AA Model of Market Making and Price Impact
Angad Singh ∗ January 6, 2021
Abstract
Traders constantly consider the price impact associated with changing their positions. This paper seeksto understand how price impact emerges from the quoting strategies of market makers. To this end, marketmaking is modeled as a dynamic auction using the mathematical framework of Stochastic DifferentialGames. In Nash Equilibrium, the market makers’ quoting strategies generate a price impact function thatis of the same form as the celebrated Almgren-Chriss model. The key insight is that price impact is themechanism through which market makers earn profits while matching their books. As such, price impactis an essential feature of markets where flow is intermediated by market makers.
Execution costs are an important issue to consider when implementing trading strategies. In electronic marketstraders typically use sophisticated algorithms to sequentially place small orders that minimize trading costs.Many trading costs come in the form of market impact, which can be understood by looking at the followingstylized snapshot of a limit order book (LOB).Figure 1: A stylized limit order book [CJP15]In Figure 1, the blue and red orders are outstanding limit orders, which correspond to outstanding quotesto trade by market participants. The ask side consists of quotes to sell, and the bid side consists of quotes ∗ Department of Mathematics, California Institute of Technology. Email: [email protected] a r X i v : . [ q -f i n . M F ] J a n o buy. The green order is a market order, which in this case is to sell. A market order specifies a quantityto buy or sell and is executed against the best outstanding quotes. In general, limit order are thought of asproviding liquidity, as they make it possible for others to trade. They are usually placed by market makers.Market orders are thought of as taking liquidity, as their placement requires someone else to also trade.The market order in this figure suffers from two important forms of market impact. Firstly, because theorder is somewhat large, it cannot all be executed at the best bid price (at the touch), and so the average pricereceived per share in the order is strictly smaller than $ q t over a time interval [0 , T ]. A positive(negative) trading rate corresponds to selling (buying) q t dt shares in the time instant [ t, t + dt ]. The trader’stransaction prices are then given by p t = D t − Γ (cid:90) t q s ds − Λ q t . Here Γ > , t ). Λ > { D t } can be thought of as thefundamental value, or unaffected midprice. It is what the midprice would have been in the absence of thetrader’s selling, and it is often modeled as an arithmetic Brownian Motion.This paper seeks to understand how a price impact function such as the one above would emerge fromthe quoting strategies of market makers, as they are the ones placing the orders against which the trader aboveis executing. The paper takes as given an exogenous stream of market orders hitting the limit order book, andthen consider an auction between market makers to clear this flow. Quoting between the market makers ismodeled as submitting demand functions to trade, i.e. announcing a set of acceptable price quantity pairs.The paper presents a static one period model (Section 2) as well as a dynamic continuous time model(Section 3). The one period model transparently highlights the tradeoffs faced by the market makers, as wellas the mechanism linking price impact to their profits. The dynamic model allows for a more realistic modelwhere liquidity traders’ flow mean reverts in the long run. This means that market makers are able to matchtheir books over time, and on average have flat exposures. In the short run, however, they have one-sidedexposures, and thus are exposed to fluctuations in fundamental value. Price impact is the mechanism throughwhich market makers ensure that they don’t lose money while taking on these short term exposures.2ll in all, the models in this paper deliver the following three insights regarding market making andprice impact. Firstly, in order to be profitable, a market maker should make sure that the following heuristicis true for her trades: sign ( trade )( P trade − P after ) < . When a market maker buys, sign ( trade ) >
