How much market making does a market need?
aa r X i v : . [ q -f i n . M F ] J un How much market making does a market need?
V´ıt Perˇzina ∗ Jan M. Swart † October 12, 2018
Abstract
We consider a simple model for the evolution of a limit order book in which limit ordersof unit size arrive according to independent Poisson processes. The frequencies of buylimit orders below a given price level, respectively sell limit orders above a given level aredescribed by fixed demand and supply functions. Buy (resp. sell) limit orders that arriveabove (resp. below) the current ask (resp. bid) price are converted into market orders.There is no cancellation of limit orders. This model has independently been reinvented byseveral authors, including Stigler in 1964 and Luckock in 2003, who was able to calculatethe equilibrium distribution of the bid and ask prices. We extend the model by introducingmarket makers that simultaneously place both a buy and sell limit order at the currentbid and ask price. We show how the introduction of market makers reduces the spread,which in the original model is unrealistically large. In particular, we are able to calculatethe exact rate at which market makers need to place orders in order to close the spreadcompletely. If this rate is exceeded, we show that the price settles at a random level thatin general does not correspond the Walrasian equilibrium price.
MSC 2010.
Primary: 82C27; Secondary: 60K35, 82C26, 60K25
Keywords.
Continuous double auction, limit order book, Stigler-Luckock model, rank-basedMarkov chain.
Acknowledgments.
Work sponsored by GA ˇCR grant 15-08819S.
Contents ∗ Univerzita Karlova, Matematicko-fyzik´aln´ı fakulta, Ke Karlovu 3, 121 16 Praha 2, Czech Republic; [email protected] † The Czech Academy of Sciences, Institute of Information Theory and Automation, Pod vod´arenskou vˇeˇz´ı4, 182 08 Praha 8, Czech Republic; [email protected] Introduction
We will be interested in a simple mathematical model for the evolution of a limit order book, asused on a stock market or commodity market. The basic model we are interested in has beenindependently (re-)invented at least four times, by [Sti64, Luc03, Pla11, Yud12b]. The aim ofthe model is not so much to identify optimal strategies for traders, but rather to identify, in asimplified set-up, the basic mechanisms that lie behind the observed shape and time evolutionof real order books.Even in regard to this modest aim, the original model as first formulated in [Luc03] isnot particularly succesful. Indeed, it leads to a highly unrealistic order book, in which thespread is very large, while far from the equilibrium price the number of limit orders growswithout bounds. In the present paper we propose an extension of the model that fixes oneunrealistic aspect of the original model, by closing the spread (at least for a special choiceof the parameters), but retains other unrealistic features. Nevertheless, it is hoped that byidentifying the basic mechanisms that lie behind the behavior of simple models, eventually amore realistic model can be developed that leads to a better understanding of the mechanismsthat shape real order books.Since our aim is not to identify trading strategies, we allow traders to behave in a waythat can be far from their optimal strategy, which in a setting where time is continuous andtrading is open ended may anyway be hard to identify. Also, we do not identify individualtraders, i.e., we allow for the possibility that some of the orders arriving at different timesmay in fact be placed by one and the same trader, but do not record this information.Our starting point is the model as first formulated in full generality in [Luc03]. In thismodel, limit orders of unit size arrive according to independent Poisson processes. The fre-quencies of buy limit orders below a given price level, respectively sell limit orders above agiven level are described by fixed demand and supply functions. Buy (resp. sell) limit ordersthat arrive above (resp. below) the current ask (resp. bid) price are converted into marketorders. There is no cancellation of limit orders. Following [Swa18], we add a second typeof traders, who always place market orders, regardless of the current price levels. From amodeling point of view, we can view these orders as buy (resp. sell) that arrive at such a high(resp. low) prices that they are always converted into market orders, except when there arecurrently no matching sell (resp. buy) limit orders in the order book. From a mathematicalpoint of view, the addition of this kind of orders is useful since it allows for positive recurrentbehavior, which is never possible in the original model.The novelty of the present paper kies in the introduction a new type of trader, who is amarket maker or more general any liquidity supplier, who instead of only buying or sellingdoes both, with the aim of making a profit from the spread. We model the effect of suchmarket makers by saying that according to a fixed Poisson rate, a buy and sell limit order ofunit size are simultaneously placed at the current bid and ask prices.In Section 3.2, we adapt the method of Luckock [Luc03] for calculating the spread to thegeneralized model (Theorem 3) and show that the introduction of market makers reducesthe spread, until it closes completely if the rate at which market makers place orders equalsthe Walrasian volume of trade. In Section 3.3 we show that if the rate of market making isincreased beyond this point, then the bid and ask prices converge to a random limit that doesnot need to correspond to the Walrasian equilibrium price (Theorem 4).In the remainder of this introduction, we formulate our model precisely and settle notation(Subsection 1.2) and discuss its history (Subsection 1.3). Section 2 is devoted to the originalmodel due to Stigler and Luckock while Section 3 discusses the new phenomena due to theintroduction of market making. 2 .2 Definition of the model
