞ ( F − ∞ ( v ) y ), and themeasure N y N − dy on [0 ,
1] converges weakly to the point mass at 1, we havelim N →∞ N Z ( v − F N ( F − N ( v ) y ) y N − dy = v − F ∞ ( F − ∞ ( v )) = 0 . This completes the proof. (cid:3)
Proof of Lemma 3.1.
We give the proof for the solutions of (3.11), the analogousproperty of the solutions of (3.14) can be proven similarly.Monotone convergence theorem in conjunction with Assumption 3.1 implieslim x →∞ Z ∞−∞ q ( σ, z ) φ F ( x + z ) dz = Z ∞−∞ q ( σ, z ) lim x →∞ φ F ( x + z ) dz. Moreover, lim x →∞ φ F ( x + z ) = Ψ + ( F ( ∞ ) − ). To see this, first note that φ + F ( z + x ) = R ∞−∞ Ψ + ( F ( u ))Π + ( F ( u )) √ πσ exp (cid:16) − ( x + z − u ) σ (cid:17) du R ∞−∞ Π + ( F ( u )) √ πσ exp (cid:16) − ( x + z − u ) σ (cid:17) du . Next, the measure Π + ( F ( u ) q ( σ, z + x − u ) du R ∞−∞ Π + ( F ( u ) q ( σ, z + x − u ) du converges to the point mass at ∞ . Indeed, for any 0 < a < ∞ , we have R a −∞ Π + ( F ( u ) q ( σ, z + x − u ) du R ∞−∞ Π + ( F ( u ) q ( σ, z + x − u ) du = R a −∞ Π + ( F ( u ) exp (cid:0) − u σ + u ( z + x ) σ (cid:1) du R ∞−∞ Π + ( F ( u ) exp (cid:0) − u σ + u ( z + x ) σ (cid:1) du ≤ exp( a ( z + x ) σ ) R a −∞ Π + ( F ( u ) exp( − u σ ) du R ∞ a Π + ( F ( u ) exp (cid:0) − u σ + u ( z + x ) σ (cid:1) du ≤ exp( − a ( z + x ) σ ) R a −∞ Π + ( F ( u ) exp( − u σ ) du R ∞ a Π + ( F ( u ) exp (cid:0) − u σ (cid:1) du , which converges to 0 as x → ∞ .Thus, using the representation of F via (3.12), we deduce F ( ∞ ) = Ψ + ( F ( ∞ ) − ) . Note that changing the order of integration is justified thanks to Assumption 3.1. Onthe other hand, Ψ + ( x − ) > x for any x < M since P ( V > x ) > x < M .This in turn implies F ( ∞ ) = M . Similarly, lim x →−∞ F ( x ) = m . Thus, F is strictlyincreasing in view of (3.12) since m = M and, therefore, F is not constant. (cid:3) Proof of Theorem 3.2.
The proof will be based on an application of Schauder’s fixedpoint theorem to a suitable mapping defined on the space of non-decreasing continuousfunctions, i.e. the candidate functions for the solution of (3.11). Note that since V takes values in [ m, M ], so does Ψ ± . This justifies the representation (3.12) for F .Since in equilibrium h must also be taking values in [ m, M ], we must expect F totake values in [ m, M ], too. Thus, we can concentrate on functions on R that takesvalues in [ m, M ]. Moreover, F will possess a derivative that is bounded by K := ( | m | + M ) Z ∞−∞ | z | e − z σ σ √ π dz (cid:18) N + N − N (cid:19) < ∞ . NFORMED TRADING AND THE LIMIT ORDER BOOK 45