0, she should make sure to do so at a price slightly lower thanwhat (she rationally thinks) the midprice will be right after the trade, and vice versa. The reason is that themarket maker is hoping to make the opposite trade and sell right after, and in order for this to be profitablethe inequality above must hold (on average). This is discussed in more detail when presenting equation (28)below.Secondly, in order to implement a quoting strategy that achieves the inequality above, the market makershould proceed as follows. At any given moment, the market maker can look at her outstanding quotes andcompute the function P ( q ), which is the trading price she will get if she ends up trading a quantity of q . P(q)should reflect expectations of future prices conditional on the fact that a trade of size q is occurring, and thisshould hold for every q . This is necessary to make sure the market maker makes money (on average) regardlessof which of her quotes end up executing today. A quoting strategy achieving this can be implemented only iforders have market impact, i.e. if transaction prices depends on order sizes. This is discussed in more detailwhen presenting Proposition 5.4 below.Finally, given this theoretical foundation of price impact from the perspective of market makers, we canprovide insight on two common execution cost measures used in practice. The first is effective cost, measuredheuristically when selling by P before − P trade . The second is realized cost, measured heuristically when selling by P after − P trade . The first measure looks at the entire price impact of the trade, and the second measure looks only at thesubsequent price reversal. In general the first measure is larger due to the long lasting impact of the currenttrade on the price [Ped15]. The theory in this paper supports this, as in the model the second measure differsfrom the first because of market makers expectations of when they will eventually be able to match theirbook. Furthermore, the theory suggests that the measures also differ because market making is not a perfectlycompetitive activity. It’s certainly true that there are barriers to entry (e.g. technological and regulatory) inthe market making industry, and it would be an interesting empirical exercise to the test the hypothesis thatthis drives the wedge between the two measures.
The contents of this paper relate to three strands of existing academic literature. The first is the theoreticaleconomics literature on market microstructure. The second is the mathematical finance literature on optimal3xecution. The third is the finance literature on asset pricing.The literature on market microstructure is fairly classical and considers various one-period equilibriumauction models. Thorough reviews are given in [OHa11] and [FPR13]. The demand schedule auction used inthis thesis was first introduced in [Wil79]. [Kyl89] uses the demand schedule auction to study price impact asa consequence of adverse selection. The contents of section 2 are similar to these papers, though the specifictheorems proven are new. One major way in which section 2 (and the rest of the paper) deviates from [Kyl89]is that price impact is not a consequence of adverse selection. Instead, price impact compensates marketmakers for absorbing short term imbalances. This is in line with the idea of liquidity as the price of immediacy,as introduced in [GM88].The literature on optimal execution considers dynamic partial equilibrium problems for an individualagent trading against a price impact function. This literature was pioneered in [AC01] and has since beenextended in a variety of directions, as reviewed in [CJP15] and [Gu´e16]. The optimal response problems forthe model in section 3 is very much analogous to the optimization problems in this literature. However, thisliterature always takes the market price impact function as exogenously given, whereas in this paper the priceimpact function is endogenous.Asset pricing is the standard framework used in the academic finance community to model market prices.Thorough introductions to the subject are given in [Coc05] and [Bac17]. This literature does not explicitlymodel auctions, as the market microstructure literature does, but instead uses a general equilibrium approach.Furthermore, trading costs are usually ignored in these models. [GP16] and [Bou+18] do consider trading costsin asset pricing settings, but these costs are exogenously specified. Also, the trading costs in these modelsare not in the form of market price impact, but instead are transaction costs paid on top of market prices.Many of the modeling techniques in this paper are inspired by these papers, and also by the continuous timeCARA-Normal frameworks used in [CK93] and [Bar+15]
This section formulates and discusses some one period models of auctions for shares . The type of auctionconsidered is typically referred to as a conditional uniform price auction, or as the demand schedule game. Atthe beginning of the auction, each agent submits a demand schedule D : R → R , which is a commitment topurchase D ( p ) shares if the price per share is p . The auctioneer then observes all the schedules, aggregatesthem into a total demand schedule, and chooses a price such that the auction clears. Agents receive sharesbased on their individual demand schedules, and finally they derive utility from the payoff associated to eachshare.This type of auction is interesting because it’s a good way to model agents trading through a limitorder book. At a very high level, the decision to place market and limit orders amounts to deciding how For our purposes a share is simply a unit of an infinitely divisible good.