Let I = ( I − , I + ) ⊂ R be a nonempty open interval, modelling the possible prices of limitorders, and let I = [ I − , I + ] ⊂ [ −∞ , ∞ ] denote its closure. Recall that a counting measure on I is a measure µ that can be written as a countable sum of delta measures. At any giventime, we represent the state of the order book by a pair ( X − , X + ) of counting measures on I ,where we interpret the delta measures that X − (resp. X + ) is composed of as buy (resp. sell)limit orders of unit size at a given price. We assume that:(i) there are no x, y ∈ I such that x ≤ y and X + ( { x } ) > X − ( { y } ) > X − (cid:0) [ x, I + ) (cid:1) < ∞ and X + (cid:0) I − , x ] (cid:1) < ∞ for all x ∈ I .Here, the first condition says that the order book cannot simultaneously contain a buy andsell limit order when the ask price of the seller is lower than or equal to the bid price of thebuyer. The second condition guarantees that M − := max (cid:0) { I − } ∪ { x ∈ I : X − ( { x } ) > } (cid:1) ,M + := min (cid:0) { I + } ∪ { x ∈ I : X + ( { x } ) > } (cid:1) , (1.1)are well-defined, which can be interpreted as the current bid and ask prices. Note that M ± := I ± if the order book contains no limit orders of the given type. It is often convenient torepresent the order book by the signed counting measure X := X + − X − . We let S ord denotethe space of all signed measures of this form, with X − and X + satisfying the conditions (i)and (ii) above.The dynamics of the model are described by two functions λ ± : I → R , which we call the demand function λ − and supply function λ + , and a nonnegative constant ρ ≥
0, which willrepresent the rate of market makers . We assume that:(A1) λ − is nonincreasing, λ + is nondecreasing,(A2) λ ± are continuous functions,(A3) λ + − λ − is strictly increasing,(A4) λ ± > I .We let d λ ± denote the measures on I defined by d λ ± (cid:0) [ x, y ] (cid:1) := λ ± ( y ) − λ ± ( x ) ( x, y ∈ I, x ≤ y ). In particular, d λ − is a negative measure and d λ + is a positive measure. We considera continuous-time Markov process ( X t ) t ≥ that takes values in the space S ord and whosedynamics have the following description. Buy orders inside the interval
With Poisson local rate − d λ − , a trader comes andplaces a buy limit order at a price x , or takes the best available sell limit order at a price ≤ x , if there is one, i.e., X 7→ X − δ x ∧ M + . Buy orders outside the interval
With Poisson rate λ − ( I + ), a trader comes and takesthe best available sell limit order, if there is one, i.e., X 7→ X − δ M + if M + < I + andnothing happens otherwise. Sell orders inside the interval
With Poisson local rate d λ + , a trader comes andplaces a sell limit order at a price x , or takes the best available buy limit order at a price ≥ x , if there is one, i.e., X 7→ X + δ x ∨ M − . Sell orders outside the interval
With Poisson rate λ + ( I − ), a trader comes and takesthe best available buy limit order, if there is one, i.e., X 7→ X + δ M − if M − > I − andnothing happens otherwise. 3 arket makers With Poisson rate ρ , a market maker arrives who places both a buyand sell limit order at the current ask and bid prices, provided these lie inside I , i.e., X 7→ X − { M − >I − } δ M − + 1 { M + λ ± ( I ∓ ) > In the days before electronic trading, market makers on the floor of the exchange wouldmatch buy and sell orders. Even though nowadays, market makers are not formally separatedfrom other traders, they still exist in the form of liquidity suppliers that are distinguished fromother traders by having a different motivation for trading. Rather than being interested inbuying or selling an asset or speculating on the future development of its price, market makersplace both buy and sell orders, at a high volume, with the aim of profiting from the smalldifference between the bid and ask prices. The strategy we have chosen for market makers isextremely simple. Depending on the current state of the order book and the expected behaviorof the other traders, more intelligent choices may be possible. We will see, however, that thepresence of market makers in itself has a huge effect on the shape of the order book. Afterthis is taken into account, their present strategy may prove not to be too unrealistic.From a purely mathematical prespective, the Stigler-Luckock model is similar to a numberof other models that are motivated by other applications. We mention in particular the BakSneppen model [BS93] and its modification by Meester and Sarkar [MS12], a model for canyonformation [Swa17], as well as the queueing models for email communication of Barab´asi [Bar05]and Gabrielli and Caldarelli [CG09]. All these models are “rank-based” in the sense that thedynamics are based on the relative order of the particles and all models contain some ruleof the form “kill the lowest (or highest) particle”. For the model of [CG09], the shape ofthe stationary process near the critical point has been studied in [FS16] and these authorsconjecture that their results also hold for the Stigler-Luckock model. If in the dynamics of the Stigler-Luckock model, one replaces the infinite lifetime of limit orders by anexponential one, then the model becomes positive recurrent for any value of the parameters and the competitivewindow (see Section 2.1) becomes ill-defined. Nevertheless, as long as the cancellation rate is small comparedto the arrival rate of orders, the quasi-stationary behavior of such a model is well approximated by a modelwithout cancellation, and the competitive window can be understood in a limiting sense. I = [0 , λ − ( x ) = 1 − x ,and λ + ( x ) = x . Shown is the state of the order book after the arrival of 10 ,
000 traders(starting from an empty order book).