To see this first observe that ∂ ¯ q ( σ, x, z ) ∂x = q ( σ, x − z ) − ¯ q ( σ, x, z ) x = 1 x Z x { q ( σ, x − z ) − q ( σ, y − z ) dy } = 1 x Z x uq x ( σ, u − z ) du. Therefore, (cid:12)(cid:12)(cid:12)(cid:12) ddx F ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z ∞−∞ | φ F ( z ) | ( | q x ( σ, x − z ) | N + N − N x Z | x | u | q x ( σ, u − z ) | du ) dz ≤ ( | m | + M ) N Z ∞−∞ | z − x | e − ( z − x )22 σ σ √ π dz + N − N x Z | x | duu Z ∞−∞ | z − u | e − ( z − u )22 σ σ √ π dz = ( | m | + M ) Z ∞−∞ | z | e − z σ σ √ π dz (cid:18) N + N − N (cid:19) . We shall show the existence of a fixed point in the normed space X := L ( R , µ ), i.e.the space of Borel measurable functions that are square integrable with respect to µ ,where µ ( dx ) = 1 √ π e − x dx. Note that µ is equivalent to the Lebesgue measure on R . Next define D : { g | g : R [ m, M ] is such that | g ( x ) − g ( y ) | ≤ K | x − y | , ∀ x, y ∈ R } and let D := { g ∈ X | g = g , µ -a.e. for some g ∈ D } . It is easy to see that D is a convex subset of X .Next define the operator T on X via T g ( x ) := Z ∞−∞ (cid:26) N q ( σ, x − z ) + N − N ¯ q ( σ, x, z ) (cid:27) φ ¯ g ( z ) dz, where ¯ g := ( g ∨ m ) ∧ M and φ g and ¯ q are as defined in (3.10) and (3.13), respectively.Note that for each x (cid:26) N q ( σ, x − z ) + N − N ¯ q ( σ, x, z ) (cid:27) dz is a probability measure on R . Step 1 ( T maps D into itself ): It is easy to verify that
T g is continuous and takesvalues in [ m, M ] in view of Lemma A.2 and that Z ∞−∞ (cid:26) N q ( σ, x − z ) + N − N ¯ q ( σ, x, z ) (cid:27) dz = 1 . Moreover, T g is differentiable with a derivative bounded by K by using theabove computations that led to the estimate K . Thus, T g ∈ D ⊂ D .Step 2 ( D is compact): Let ( g n ) ⊂ D . Then there exists ( g n ) ⊂ D such that µ -a.e.we have g n = g n for each n ≥
1. Then by Arzela-Ascoli Theorem there existsa subsequence that converges uniformly on compacts to a continuous function g . Without loss of generality let us assume that g n converges to g . Note thatnecessarily | g ( x ) − g ( y ) | ≤ K | x − y | for all x, y ∈ R , i.e g ∈ D . Finally,as g n s are uniformly bounded and µ is a probability measure, the dominatedconvergence theorem yields g n → g in L ( R , µ ).Step 3 ( T : D → D is continuous): Suppose g n → g in D as n → ∞ . In view ofthe definition of D we may assume without loss of generality that that g n sare continuous since changing g n on a Lebesgue null set does not alter thevalue of T g n . By another application of Arzela-Ascoli theorem there exists asubsequence that converges pointwise to some continuous function, which wemay identify with g due to the uniqueness of L -limits up to a null set. Thus,we may assume g is continuous, too.Moreover, the same argument shows that every subsequence of g n has afurther subsequence that converges to g pointwise since continuous functionsthat agree on Lebesgue null sets should agree at every point. Thus, g n → g pointwise as n → ∞ .Next, since φ g n is uniformly bounded, the dominated convergence theoremyieldslim n →∞ T g n ( x ) = Z ∞−∞ dz (cid:26) N q ( σ, x − z ) + N − N ¯ q ( σ, x, z ) (cid:27) lim n →∞ φ g n ( z ) . On the other hand, Lemma A.2 and Girsanov theorem imply for z > φ g n ( z ) = E [Ψ + ( g n ( σβ + z ))Π + ( g n ( σβ + z ))] u n (0 , z ) , where E is the expectation operator for P under which β is a standard Brow-nian motion, and u n is the function u + in Lemma A.2 defined by the terminalcondition Π + ( g n ). Since Ψ + and Π + are continuous except on a Lebesgue nullset, the dominated convergence theorem yieldslim n →∞ E (cid:2) Ψ + ( g n ( σβ + z ))Π + ( g n ( σβ + z )) (cid:3) = E (cid:2) Ψ + ( g ( σβ + z ))Π + ( g ( σβ + z )) (cid:3) . Similarly,lim n →∞ u n (0 , z ) = lim n →∞ E (cid:2) Π + ( g n ( σβ + z )) (cid:3) = E (cid:2) Π + ( g ( σβ + z )) (cid:3) . NFORMED TRADING AND THE LIMIT ORDER BOOK 47