There are N ≥ S outstanding shares. Each share will, after the auction,provide a random payoff of ˜ µ ∼ N ( µ, σ ). Market makers are assumed to have CARA preferences over payoffs,with risk aversion parameter γ >
0. Thus if a market maker purchases q shares in the auction for a priceof p per share, her expected utility is E [ − e − γq (˜ µ − p ) ]. We will assume that market makers can only submitaffine and decreasing demand schedules . So a demand schedule is characterized by a pair ( a, b ) ∈ R × (0 , ∞ ),corresponding to the commitment to purchase a − bp shares when the price per share is p . In addition to the N market makers there are liquidity traders who have a random perfectly inelasticdemand for − ˜ u shares. Thus the total number of shares the N market makers must clear in the auction is The affine assumption is not strictly necessary, as an optimal response to a profile of affine schedules is also given by an affineschedule. b = ∞ corresponds to placing an order that specifies a price but no quantity. Such an order is not possible on an exchange,and thus is not allowed here. Technically we should allow b = 0, since it corresponds to placing a market order. However, we onlyconsider symmetric equilibria below, and b = 0 can trivially never be such an equilibrium. Thus for the sake of exposition b = 0 isomitted from the outset. + ˜ u . By adjusting the mean of ˜ u we can simply assume that the total number of shares being auctioned israndom and equal to ˜ u . We assume that ˜ u is independent of ˜ µ , but make no other assumptions about itsdistribution.Given demand schedules ( a , b ) , · · · , ( a N , b N ), the auction price p and the shares q , · · · , q N bought byeach market maker are implicitly given by a n − b n p = q n (1) q + · · · + q N = ˜ u. (2)This describes a game in normal form, and we now proceed to study its Nash equilibria. We will focus only onsymmetric equilibria, that is on equilibria where all market makers submit the same demand schedule. Notethat in a symmetric equilibrium all market makers purchase the same number of shares in the auction, i.e. q = · · · = q N = ˜ uN . Theorem 2.1.
Fix exogenous parameters N ≥ , γ, σ > and µ ∈ R .If ˜ u is degenerate, i.e. ˜ u = u ∈ R a.s., then there is a one-to-one correspondence between symmetric equilibriaand λ > . In equilibrium the price is p = µ − γσ N u − λN u. (3) If ˜ u is non-degenerate and N ≥ , then there is a unique symmetric equilibrium with price p = µ − γσ N ˜ u − γσ N ( N −
2) ˜ u. (4) If ˜ u is non-degenerate and N = 2 then a symmetric equilibrium does not exist.Proof. Fix a strategy ( a, b ) ∈ R × (0 , ∞ ) to be played by all but one agent, and consider the optimal responseproblem faced by the remaining agent. If this agent plays strategy ( α, β ) ∈ R × (0 , ∞ ), then her expectedutility will be E [ − e − γq (˜ µ − p ) ] with p and q given implicitly by q = α − βp (5)˜ u − qN − a − bp. (6)Rearranging we obtain p = F − λ ˜ u + λq (7) q = (cid:20) α λβ − β λβ F (cid:21) + λβ λβ ˜ u, (8)where F := ab and λ := b ( N − .We note a few things about these equations. Firstly, the parameters F and λ characterize the symmetricprofile given by ( a, b ) and are independent of the remaining agent’s demand schedule. Secondly, equation(7) tells us that the price the remaining agent receives is uniquely determined by the number of shares shereceives. Thus the agent is indifferent between schedules that lead to the same quantity of shares. Thirdly,6rom equation (8) we see that choosing ( α, β ) ∈ R × (0 , ∞ ) amounts to choosing to receive the quantity q ∈ V where V := { A + B ˜ u : ( A, B ) ∈ R × (0 , } . (9)Fourthly, if the agent submits the schedule ( a, b ) then she receives the quantity q = ˜ uN .All this goes to show that in forming an optimal response, the remaining agent can maximize directlyover q ∈ V , with the price given by (7), and in equilibrium the maximum must be attained at ˜ uN . Thussymmetric equilibria correspond to F ∈ R and λ > uN ∈ arg max q ∈ V E (cid:20) − e − γq (cid:0) ˜ µ − F + λ ˜ u − λq (cid:1)(cid:21) , (10)and in equilibrium the price is p = F − N − N λ ˜ u. (11)Now, note that ˜ u is independent of ˜ µ and every q ∈ V is ˜ u measurable. Since the MGF of ˜ µ is E [ e t ˜ µ ] = e µt + σ t , we can compute the expectation in (10) for any q ∈ V as E (cid:34) − e − γq (cid:16) µ − F + λ ˜ u − ( λ + γσ ) q (cid:17)(cid:35) . (12)For any F ∈ R and λ >