Consider a Stigler-Luckock model with λ ± ( I ∓ ) = 0 and without market makers (i.e., ρ = 0).Assumptions (A1)–(A4) imply that there exists a unique price x W ∈ I and constant V W > λ − ( x W ) = λ + ( x W ) =: V W . (2.1)Classical economic theory going back to Walras [Wal74] says that in a perfectly liquid marketin equilibrium, a commodity with demand and supply functions λ ± is traded at the price x W and the volume of trade is given by V W . We call x W the Walrasian price and V W the Walrasian volume of trade .Perhaps not surprisingly, in the absence of market makers, Stigler-Luckock models turnout to be highly non-liquid. Indeed, buyers willing to pay a price above the Walrasian price x W and sellers willing to sell for a price below x W may have to wait a considerable timebefore they get their trade, since the bid and ask prices do not settle at x W but instead keepfluctuating in a competitive window ( x − , x + ) which satisfies λ − ( x − ) = λ + ( x + ). As a result, Luckock’s volume of trade V L := λ − ( x − ) = λ + ( x + ) is larger than the Walrasian volume oftrade V W and in fact larger than it could be at any fixed price level.Figure 1 shows the result of a numerical simulation of the uniform model with I = [0 , λ − ( x ) = 1 − x , and λ + ( x ) = x . Depicted is the state of the order book, started from the emptyinitial state, after the arrival of 10,000 traders. This and more precise simulations suggestthat the boundaries of the competitive window are given by x − ≈ .
218 and x + ≈ . x − and sell limit orders at prices above x + stay in the order book forever, while all other orders are eventually matched. As a result,Luckcock’s volume of trade V L ≈ .
782 is considerably higher than the Walrasian volumeof trade V W = 0 .
5. Luckock [Luc03] described a method how to calculate x − , x + , and V L .In particular, for the uniform model, his method predicts that V L = 1 /z with z the uniquesolution of the equation e − z − z + 1 = 0. To explain Luckock’s formula for V L , we need to lookat restricted models. Let ( X t ) t ≥ be a Stigler-Luckock model defined by demand and supply functions λ ± : I → R and rate of market makers ρ ≥
0. Let ( J − , J + ) = J ⊂ I be an open subinterval of I and let λ ′± : J → R be the restrictions of the functions λ ± to J . Let ( X ′ t ) t ≥ be the Stigler-Luckockmodel on J defined by the by the demand and supply functions λ ′± and the rate of market6akers ρ . We call ( X ′ t ) t ≥ the restricted model on J . Its dynamics are the same as for theoriginal model ( X t ) t ≥ , except that limit orders arriving outside J cannot be written into theorder book. Instead, buy limit orders arriving at prices in [ J + , I + ) are converted into buymarket orders while buy limit orders arriving at prices in ( I − , J − ] have no effect. Similar rulesapply to sell limit orders. Note that as long as the bid and ask prices M ± t stay inside J , theevolution of both models inside J is the same, i.e., restricting the measure X t to J yields X ′ t .Consider, in particular, a Stigler-Luckock model with λ ± ( I ∓ ) = 0 and without marketmakers (i.e., ρ = 0). Let λ − − : [0 , λ − ( I − )] → I and λ − : [0 , λ + ( I + )] → I denote the left-continuous inverses of the functions λ − and λ + , respectively, i.e., λ − − ( V ) := sup { x ∈ I : λ − ( x ) ≥ V } and λ − ( V ) := inf { x ∈ I : λ + ( x ) ≥ V } . (2.2)Let V max := λ − ( I − ) ∧ λ + ( I + ) denote the maximal possible volume of trade. To avoid triviali-ties, let us assume that(A5) V W < V max .By the continuity of the demand and supply functions, for each V ∈ ( V W , V max ], setting J ( V ) := ( λ − − ( V ) , λ − ( V )) defines a subinterval J ( V ) ⊂ I such that λ − ( J − ( V )) = V = λ + ( J + ( V )). For later use, we define a continuous, strictly increasing function Φ : [ V L , V max ] → R with Φ(0) = 0 byΦ( V ) := Z VV W n λ + (cid:0) λ − − ( W ) (cid:1) + 1 λ − (cid:0) λ − ( W ) (cid:1) o W d W. (2.3)By definition, a Stigler-Luckock model is positive recurrent if started from an empty orderbook, it returns to the empty state in finite expected time. The following facts have beenproved in [Swa18]. Proposition 1 (Luckock’s volume of trade)