Thus, we have shown lim n →∞ φ g n ( z ) = φ g ( z ) for z >
0. Analogous argumentsyields the convergence for z ≤
0, which in turn establishes the pointwiseconvergence of
T g n to T g ; i.e.lim n →∞ T g n ( x ) = T g ( x ) , x ∈ R . This yields the claim by an application of the dominated convergence theoremsince
T g n s are uniformly bounded.Therefore, T admits a fixed point in D by Schauder’s fixed point theorem. That is,there exists a g ∈ D such that g = T g . Hence, there exists a solution to (3.11). Theclaim now follows from Theorem 3.1. (cid:3)
Proof of Theorem 3.3. (1) Let D be the space of nondecreasing functions on R that take values in [ m, M ] and define an operator T on D via T r ( x ) = Z ∞ dz (cid:26) N q ( σ, x − z ) + N − N ¯ q ( σ, x, z ) (cid:27) Z ∞−∞ Ψ + ( r ( y )) q ( σ, z − y ) dy + E [ V ] Z −∞ dz (cid:26) N q ( σ, x − z ) + N − N ¯ q ( σ, x, z ) (cid:27) . First observe that
T r ≥ T r if r ≤ r since Ψ + is non-decreasing. Thus,setting r ≡ m and r n = T r n − for n ≥
1, we observe that r n is an increasingsequence of nondecreasing continuous functions taking values in [ m, M ]. Thus,the limit, r ∞ exists, is nondecreasing, and continuous by Dini’s theorem. Italso follows from the dominated convergence theorem and that Ψ + has onlycountably many discontinuities that r ∞ is a solution of (3.17).Now let E denote the set of all solutions of (3.17) in D and define R ∗ ( x ) := sup r ∈E r ( x ) . Clearly, R ∗ takes values in [ m, M ]. Due to the aforementioned monotonicity T R ∗ ≥ T r = r for all r ∈ E . Consequently, T R ∗ ≥ R ∗ . Setting g = R ∗ and g n = T g n − for n ≥
1, we again obtain an increasing sequence which convergesto some element r of E . Moreover, r ≥ R ∗ , which in turn implies r = R ∗ bythe construction of R ∗ .Note that if R ∗ is constant, it has to equal M since R ∗ ( ∞ ) = M . However,under this assumption, the right side of (3.17) equals M Z ∞ dz (cid:26) N q ( σ, x − z ) + N − N ¯ q ( σ, x, z ) (cid:27) + E [ V ] Z −∞ dz (cid:26) N q ( σ, x − z ) + N − N ¯ q ( σ, x, z ) (cid:27) < M as E [ V ] < M . Thus, R ∗ cannot be constant. To show the final assertion let F be a solution of (3.11). Since Ψ − ≤ E [ V ]and that φ + F ( z ) ≤ R ∞−∞ Ψ + ( F ( y )) q ( σ, z − y ) dy by (A.37), we deduce that F ( x ) ≤ T F ( x ) . As above, setting g = F and g n = T g n − for n ≥
1, we again obtain anincreasing sequence which converges to a member of E , which proves the claim.(2) Since Ψ − ≤ E [ V ], the proof follows the similar lines as above and, hence,omitted (cid:3) Proof of Theorem 4.1.