0, the function of q in (12) is strictly concave over the convex set V . Thus theargmax in (10) consists of at most one point. Furthermore, for any F ∈ R , λ > u ∈ R , the much relaxedproblem of maximizing the exponent in (12)max q ∈ R q ( µ − F + λu − (cid:16) λ + γσ (cid:17) q )has unique solution ˆ q = µ − F λ + γσ + λ λ + γσ u .It follows that (10) holds if and only if˜ uN = µ − F λ + γσ + λ λ + γσ ˜ u. (13)Thus symmetric equilibria correspond to F ∈ R and λ > u is degenerate and equal to u ∈ R a.s., then (13) simply reads F = µ − γσ N u + N − N λu. (14)Thus there a one-to-one correspondence between symmetric equilibria and λ >
0, with F given by (14). Thefirst statement in the theorem follows by using (14) to substitute for F in (11).Next suppose that ˜ u is non-degenerate. Then (14) holds if and only if F = µ and λ > N − λ = γσ . (15)If N = 2 then no λ > N ≥ λ > λ = γσ N − , and so there is a unique symmetricequilibrium. The second statement in the theorem now follows by plugging in F = µ and λ = γσ N − in (11).7 .1.1 Discussion The idea in the proof is to characterize symmetric profiles in terms of the parameters F and λ of the inducedthe optimal response problem. The optimal response problem is to choose an expected utility maximizingquantity on the linear supply curve (7), which has intercept F − λ ˜ u and slope λ >
0. This is a concavemaximization problem with a unique solution. The symmetric equilibrium condition on F and λ is thatthis solution is ˜ uN . In the non-degenerate case this condition uniquely determines F and λ whereas in thedegenerate case it only specifies F as a function of λ .What happens in the degenerate case is that the market makers are able to form an agreement tomisprice the asset and then take equal shares of the profit. Consider for example the case when ˜ u = 1, so aunit share is being cleared at the auction. Then (3) states that the equilibrium price can take on any valuebelow µ − γσ N , which is the price the asset would trade at in a competitive equilibrium. Thus we see that theasset is being priced relatively low, and since each market maker takes N shares, they split the profits equally.Since the game is non-cooperative, in order to form an agreement the market makers must have a wayto prevent others from taking more than an equal share of the profits. The key point is that all market makerssubmit entire demand schedules, which specify what the price must be contingent on the quantity the marketmaker receives. So if one market maker were to take more than N shares, some other market makers wouldreceive less than N shares, and this would cause the price to move, thus dissuading any one market makerfrom trying to take extra shares in the first place.The amount by which the price would move if a market maker took more then N shares is governed bythe parameter λ , which corresponds to the quantity elasticity b of the equilibrium demand schedule. The lowerthe equilibrium price p , the more of an incentive a market maker has to acquire more than N shares, and thusthe higher λ needs to be to prevent the market maker from doing so. (3) says exactly that low equilibriumprices correspond to high values of λ .The problem in the degenerate case is that market makers suffer no cost from being quantity elastic,since there will be no surprise trades in equilibrium. Hence λ can take on any positive value in equilibrium.In the non-degenerate case, market makers suffer costs from being quantity elastic in equilibrium, becausethere is uncertainty in the quantity to be cleared. These costs manifest in how the parameter λ effects theuncertainty of equilibrium prices. Since market makers care about the uncertainty of prices, this pins downthe unique equilibrium value of λ as γσ N − .The coefficient of ˜ u in (4) is N − N ( N − γσ , which is the price impact