Assume (A1)–(A5), λ ± ( I ∓ ) = 0 and ρ = 0 .Then, for each V ∈ ( V W , V max ) , the restricted Stigler-Luckock model on J ( V ) is positiverecurrent if and only if Φ( V ) < /V . Proof
This follows from Proposition 2, Theorem 3, and formula (1.22) in [Swa18].Proposition 1 suggests that Luckcock’s volume of trade should be given by V L = sup (cid:8) V ∈ [ V W , V max ) : Φ( V ) < /V (cid:9) , (2.4)and that the competitive window is given by ( x − , x + ) = J ( V L ) = (cid:0) λ − − ( V L ) , λ − ( V L ) (cid:1) . Theseformulas agree well with numerical simulations and also agree with the (somewhat more com-plicated) method for calculating V L described in [Luc03]. For the uniform model, one cancheck that one obtains for V L the constant described at the end of the previous subsection.Under certain additional technical assumptions on λ ± , which include the uniform model, ithas been proved in [KY18, Thms 2.1 and 2.2] that the limit inferior and limit superior of thebid and ask prices are a.s. given by the boundaries of the competitive window, as we have justcalculated it.We note that V L > V W always but it is possible that V L = V max . In the latter case, thecompetitive window is the whole interval I . For example, this happens for the model with I = [0 , λ − ( x ) = (1 − x ) α , and λ + ( x ) = x α if 0 < α ≤ /
2. In the next subsection, we willsee that if V L < V max and one assumes that the restricted model on the competitive windowhas an invariant law, then the equilibrium distributions of the bid and ask prices are given bythe unique solutions of a certain differential equation.7 .3 Stationary models By definition, an invariant law for a Stigler-Luckock model is a probability law on S ord so thatthe process started in this initial law is stationary. We let S finord := (cid:8) X ∈ S ord : X − and X + are finite measures (cid:9) (2.5)denote the subspace of S ord consisting of all states in which the order book contains onlyfinitely many orders. If a Stigler-Luckock model is positive recurrent, then it has a uniqueinvariant law that is moreover concentrated on S finord (see [Swa18, Thm 3]). In particular, thisapplies to the restricted model on J ( V ) for any V < V L . If V L < V max , then it is believedthat the restricted model on the competitive window J ( V L ) also has a unique invariant law,but this invariant law is not concentrated on S finord . Instead, in equilibrium, the competitivewindow contains a.s. infinitely many limit orders of each type. In [FS16], a precise conjectureis made about the asymptotics of X − near J − ( V L ) and X + near J + ( V L ) in equilibrium.On a rigorous level, even just proving existence of an invariant law for the restricted modelon J ( V L ) is so far an open problem. However, postulating the existence of such an invariantlaw, Luckock was able to calculate the equilibrium distribution of the bid and as prices. Wecite the following result from [Swa18, Thm 1]. Essentially, this goes back to [Luc03, formulas(20) and (21)], although he only considers the case λ ± ( I ∓ ) = 0. Theorem 2 (Luckock’s differential equation)
Assume that a Stigler-Luckock model withdemand and supply functions satisfying (A1)–(A4) and ρ = 0 has an invariant law. Let ( X t ) t ≥ denote the process started in this invariant law, and let M ± t = M ± ( X t ) denote the bidand ask price at time t ≥ . Define functions f ± : I → R by f − ( x ) := P (cid:2) M − t ≤ x (cid:3) and f + ( x ) := P (cid:2) M + t ≥ x (cid:3) ( x ∈ I ) , (2.6) which by stationarity do not depend on t ≥ . Then f ± are continuous and solve the equations (i) f − d λ + + λ − d f + = 0 , (ii) f + d λ − + λ + d f − = 0 , (iii) f − ( I + ) = 1 = f + ( I − ) , (2.7) where f − d λ + denotes the measure d λ + weighted with the density f − , and the other terms havea similar interpretation. Consider a Stigler-Luckock model satisfying (A1)–(A5), λ ± ( I ∓ ) = 0 and ρ = 0. Let J bea subinterval such that J ⊂ I . Then it has been shown in [Swa18, Prop. 2] that Luckock’sequation (2.7) for the restricted model ( X ′ t ) t ≥ on J has a unique solution ( f − , f + ). ByTheorem 2, if the restricted model on J has an invariant law, then f − ( J − ) = P (cid:2) X ′ − t = 0 (cid:3) and f + ( J + ) = P (cid:2) X ′ + t = 0 (cid:3) (2.8)are the equilibrium probabilities that the restricted model ( X ′ t ) t ≥ contains no buy or sell limitorders, respectively. In particular, if the restricted model on J has an invariant law, then thesequantities must be ≥