We shall give a proof of the first statement as the second onecan be proven along similar lines. We can assume without loss of generality that E [ V ] = 0 since, otherwise, we can replace V by V − E [ V ] and redefine Ψ + and R accordingly.By means of a straightforward change of variable we obtain for x > G ( αx ) G ( α ) = 1 N Z ∞ dzq (cid:16) σα , x − z (cid:17) Z ∞−∞ dyq (cid:16) σα , z − y (cid:17) M − Ψ + ( R ( αy )) M − R ( α )+ N − N Z ∞ dz ¯ q (cid:16) σα , x, z (cid:17) Z ∞−∞ dyq (cid:16) σα , y − z (cid:17) M − Ψ + ( R ( αy )) M − R ( α ) . (B.40)+ MM − R ( α ) Z −∞ dz (cid:26) N q (cid:16) σα , x − z (cid:17) + N − N ¯ q (cid:16) σα , x, z (cid:17)(cid:27) Observe that the measure q ( σα , x − z ) dz converges to the Dirac measure at point x > α → ∞ .Step 1: For y > M − Ψ + ( R ( αy )) M − R ( α ) = ( M − R ( αy ))Ψ + x ( y ∗ ) M − R ( α )for some y ∗ ≥ R ( αy ) by the Mean Value Theorem, where Ψ + x stands for thederivative of Ψ + , since Ψ + ( M ) = M . Thus,lim α →∞ ( M − Ψ + ( R ( αy )) M − R ( α ) = ( M − R ( αy ))Ψ + x ( y ∗ ) M − R ( α ) = Ψ + x ( M ) lim α →∞ G ( αy ) G ( α ) (B.41)as R ( ∞ ) = M .Step 2: Let γ ∗ ( x ) := lim inf α →∞ G ( αx ) G ( α ) . Then, in view of the first step, Fatou’s lemmayields γ ∗ ( x ) ≥ Ψ + x ( M ) N γ ∗ ( x ) + N − N x Ψ + x ( M ) Z x γ ∗ ( y ) dy. Thus, if γ ∗ ( x ) = 0 for some x >
0, it must be 0 for almost all x >
0. However, γ ∗ ( x ) ≥ x ∈ (0 ,
1) since G is decreasing. Thus, γ ∗ > x > γ ∗ is bounded away from 0 on [0 , n ] for any n ≥ NFORMED TRADING AND THE LIMIT ORDER BOOK 49
Step 3: It follows from Step 2 and Corollary 2.0.5 in [6] that x d C ≤ G ( αx ) G ( α ) ≤ Cx c for x ≥ α ≥ α for some constants c, d, C and α . Moreover, since for x < G ( αx ) G ( α ) = (cid:18) G ( αxx − ) G ( αx )) (cid:19) − , and αx > α for large enough α , we deduce that the mapping ( α, x ) G ( αx ) G ( α ) is bounded when x belongs to bounded intervals in (0 , ∞ ).Step 4: Moreover,( M − R ( α )) α ≥ N − N Z α du Z ∞ dzq ( σ, u − z ) Z ∞−∞ dy ( M − Ψ + ( R ( y ))) q ( σ, z − y ) , which in turn implies1 M − R ( α ) ≤ Kα, α >
K < ∞ . (B.42)Since G ( αx ) G ( α ) is bounded when x belongs to bounded intervals in (0 , ∞ ) by Step3, we obtain, for any ε > α →∞ Z ∞−∞ dzq (cid:16) σα , x − z (cid:17) Z z + εz − ε dyq (cid:16) σα , z − y (cid:17) M − Ψ + ( R ( αy )) M − R ( α ) = Ψ + x ( M ) γ ( x ) , (B.43)where γ ( x ) := lim α →∞ G ( αx ) G ( α ) in view of (B.41) provided that the limit exists.Furthermore, in view of (B.42) we also have Z ∞−∞ dzq (cid:16) σα , x − z (cid:17) Z R \ ( z − ε,z + ε ) dyq (cid:16) σα , z − y (cid:17) M − Ψ + ( R ( αy )) M − R ( α ) ≤ KM Z ∞−∞ dzq (cid:16) σα , x − z (cid:17) Z R \ ( z − ε,z + ε ) dyαq (cid:16) σα , z − y (cid:17) → α → ∞ . (B.44)Step 5: Applying the arguments of Step 4 to the second and the third integrals in(B.40) now shows that for x > γ ( x ) = Ψ + x ( M ) N γ ( x ) + N − N x Ψ + x ( M ) Z x γ ( y ) dy. (B.45)In particular, lim α →∞ G ( αx ) G ( α ) exists. Using the initial condition that γ (1) = 1,direct manipulations show that γ ( x ) = x ρ + . (cid:3) Proof of Theorem 4.2.