of the liquidity traders’ order. If theliquidity traders sell (cid:15) more shares, so the realization of ˜ u is (cid:15) higher, then the equilibrium price is N − N ( N − γσ (cid:15) lower. Thus orders walk the book: the larger an order, the lower the transaction price.The decomposition of price impact into the two terms is motivated by considering the competitive limitas N → ∞ and γN is held fixed. In the limit the second term vanishes and only the first remains. Thus µ − γσ N ˜ u is the competitive benchmark, and the second term is the deviation due to imperfect competition.As in the competitive case, the term γσ uN is the risk compensation each market maker requires to take the8quilibrium exposure of ˜ uN .The interpretation is that price impact arises for two reasons in the model. Firstly to make sure marketmakers are appropriately compensated for bearing risk, and secondly because market makers have marketpower. The first reason persists even in the competitive limit, and as a result price impact does not vanish inthe limit. This will be a recurring theme throughout the paper. The continuous time model considered in the next section essentially consists of the auction above at eachinstant, with the addition of certain state variables that need to be carried from instant to instant. The statevariables are the existing inventories of shares that market makers have accumulated from trading in the past.This section introduces these state variables in a static setting as types, thus generalizing the model above toa Bayesian game.In addition to forming a tighter connection with the continuous time model to follow, the rephrasedmodel in this section has two other appealing features. Firstly, in the previous subsection the total number ofoutstanding shares played no distinct role from the liquidity traders’ order. This is perhaps counterintuitive,as the liquidity traders’ order should have price impact, whereas the total number of outstanding shares shouldbe a fixed component of the price. In this section the total number of outstanding shares will show up as afixed component of the price. Secondly, in the previous section all market makers purchased the same numberof shares in equilibrium. In this section their purchases will be heterogenous.The model is exactly as before except that each of the N market makers starts out with an existinginventory of X n ∈ R shares. Thus if market maker n purchases q n shares in the auction for a price of p pershare, then her expected utility is E (cid:104) − e − γ (cid:0) ( X n + q n )˜ µ − pq n (cid:1)(cid:105) . (16)The liquidity traders start out with zero shares, and the total number of outstanding shares is S , so (cid:80) Nn =1 X n = S .More formally, we work on a probability space with a single objective probability measure. There are N + 2 real-valued random variables defined on this probability space: ˜ µ , ˜ u , and X , · · · , X N . There areexogenous constants µ, S ∈ R and σ > µ ∼ N ( µ, σ ) and (cid:80) Nn =1 X n = S . Furthermore, ˜ µ and ˜ u are independent of each other as well as X , · · · , X N .The type (or private information) of market maker n is X n . A strategy is a measurable function mappingthe realization of a market maker’s type to a choice of demand schedule. As before demand schedules arerestricted to be affine and strictly decreasing, so a strategy for market maker n is a measurable mapping( a n , b n ) : R → R × (0 , ∞ ), X (cid:55)→ ( a n ( X ) , b n ( X )). Given a strategy profile and a realization of ( X , · · · , X N ),prices and quantities are determined from (1) and (2) with a n = a n ( X n ) and b n = b n ( X n ). In the previous section, the total number of outstanding shares was absorbed into the mean of ˜ u . in this game, but we will focus on equilibria that have a very specific structure. Definition 2.2.
A strategy s : R → R × (0 , ∞ ) is called linear if there exist constants a, ξ ∈ R and b ∈ (0 , ∞ )such that s ( X ) = (cid:0) aX + ξ, b (cid:1) ∀ X ∈ R . Our focus will be on linear symmetric equilibria, that is on equilibria where all market makers play thesame linear strategy.