0, and if the restricted model is positive recurrent they must be >
0. In[Swa18, Thm 3] it is shown that conversely, if f − ( J − ) ∧ f + ( J + ) >
0, then the restricted modelon J is positive recurrent. For intervals of the form J ( V ) = (cid:0) λ − − ( V ) , λ − ( V ) (cid:1) as in (2.2), itis shown in [Swa18, Prop 2 and formula (1.22)] that • If Φ( V ) < /V , then f − ( λ − − ( V )) > f + ( λ − ( V )) > • If Φ( V ) = 1 /V , then f − ( λ − − ( V )) = 0 = f + ( λ − ( V )).(Here Φ is the function defined in (2.3).) In particular, if V L < V max , then Luckock’s equationhas a unique solution ( f − , f + ) on the competitive window J ( V L ), and this solution satisfies f − ( J − ( V L )) = 0 = f + ( J + ( V L )), which indicates that the bid and ask prices never leave thecompetitive window. 8 0.2 0.4 0.6 0.8 1 ρ = 0 . ρ = 0 . ρ = 0 .
60 0.2 0.4 0.6 0.8 1 ρ = 0 0 0.2 0.4 0.6 0.8 1 ρ = 0 . ρ = 0 . ρ of market makers. Shown is the state of the order book after the arrival of 10 , ρ = 0 . In Figure 2, we show the results of numerical simulations of the “uniform” Stigler-Luckockmodel with I = [0 , λ − ( x ) = 1 − x , and λ + ( x ) = x , for different rates ρ of market makers.We observe that as ρ is increased, the size of the competitive window decreases, until for ρ = ρ c = 0 .
5, it closes completely and the bid and ask prices settle at the Walrasian price x W . If the rate ρ of market makers is increased even more beyond this point, we observe aninteresting phenomenon. In this regime, the bid and ask prices converge to a random limitwhich is different each time we run the simulation, and which in general also differs from theWalrasian price x W . The reason for this is a huge surplus of limit buy and sell orders placedby market makers on the current bid and ask prices, which is capable of “freezing” the priceat a random position.In the coming subsections, we will demonstrate that the critical rate ρ c of market makersfor which the competitive window closes completely is for continuous models given by ρ c = V W ,the Walrasian volume of trade. We will argue that for ρ < V W , the equilibrium distributions ofthe bid and ask prices are still given by the unique solutions of a differential equation, similarto the one for the model with λ ± ( I ∓ ) = 0. For ρ ≥ V W , we will prove that the bid and askprice converge to a common limit and determine the subinterval of possible prices where thislimit can take values. In the present subsection, we show how for 0 < ρ < V W , one can calculate the competitivewindow and the equilibrium distributions of the bid and ask prices by methods similar to thosefor ρ = 0. We first investigate how Luckock’s differential equation changes in the presence ofmarket makers. Theorem 3 (Luckock’s differential equation)
Theorem 2 generalizes to ρ ≥ provided e modify Luckock’s equation (2.7) to (i) f − d λ + + ( λ − − ρ )d f + = 0 , (ii) f + d λ − + ( λ + − ρ )d f − = 0 , (iii) f − ( I + ) = 1 = f + ( I − ) . (3.1) Proof
We first show that f ± are continuous. By symmetry, it suffices to do this for f − . Rightcontinuity is immediate from the continuity of the probability measure P . To prove continuity,it suffices to prove that P [ M − t = x ] = 0 for all x ∈ ( I − , I + ]. This is clear for x = I + . Imaginethat P [ M − = x ] > x ∈ ( I − , I + ). Since X ∈ S ord , there are initially finitely manybuy limit orders in [ x, I + ). By assumption (A4), there is a positive probability that thesebuy limit orders are all removed at some time before time one, while by assumption (A2), theprobability of a new buy limit order arriving at x after such a time is zero. This proves that P [ M − = x ] < P [ M − = x ], contradicting