We shall again only give the proof the first statement andassume without loss of generality that E [ V ] = 0.For any α > r ( α, x ) := R ( αx ) − R ( α )( M − R ( α )) n +1 . Straightforward manipulations similar to the ones employed in the proof of Theorem4.1 leads to r ( α, x ) = − R ( α )( M − R ( α )) n +1 Z −∞ dz (cid:26) N q (cid:16) σα , x − z (cid:17) + N − N ¯ q (cid:16) σα , x, z (cid:17)(cid:27) + Z ∞ dz (cid:26) N q (cid:16) σα , x − z (cid:17) + N − N ¯ q (cid:16) σα , x, z (cid:17)(cid:27) Z ∞ dyq (cid:16) σα , z − y (cid:17) Ψ + ( R ( αy )) − R ( α )( M − R ( α )) n +1 + Z ∞ dz (cid:26) N q (cid:16) σα , x − z (cid:17) + N − N ¯ q (cid:16) σα , x, z (cid:17)(cid:27) Z −∞ dyq (cid:16) σα , z − y (cid:17) Ψ + ( R ( αy )) − R ( α )( M − R ( α )) n +1 ≤ Z ∞ dz (cid:26) N q (cid:16) σα , x − z (cid:17) + N − N ¯ q (cid:16) σα , x, z (cid:17)(cid:27) Z ∞ dyq (cid:16) σα , z − y (cid:17) Ψ + ( R ( αy )) − R ( α )( M − R ( α )) n +1 + Z ∞ dz (cid:26) N q (cid:16) σα , x − z (cid:17) + N − N ¯ q (cid:16) σα , x, z (cid:17)(cid:27) Z −∞ dyq (cid:16) σα , z − y (cid:17) Ψ + ( R ( α )) − R ( α )( M − R ( α )) n +1 ≤ K + Z ∞ dz (cid:26) N q (cid:16) σα , x − z (cid:17) + N − N ¯ q (cid:16) σα , x, z (cid:17)(cid:27) Z ∞ dyq (cid:16) σα , z − y (cid:17) Ψ + ( R ( αy )) − R ( α )( M − R ( α )) n +1 for some K > α , where the first inequality follows from that R ≥ E [ V ] = 0 and R ( αy ) ≤ R ( α ) for y ≤ Ψ + ( R ( αy )) − R ( α )( M − R ( α )) n +1 .Moreover, for y > + ( R ( αy )) − R ( α )( M − R ( α )) n +1 = Ψ + ( R ( αy )) − R ( αy )( M − R ( α )) n +1 + r ( α, y ) ≤ Ψ + ( R ( αy )) − R ( αy )( M − R ( αy )) n +1 + r ( α, y )due to the monotonicity of R . Thus, utilising the boundedness of Ψ + ( R ( αy )) − R ( α )( M − R ( α )) n +1 oncemore, we arrive at r ( α, x ) ≤ κ + Z ∞ dz (cid:26) N q (cid:16) σα , x − z (cid:17) + N − N ¯ q (cid:16) σα , x, z (cid:17)(cid:27) Z ∞ dyq (cid:16) σα , z − y (cid:17) r ( α, y ) . (B.46)for some κ ∈ (0 , ∞ ) that is independent of α .Step 1: Let κ be as above and consider the operator T : C ([1 , ∞ ) , [0 , ∞ )) → C ([1 , ∞ ) , [0 , ∞ ))defined by T f ( x ) = κ + Z ∞ dz (cid:26) N q (cid:16) σα , x − z (cid:17) + N − N ¯ q (cid:16) σα , x, z (cid:17)(cid:27) Z ∞ dyq (cid:16) σα , z − y (cid:17) f ( y ) . NFORMED TRADING AND THE LIMIT ORDER BOOK 51
Clearly T is increasing, i.e. T f ≥ T g if f ≥ g .Step 2: For z ≥ c ∈ R , Z ∞ c dyq (cid:16) σα , z − y (cid:17) y = σ α q (cid:16) σα , z − c (cid:17) + z Z ∞ c − z dyq (cid:16) σα , y (cid:17) ≤ z + σα √ π . (B.47)Thus, if f ( x ) = βx + δ for some β > δ ≥
0, we have