Theorem 2.3.
Fix exogenous parameters N ≥ , γ, σ > and µ, S ∈ R .If ˜ u is non-degenerate then there is a unique linear symmetric equilibrium. In equilibrium, the price is p = µ − γσ N S − N − N ( N − γσ ˜ u (17) and the quantities purchased by each market maker are q n = − N − N − X n − SN ) + ˜ uN . (18) Proof.
Fix a linear strategy given by a, ξ ∈ R and b ∈ (0 , ∞ ) to be played by all but one agent, andconsider the optimal response problem faced by the remaining agent. Suppose the remaining agent plays thestrategy R → R × (0 , ∞ ) , x (cid:55)→ ( α ( x ) , β ( x )). If the agent’s initial inventory is X , then her expected utility is E [ − e − γ (( X + q )˜ µ − pq ) ], where p and q are given implicitly by aN − S − X ) + ξ − bp = ˜ u − qN − α ( X ) − β ( X ) p = q. (20)Rearranging we obtain p = F + C ( S − X ) − λ ˜ u + λq (21) q = (cid:20) α ( X )1 + λβ ( X ) − β ( X )1 + λβ ( X ) (cid:16) F + C ( S − X ) (cid:17)(cid:21) + λβ ( X )1 + λβ ( X ) ˜ u, (22)where F := ξb , C := ab ( N − and λ := b ( N − .We note a few things about theses equations. Firstly, the parameters F , C and λ characterize thesymmetric profile given by a, ξ and b , and they are independent of the remaining agent’s demand schedule.Secondly, equation (21) tells us that if we hold the remaining agent’s initial inventory fixed, then the pricethe remaining agent receives is uniquely determined by the quantity she trades. Thus the agent is indifferentbetween demand schedules that lead to the same quantity of shares. Thirdly, from equation (22) we see thatchoosing a strategy ( α, β ) amounts to choosing functions A : R → R and B : R → (0 ,
1) such that the agent’s What we call a Bayesian Nash equilibrium is sometimes called a strong Bayesian Nash equilibrium. We require market maker n to choose a strategy that maximizes (16) conditional on X n for every realization of X n . This is in contrast to the weakerrequirement of choosing a strategy that just maximizes (16), which averages over realizations of X n . q = A ( X ) + B ( X )˜ u . Fourthly, if the agent uses the linear strategy ( aX + ξ, b ) then hertraded quantity is q = Cλ (cid:0) X − SN (cid:1) + ˜ uN .All this goes to show that linear symmetric equilibria correspond to F, C ∈ R and λ > Cλ (cid:0) x − SN (cid:1) + ˜ uN ∈ arg max q ∈ V E (cid:20) − e − γ (cid:16) x ˜ µ + q (cid:0) ˜ µ − F − C ( S − x )+ λ ˜ u − λq (cid:1)(cid:17)(cid:21) ∀ x ∈ R , (23)where V is as in (9). In equilibrium the price is p = F + N − N CS − N − N λ ˜ u, (24)and agents’ trades are q n = Cλ (cid:0) X n − SN (cid:1) + ˜ uN . (25)Now, note that ˜ u is independent of ˜ µ and every q ∈ V is ˜ u measurable. Since the MGF of ˜ µ is E [ e t ˜ µ ] = e µt + σ t , we can compute the expectation in (23) for any q ∈ V and x ∈ R as e − γµx E (cid:20) − e − γ (cid:16) q (cid:0) µ − F − C ( S − x )+ λ ˜ u − λq (cid:1) − γσ ( x + q ) (cid:17)(cid:21) . (26)For any F, C, x ∈ R and λ >
0, the function of q in (26) is strictly concave over the convex set V . Thusthe argmax in (23) consists of at most one point. Furthermore, for any F, C, x ∈ R , λ > u ∈ R , themuch relaxed problem of maximizing the exponent inside the expectation in (26)max q ∈ R q (cid:0) µ − F − C ( S − x ) + λ ˜ u − λq (cid:1) − γσ x + q ) has unique solution ˆ q = µ − F − CS λ + γσ + C − γσ λ + γσ x + λ λ + γσ u .It follows that (23) holds if and only if Cλ (cid:0) x − SN (cid:1) + ˜ uN = µ − F − CS λ + γσ + C − γσ λ + γσ x + λ λ + γσ ˜ u ∀ x ∈ R . (27)Since ˜ u is non-degenerate, (27) holds if and only if F = µ, C = − γσ N − , and λ = γσ N − . The theorem now followsby plugging these values is (24) and (25). The proof is similar to the one in the previous subsection, with the idea being to characterize symmetricprofiles in terms of the parameters F , C and λ of the induced optimal response problem. The additionalparameter C governs how the intercept of the supply curve in the optimal response problem depends on theoptimizing agent’s initial inventory. More specifically, C captures the dependence of the intercept on the sumof all other agents’ inventories, which the optimizing agent can compute by subtracting her own