stationarity.To prove (3.1), we observe that by stationarity, for each measurable A ⊂ I that is boundedaway from I − , sell limit orders are added in A at the same rate as they are removed. Thisyields the equation Z A P [ M − t < x ] d λ + (d x ) + ρ Z A P [ M + t ∈ d x ] = Z A λ − ( x ) P [ M + t ∈ d x ] . (3.2)Here, the first term on the left-hand side is the frequency at which sell limit orders are addedat a price x ∈ A while the current bid price is lower than x , the second term on the left-handside is the frequency at which market makers add sell limit orders at the current ask price,and the right-hand side is the frequency at which sell limit orders at the current ask price areremoved because of the arrival of a buy limit order at a lower price or the arrival of a buymarket order. Using also continuity of f − , (3.2) proves (3.1) (i). The proof of (ii) is similarwhile the boundary conditions (iii) follow from the fact that M − t < I + and M + t > I − a.s.Assume (A1)–(A5), fix ρ and define ˜ λ ± := λ ± − ρ . Then d˜ λ ± = d λ ± and hence (3.1) isjust Luckock’s original equation (2.7) with λ ± replaced by ˜ λ ± . In particular, if ρ < V W , then˜ V max := sup (cid:8) V ≥ V W : ˜ λ − ( λ − ( V )) ∧ ˜ λ + ( λ − − ( V )) > (cid:9) (3.3)satisfies V W < ˜ V max , and for each V ∈ ( V W , ˜ V max ), the functions ˜ λ ± are positive on thesubinterval J ( V ) = ( λ − − ( V ) , λ − ( V )). This suggests that for the model with market makers,Luckock’s volume of trade should be given by (2.4) but with V max replaced by ˜ V max and withthe functions λ ± in the definition of Φ in (2.3) replaced by ˜ λ ± .Defining V L by this formula, if V L < ˜ V max , then [Swa18, Prop. 2] tells us that (3.1) hasa unique solution ( f − , f + ) on the competitive window J ( V L ) = ( λ − − ( V L ) , λ − ( V L )), whichshould give the equilibrium distribution of the bid and ask prices. Moreover, since ˜ V max (which depends on ρ ) decreases to V W as ρ ↑ V W , we see that V L ↓ V W and the size of thecompetitive window decreases to zero as ρ ↑ V W . In the previous subsection, we have argued that the competitive window has a positive lengthfor each ρ < V W but its length decreases to zero as ρ ↑ V W . In the present subsection, we lookat the regime ρ ≥ V W . It will be necessary to strengthen assumptions (A1) and (A3) on thedemand and supply functions λ ± , to:(A6) λ − is strictly decreasing on I and λ + is strictly increasing on I .10e have argued in Subsection 1.2 that the assumptions (A1)–(A3) can basically be madewithout loss of generality. Moreover, (A4) and (A5) only exclude trivial cases. Assumption(A6) is restrictive, however. As explained at the end of Subsection 1.2, we can include modelswhere prices assume only discrete values in our analysis by constructing such models as func-tions of other models which satisfy (A1)–(A3). However, as is clear from (1.2), these modelswill not satisfy (A6), so our result Theorem 4 below does not apply to discrete models.For models with λ ± ( I ∓ ) = 0, we generalize our previous definition of the Walrasian volumeof trade V W by setting V W := sup x ∈ I (cid:0) λ − ( x ) ∧ λ + ( x ) (cid:1) . (3.4)Under the assumptions (A2) and (A6), the function λ − ∧ λ + assumes its maximum over I ina unique point x W , which we call the Walrasian price. For models with λ ± ( I ∓ ) = 0, thesedefinitions agree with our earlier definitions. The following theorem describes the behavior ofStigler-Luckcock models with ρ ≥ V W . Theorem 4 (Fixation of the price)
Let ( X t ) t ≥ be a Stigler-Luckock model with demandand supply functions λ ± satisfying (A2), (A4), and (A6), and rate of market makers ρ satis-fying ρ ≥ V W , started in an initial state in S ord . Let M ± t = M ± ( X t ) denote the bid and askprice at time t ≥ . Then there exists a random variable M ∞ such that lim t →∞ M − t = lim t →∞ M + t = M ∞ a . s . (3.5) Moreover, the support of the law of M ∞ is given by { x ∈ I : λ − ( x ) ∨ λ + ( x ) ≤ ρ } . In particular,if ρ = V W , then M ∞ = x W a.s. We prepare for the proof of Theorem 4 with a number of lemmas, some of which are ofindependent interest.