T f ( x ) ≤ δ + κ + √ σβα √ π + βxN + β N − xN Z x ydy = δ + κ + √ σβα √ π + βx ( N + 1)2 N ≤ δ + κ + β √ σα √ π + 3 x ! , where the last inequality is due to the hypothesis that N ≥ β = γκ for some γ > T f ≤ f whenever α ≥ √ σγ √ π ( γ − .Step 3: Define D ( α ) := { f : [1 , ∞ ) → [0 , ∞ ) : f is continuous and f ( x ) ≤ κx + 1( M − R ( α )) n ∀ x ≥ } . and observe that the restriction of r ( α, · ) to [1 , ∞ ) belongs to D ( α ) for all α > T D ( α ) ⊂ D ( α ) for large enough α . Thus, itadmits a fixed point in D ( α ) for large enough α by Tarski’s theorem (Theorem1 in [25]). In fact, it admits a unique fixed point. Indeed, if f and g are twofixed points of T in D ( α ),( f − g )( x ) = Z ∞ dz (cid:26) N q (cid:16) σα , x − z (cid:17) + N − N ¯ q (cid:16) σα , x, z (cid:17)(cid:27) Z ∞ dyq (cid:16) σα , z − y (cid:17) ( f − g )( y ) . Thus,inf x ≥ ( f − g )( x ) ≥ inf x ≥ ( f − g )( x ) inf x ≥ Z ∞ dz (cid:26) N q (cid:16) σα , x − z (cid:17) + N − N ¯ q (cid:16) σα , x, z (cid:17)(cid:27) Z ∞ dyq (cid:16) σα , z − y (cid:17) = c inf x ≥ ( f − g )( x )for some c ∈ (0 , x ≥ ( f − g )( x ) < ∞ , the above implies inf x ≥ ( f − g )( x ) ≤
0. Applying the same argument to g − f , we deduce f = g .Moreover, taking f ∗ ( x ) = 5 κx and utilising the previous step we deducethat the unique fixed point is bounded from above by f ∗ . Theorem 1 in [25]now yields that r ( α, · ) ≤ f ∗ for large enough α in view of (B.46).Step 4: In view of Step 3 we have that r ( α, x ) ≤ κ for all x ∈ [1 ,
2] for large enough α . Thus, Theorem 3.1.5 in [6] yields r ( α, · ) is bounded in bounded subintervalsof (0 , ∞ ) uniformly in α . Step 5: In view of Step 4 and using arguments similar to the ones used in Steps 4 and5 of the proof of Theorem 4.1, we arrive at γ ( x ) = 1 k + γ ( x ) N + N − xN Z x γ ( y ) dy, where γ ( x ) := lim α →∞ r ( α, x ) for x >
0. The unique solution of the aboveequation with γ (1) = 0 is given by γ ( x ) = Nk ( N −
1) log x. Step 6: Consider g ( x ) := exp (cid:0) ( M − R ( x )) − n (cid:1) . Observe that g ( αx ) g ( α ) = exp (cid:18) ( M − R ( α )) n − ( M − R ( αx )) n ( M − R ( α )) n ( M − R ( αx )) n (cid:19) = exp ( R ( αx ) − R ( α )) P n − i =0 ( M − R ( α ) n − − i ( M − R ( αx ) i ( M − R ( α )) n ( M − R ( αx )) n ! = exp R ( αx ) − R ( α )( M − R ( α )) n +1 M − R ( α ) M − R ( αx ) n − X i =0 (cid:18) M − R ( α ) M − R ( αx ) (cid:19) n − − i ! , which converges to exp( nγ ( x )) as α → ∞ since M − R is slowly varying at ∞ . Thus, g ( x ) = x NN − nk s ( x ) , x > . where s is a slowly varying function at ∞ . That is,( M − R ( x )) − n = NN − nk log x + log s ( x ) . Since lim x →∞ log s ( x )log x = 0 (cf. Proposition 1.3.6 (i) and (iii) in [6]), we have M − R ( x ) ∼ (cid:18) NN − nk (cid:19) − n (log x ) − n . x → ∞ . (cid:3) Proof of Corollary 4.1.