inventoryfrom the total number of outstanding shares, i.e. S − X .The supply curve represents the prices at which the other agents are willing to clear the joint order ofthe liquidity traders and the optimizing agent. These prices must depend on the preexisting exposures of the11emaining agents, hence the presence of the parameter C . For symmetric profiles, the others’ exposure can beaggregated instead of considering individual exposures, which greatly simplifies the problem.The constant term in the equilibrium price is µ − γσ N S , as opposed to just µ in the previous theorem.Thus there is a constant discount in the price reflecting the total number of outstanding shares. Intuitively thisdiscount appears here because the market makers are already in possession of S shares prior to the auction,whereas in the previous section they initially posses no shares. The coefficient of ˜ u in the equilibrium price is N − N ( N − γσ , exactly as in the previous subsection.A unified way to write the equilibrium price in the two theorems is in terms of the aggregate inventoryof the market makers after the auction, denoted S post . In the first subsection’s model we have S post = ˜ u andin the second subsection we have S post = S + ˜ u . In both cases, the equilibrium price is p = µ − γσ N S post − γσ N ( N −
2) ˜ u. (28)The first two terms here are the competitive benchmark, and the last term is the deviation due to imperfectcompetition.At first glance it might seem surprising and counterintuitive that the deviation due to imperfectcompetition depends on ˜ u and not S post . For example, if S post > u <
0, then the market makers are inaggregate long the asset, but the price is high relative to the competitive benchmark. The deviation due toimperfect competition should always favor the market makers, so one might expect it to make the price lowwhen they are going long and high when they are going short. However this reasoning is flawed because theprice in (28) is not the price at which the market makers enter their aggregate position of S post . It is merelythe price at which the market makers shift their aggregate position from S to S post . Said another way, (28) isnot the denominator in the market makers’ aggregate return, and as such the low/high long/short reasoningdoes not apply.The logic behind (28) is that the price is low when the liquidity traders are selling, ˜ u >
0, and high whenthey are buying, ˜ u <
0. Thus, roughly speaking, the liquidity traders are always ”getting ripped off.” This canbe made more precise by recalling the analogy between the auction and a limit order book. Based on thisanalogy, the price in (17) can be interpreted as saying that the is µ − γσ N S , and orders walk the book at arate of γσ N ( N − per share.Now, suppose the auction is repeated an instant later (prior to the realization of payoffs). Since theaggregate inventory of the market makers will be S post an instant later, the will be µ − γσ N S post . So, (28) saysthat if the market makers buy in the first auction, ˜ u >
0, then they do so at a price lower than the in thesecond auction. Similarly if they sell in the first auction, ˜ u <
0, then they do so at a price higher than the inthe second auction. Thus market makers always trade in the first auction at prices that are favorable relativeto the in the second auction. In particular, if a market maker were to unwind the position acquired in the firstauction with a limit order in the second auction, then the roundtrip trade would earn positive profits. This isthe sense in which the deviation due to imperfect competition always favors the market makers. This logic Technically a market order could also work in the second auction if it does not suffer too much price impact. The point ˜ uN . Since the market makers must clear ˜ u in the auction, it followthat the average number of shares bought by each market maker is ˜ uN , but some market makers might buymore and some less. The total initial inventory of the market makers is S , so the average inventory held byeach market maker is SN . (18) says that the market makers with above average