Lemma 5 (Lower bound on freezing probability)
Let ( X t ) t ≥ be a Stigler-Luckock modelon an interval I with demand and supply functions λ ± satisfying (A1)–(A4) and rate of marketmakers ρ ≥ . Assume that initially M − = y where y ∈ I satisfies λ + ( y ) < ρ . Then P (cid:2) M − t ≥ y ∀ t ≥ (cid:3) ≥ − λ + ( y ) ρ . (3.6) Proof
Consider the number X − t ( { y } ) of buy limit orders that are placed exactly at the price y . At times when M − t = y , this quantity goes up by one with rate ρ and down by one withrate λ + ( y ), while at times when M − t > y , this quantity does not change at all. Thus, up tothe first time that X − t ( { y } ) = 0, this process is a random time change of the random walkon Z that jumps up one step with rate ρ and down one step with rate λ + ( y ). If λ + ( y ) < ρ ,then by the well-known gambler’s ruin, this random walk, started in 1, stays positive withprobability 1 − λ + ( y ) /ρ . Lemma 6 (Bound on the competitive window)
Let ( X t ) t ≥ be a Stigler-Luckock modelon an interval I with demand and supply functions λ ± satisfying (A1)–(A4) and rate of marketmakers ρ ≥ . Assume that x, y ∈ I satisfy λ − ( x ) > λ − ( y ) and λ + ( y ) < ρ . Then P (cid:2) lim inf t →∞ M − t < x and lim sup t →∞ M + t > y (cid:3) = 0 . (3.7) By symmetry, the same conclusion can be drawn if λ + ( x ) < λ + ( y ) and λ − ( x ) < ρ . roof If we start the process in an intial state such that M +0 ≥ y , then there is a probability p := λ − ( x ) − λ − ( y ) λ − ( I − ) + λ + ( I + ) + ρ > x, y ). By Lemma 5, there is then a probability of at least q := 1 − λ + ( y ) /ρ > M − t never drops to values ≤ x anymore. Thus, letting σ denotethe first time that a trader arrives at the market, we have that P (cid:2) M − t > x ∀ t ≥ σ | M +0 ≥ y (cid:3) ≥ pq > . (3.9)We claim that this implies (3.7). To see this, set τ := 0 and define inductively σ k := inf { t ≥ τ k : M + t ≥ y } ( k ≥ ,σ ′ k := inf { t > σ k : a trader arrives } ( k ≥ ,τ k := inf { t ≥ σ ′ k − : M − t ≤ x } ( k ≥ , (3.10)where the infimum over the empty set is := ∞ . By the strong Markov property, P [ τ k < ∞ ] ≤ (1 − pq ) k and hence P [ τ k < ∞ ∀ k ≥
0] = 0, which implies (3.7).
Lemma 7 (Freezing)
Let ( X t ) t ≥ be a Stigler-Luckock model with demand and supply func-tions λ ± satisfying (A2), (A4), and (A6), and rate of market makers ρ satisfying ρ ≥ V W .Then there exists a random variable M ∞ such that lim t →∞ M − t = lim t →∞ M + t = M ∞ a . s . (3.11) Proof
If (3.11) does not hold, then there must exist x, y ∈ I with x < y such that P (cid:2) lim inf t →∞ M − t < x and lim sup t →∞ M + t > y (cid:3) > . (3.12)By (A6), making the interval ( x, y ) smaller if necessary we can assume without loss of gen-erality that we are in one of the following two cases: I. λ + ( y ) < ρ , and II λ − ( x ) < ρ . Usingagain (A6), we see that (3.12) contradicts Lemma 6. Lemma 8 (Bound on possible limit values)
Under the assumptions of Lemma 7, therandom variable M ∞ from (3.11) satisfies λ − ( M ∞ ) ∨ λ + ( M ∞ ) ≤ ρ a . s . (3.13) Proof
By symmetry, it suffices to prove that λ + ( M ∞ ) ≤ ρ a.s. Assume the converse. Thenthere exists some z ∈ I with λ + ( z ) > ρ such that P (cid:2) M ∞ ∈ ( z, I + ] (cid:3) >
0. By the continuity of λ − , for each ε >
0, we can cover the compact interval [ z, I + ] with finitely many intervals ofthe form ( x, y ) (if y < I + ) or ( x, y ] (if y = I + ) such that λ − ( x ) − λ − ( y ) ≤ ε . In view of this,we can find x < y and u > λ + ( x ) > ρ + (cid:0) λ − ( x ) − λ − ( y ) (cid:1) and P [ x ≤ M − t ≤ M + t ≤ y ∀ t ≥ u ] > u, ∞ ), the number of buy limit orders in [ x, y ) can only increasewhen a market maker arrives or a buyer places a buy limit order in [ x, y ). On the other hand,the number of buy limit orders in [ x, y ) decreases each time a trader places a sell market orderor a sell limit order at some price in ( I − , x ], which happens at times according to a Poissonprocess with rate λ + ( x ). Since λ + ( x ) > ρ + (cid:0) λ − ( x ) − λ − ( y ) (cid:1) , by the strong law of largenumbers applied to the Poisson processes governing the arrival of different sorts of traders,12e see that a.s. on the event that x ≤ M − t ≤ M + t ≤ y ∀ t ≥ u , there must come a time whenthere are no buy limit orders left in [ x, y ), which is a contradiction. Proof of Theorem 4