Again we only prove the first statement. The hypothesis im-plies Ψ + ( x ) − x is regularly varying with index n + 1 at M . Thus, Ψ + ( R ) − R isregularly varying of index ( n + 1) ρ + at ∞ , which in particular implieslim x →∞ Ψ + ( R ( x )) − R ( x ) R ∞ x Ψ + ( R ( y )) − R ( y ) y dy = − lim x →∞ x (Ψ + x ( R ( x ) R ′ ( x ) − R ′ ( x ))Ψ + ( R ( x )) − R ( x ) = − ( n +1) ρ + , (B.48)in view of Theorem 1.5.11(ii) in [6]. NFORMED TRADING AND THE LIMIT ORDER BOOK 53
Moreover, direct manipulations yieldΠ + ( x ) = R Mx Π + ( y ) dy Ψ + ( x ) − x , which in turn implies − Π + x ( x )Π + ( x ) = Ψ + x ( x )Ψ + ( x ) − x . Therefore, lim x →∞ x Π + x ( R ( x )) R ′ ( x )Π + ( R ( x )) = − lim x →∞ xR ′ ( x )Ψ + x ( R ( x ))Ψ + ( R ( x )) − R ( x )= − ( n + 1) ρ + − lim x →∞ xR ′ ( x )Ψ + ( R ( x )) − R ( x )= − ( n + 1) ρ + − k lim x →∞ xR ′ ( x )( M − R ( x )) n +1 , where the second equality follows from (B.48).Note that if Ψ + x ( M ) < n = 0 and k = − Ψ + ( M ) (see Remark 3). In this case,lim x →∞ xR ′ ( x ) M − R ( x ) = − ρ + by Theorem Theorem 1.5.11(ii ) in [6]. Thus,lim x →∞ x Π + x ( R ( x )) R ′ ( x )Π + ( R ( x ) = ρ + ( k −
1) = − Ψ + x ( M )1 − Ψ + x ( M ) N . An application of Exercise 1.11.13 in [6] to 1 / Π + ( R ) establishes the claim.Now, suppose Ψ + x ( M ) = 1 and observe that n is necessarily bigger than 0 and ρ + = 0 in this case. Recall the function g in Step 6 of the proof of Theorem 4.2 andnote that lim x →∞ xg ′ ( x ) g ( x ) = N n ( N − k by another application of Theorem 1.5.11(i) in [6]. Since xg ′ ( x ) g ( x ) = xnR ′ ( x )( M − R ( x )) n +1 , we arrive at lim x →∞ xR ′ ( x )( M − R ( x )) n +1 = Nk ( N − . Therefore, lim x →∞ x Π + x ( R ( x )) R ′ ( x )Π + ( R ( x ) = − NN − − Ψ + x ( M )1 − Ψ + x ( M ) N , and we again conclude by means of Exercise 1.11.13 in [6]. (cid:3) Proof of Theorem 4.3.
We shall give a proof of the first statement as the second onecan be proven along similar lines.By means of a change of variable employed in earlier proofs we obtain for x > G ( αx ) G ( α ) = 1 N Z ∞ dzq (cid:16) σα , x − z (cid:17) Z ∞−∞ ν ( α, z, dy ) dy M − Ψ + ( F ( αy )) M − F ( α )+ N − N Z ∞ dz ¯ q (cid:16) σα , x, z (cid:17) Z ∞−∞ ν ( α, z, dy ) dy M − Ψ + ( F ( αy )) M − F ( α ) (B.49)+ 1 M − F ( α ) Z −∞ dz (cid:26) N q (cid:16) σα , x − z (cid:17) + N − N ¯ q (cid:16) σα , x, z (cid:17)(cid:27) φ F ( αz ) , where ν ( α, z, dy ) := Π + ( F ( αy ) q (cid:16) σα , y − z (cid:17) ) R ∞−∞ du Π + ( F ( αu ) q (cid:16) σα , u − z (cid:17) ) dy. (B.50)We shall first demonstrate that for z > ν ( α, z, dy ) converges to theDirac measure at point z as α → ∞ . Indeed, let R ∗ be the maximal solution of (3.17)and observe that for any ε > z > ν ( α, z, ( R \ ( z − ε, z + ε ))) = R R \ ( z − ε,z + ε ) dy Π + ( F ( αy ) q (cid:16) σα , y − z (cid:17)R ∞−∞ du Π + ( F ( αu ) q (cid:16) σα , u − z (cid:17) ≤ R R \ ( z − ε,z + ε ) dyq (cid:16) σα , y − z (cid:17)R ∞ du Π + ( R ( αu ) q (cid:16) σα , u − z (cid:17) , where the last inequality is due to the fact that F ≤ R ∗ by Theorem 3.3.Moreover, since Π + ( R ) is regularly varying at ∞ with some index r ≤ + ( R ( αu ) ≥ c (1 + u ) r − δ α r − δ for some c > u ≥ ν ( α, z, ( R \ ( z − ε, z + ε ))) ≤ α δ − r R R \ ( z − ε,z + ε ) dyq (cid:16) σα , y − z (cid:17) c R ∞ du (1 + u ) r − δ q (cid:16) σα , u − z (cid:17) , (B.51)the right side of which converges to 0 as α → ∞ . Similary, we can showlim α →∞ ν ( α, z, ( −∞ , z − ε )) = 0 . Step 1: Let γ ∗ ( x ) := lim inf α →∞ G ( αx ) G ( α ) . It follows from the same argument in Step 2of the proof of Theorem 4.1 that γ ∗ is bounded away from 0 on [0 , n ] for any n ≥
1. This in turn implies the mapping ( α, x ) G ( αx ) G ( α ) is bounded when x belongs to bounded intervals in (0 , ∞ ) as in Step 3 of the same proof.Step 2: Moreover, since F ≤ R ∗ , (B.42) yields1 M − R ( α ) ≤ Kα, α >
K < ∞ . NFORMED TRADING AND THE LIMIT ORDER BOOK 55
Thus, the arguments of Steps 4 and 5 of the proof of Theorem 4.1 are stillapplicable due to (B.51) and that Ψ − F ≤ E [ V ]. Consequently, γ still satisfies(B.45), where γ ( x ) = lim x →∞ G ( αx ) G ( α ) . In particular, γ ( x ) = x ρ + .Step 3: If Ψ + x ( M ), the proof of Theorem 4.2 can be applied verbatim once we showthat f ( α, · ) is bounded in [1 ,
2] uniformly in α , where f ( α, x ) := F ( αx ) − F ( α )( M − F ( α )) n +1 . Since F ( α ) is eventually larger than E [ V ] and φ F ( z ) ≤ E [ V ] for z <
0, wehave for large enough αf ( α, x ) = Z −∞ dz (cid:26) N q (cid:16) σα , x − z (cid:17) + N − N ¯ q (cid:16) σα , x, z (cid:17)(cid:27) φ F ( αz ) − F ( α )( M − F ( α )) n +1 + Z ∞ dz (cid:26) N q (cid:16) σα , x − z (cid:17) + N − N ¯ q (cid:16) σα , x, z (cid:17)(cid:27) Z ∞−∞ ν ( α, z, dy ) Ψ + ( F ( αy )) − F ( α )( M − F ( α )) n +1 ≤ Z ∞ dz (cid:26) N q (cid:16) σα , x − z (cid:17) + N − N ¯ q (cid:16) σα , x, z (cid:17)(cid:27) Z ∞−∞ q (cid:16) σα , z − y (cid:17) Ψ + ( F ( αy )) − F ( α )( M − F ( α )) n +1 , where the last inequality follows from Part (4) of Lemma A.2 since Ψ + ( F ( αy ))is increasing in y for positive α . Thus, f ( α, · ) can be shown to be bounded bythe same function that bounds r ( α, · ) introduces in the proof of Theorem 4.2.Repeating the remaining arguments therein yields the claim. (cid:3) Proof of Corollary 4.3.
It follows from Theorem 3.1 that F ∗ must solve (3.11). Inparticular F ∗ is regularly varying at ∞ of order ρ + .In view of Proposition 2.2 we have M − IS ∗ ( x ) M − F ∗ ( x ) = N Z M − F ∗ ( xy ) M − F ∗ ( x ) y N − dy = N R x ( M − F ∗ ( y )) y N − dyx N ( M − F ∗ ( x )) . Observe that lim y →∞ ( M − F ∗ ( y )) y − ρ + ε = ∞ by Proposition 1.3.6 (v) in [6]. Thus, R ∞ ( M − F ∗ ( y )) y N − dy = ∞ . This justifies the application of the L’Hospital rule toarrive at lim x →∞ M − IS ∗ ( x ) M − F ∗ ( x ) = lim x →∞ N ( M − F ∗ ( x )) x N − N x N − ( M − F ∗ ( x )) − x N F ∗ x ( x )= 11 − lim x →∞ xF ∗ x ( x ) N ( M − F ∗ ( x )) = 11 + ρ + N = NN + ρ + , where the third equality follows from Exercise 1.11.13 applied to 1 / ( M − F ).Asymptotic relationship near −∞ is proved the same way. (cid:3) Department of Statistics, London School of Economics and Political Science, 10Houghton st, London, WC2A 2AE, UK
E-mail address : [email protected] Baruch College, CUNY, NY, USA and AlgoCortex LLC
E-mail address : [email protected] .0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0X−0.0050.0000.0050.0100.015 F ( X ) Functional form of the equilibrium (log-normal, N=25) −0.011 + 0.0155 √ ln(1 + 1.3X)equilibrium.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0x0.000.250.500.751.001.251.501.75 I S ( x ) Functional fo m of IS: Student, N=25 (0.12 + 0.195x) − 0.1225/47