Lemmas 7 and 8 show that M ± t converge a.s. to a common limit M ∞ which takes values in the compact interval C := { x ∈ I : λ − ( x ) ∨ λ + ( x ) ≤ ρ } . If ρ = V W ,then by (A6), C consists of the single point C = { x W } . On the other hand, if ρ > V W ,then by (A6), C = [ C − , C + ] is an interval of positive length. To complete the proof, wemust show that in the latter case, for each C − < x < y < C + , the event M ∞ ∈ ( x, y ) haspositive probability. It is not hard to see that for each X ∈ S ord and t >
0, there is a positiveprobability that x < M − t < M + t < y . Thus, it suffices to prove that if x < M − < M +0 < y ,then P [ M ∞ ∈ ( x, y )] >
0. This is similar to Lemma 5, but we use a slightly different argument.Note that by (A6), λ − ( x ) < ρ and λ + ( y ) < ρ . As long as x ≤ M − t ≤ M + t ≤ y , the number X − t (cid:0) [ x, y ] (cid:1) of buy limit orders in [ x, y ] goes up by one with rate at least ρ and decreases byone with rate at most λ + ( y ). A similar statement holds for the number of sell limit orders in( x, y ). Let ( N − t , N + t ) t ≥ be a Markov process in Z that jumps with rates( n − , n + ) ( n − + 1 , n + ) at rate ρ, ( n − , n + ) ( n − − , n + ) at rate λ + ( y ) , ( n − , n + ) ( n − , n + + 1) at rate ρ, ( n − , n + ) ( n − , n + −
1) at rate λ − ( x ) . (3.14)Then ( N − t ) t ≥ and ( N + t ) t ≥ are independent random walks with positive drift, and hence bythe strong law of large numbers, if N − > N +0 >
0, then P [ N − t > N + t > ∀ t ≥ > . (3.15)The claim now follows from a simple coupling argument, comparing X ± t (cid:0) [ x, y ] (cid:1) with N ± t . The Stigler-Luckock model is one of the most basic and natural models for traders inter-acting through a limit order book, so natural, in fact, that it has been at least four timesindependently (re-)invented [Sti64, Luc03, Pla11, Yud12b]. Although it is based on natu-ral assumptions, its behavior is unrealistic since the bid and ask prices do not settle at theWalrasian equilibrium price but rather keep fluctuating inside an interval of positive lengthcalled the competitive window. This provides an opportunity for market makers or liquiditysuppliers who make money from buying at a low price and selling at a higher price.In this paper, we have added such market makers to the model who trade using a verysimple strategy, namely, by placing one buy and sell limit order at the current bid and askprices. We have seen that the addition of market makers makes the model more realistic in thesense that the size of the competitive window decreases. In particular, for continuous models,if the rate at which market makers place orders equals the Walrasian volume of trade, thenthe size of the competitive window decreases to zero and the bid and ask prices converge tothe Walrasian equilibrium price. If the rate of market makers is even higher, then the bidand ask prices also converge to a common limit, but now the limit price is random and ingeneral differs from the Walrasian equilibrium price. Moreover, in this regime, some of thelimit orders placed by market makers are never matched by market orders but stay in theorder book forever (on the time scale we are interested in).In reality, market makers make profit only if their limit orders are matched, and this profitis proportional to the size of the competitive window. Therefore, in real markets, there is nomotivation for market makers to trade once the size of the competitive window has shrunk tozero. In view of this, in reality, we can expect a self-regulating mechanism that makes surethat in the long run, the rate at which market makers place orders is approximately equalto the Walrasian volume of trade. The effect of this is that in the limit, all trade involves13arket makers, i.e., the buyers and sellers of the original Stigler-Luckock model do not directlyinteract with each other but make all their trade with the market makers.We conclude from this that adding market makers to the Stiger-Luckock model has pro-duced a more realistic model, especially if the rate of market makers is chosen equal to theWalrasian volume of trade. Future, better models should include a self-regulating mechanismthat links the rate at which market makers place orders to the present state of the order bookby weighing their expected profit against the costs and risks. Realistic models should alsoconsider prices that can take only discrete values since in reality the size of the competitivewindow and hence the potential for profit for market makers are bounded from below by thetick size.
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