inventories buy less, and thosewith below average inventories buy more.A more precise way to understand the traded quantities is in terms of the Pareto optimality of theinventory distribution before and after the auction. Since all market makers are identical, it would be Paretooptimal for them to hold SN shares before the auction and S post N after the auction. Individual inventories afterthe auction are X npost := X n + q n and from (18) it follows that X npost − S post N = 1 N − X n − SN ) . (29)Thus (29) says that trading in the auction moves inventories closer to efficiency by a factor of N − . When N takes its smallest value of 3, the market makers only move halfway towards efficiency, whereas they moveentirely towards efficiency in the limit as N → ∞ . Thus imperfect competition among the market makersresults in imperfect risk sharing. This section presents a continuous time model of trading. The model is set on an infinite horizon, and eachinstant in time consists of the auction from above. Traders hold cash and shares, and these fluctuate overtime based on the outcomes of the auctions. As in Section 2 there are two types of traders: market makers,endogenous, and liquidity traders, exogenous.The first subsection formulates the model as a stochastic differential game, and the second subsectionformally defines the equilibrium concept to be considered. The third subsection states the main theorem, whichprovides a complete closed-form characterization of linear symmetric equilibria. The proof of the theorem isgiven in Section 4, and an analysis of the equilibrium is given in Section 5.
Fix a filtered probability space (Ω , F , {F t } , P ) equipped with two independent Brownian motions, { B Dt } and { B ˜ St } , and satisfying the usual conditions. We consider a market on an infinite horizon where shares of a is to unwind the position in the second auction at a favorable price relative to (28). Since the in the second auction will be µ − γσ N S post , the price will always be favorable if using a limit order, and for a market order it depends on the price impact. Inthe dynamic model, equilibrium price impact will be constant over time and using market orders will not work for the marketmakers. However liquidity trader flow will mean revert, so the market makers will eventually be able to unwind their positionsusing limit orders, and this will earn positive profits. N ∈ N market makers and a collection ofliquidity traders. Each market maker starts out at time 0 holding X n shares of the asset, n = 1 , · · · , N . Theliquidity traders start out with a collective shareholding of − S , which must satisfy S = X + · · · + X N sincethe asset is in zero net supply.At each instant in time the traders transact with one another at a uniform price p t . Trading occurssmoothly, meaning that each trader has a trading rate, which is the time derivative of her current shareholdings. Trading rates and the trading price at each instant in time are determined by a demand schedule auctionbetween the traders. Thus at time t each trader submits an affine demand schedule of the form q = u t − v t p .This is a commitment to trade at rate u t − v t p at time t if the trading price is p . We denote this demandschedule by ( u t , v t ). The process { ( u t , v t ) } is required to be progressively measurable, though we will placemore stringent conditions on it below.The liquidity traders’ collective demand schedule at time t is assumed to be of the form ( −N t , −N t , interpreted as a market orderto sell N t dt shares over the time interval [ t, t + dt ]. The liquidity traders’ collective inventory at time t isdenoted by − S t , and it follows that { S t } evolves as a consequence of trading according to dS t = N t dt. (30)Furthermore, the process {N t } is exogenously specified according to N t = − φ ( S t − ˜ S t ) (31) d ˜ S t = − ψ ˜ S t dt + σ ˜ S dB ˜ St (32)where φ, ψ, σ ˜ S > − ˜ S t is the current inventory target of the liquidity traders,and φ governs the speed with which they trade towards their target. Thus when their current inventory isabove the target, − S t > − ˜ S t , the liquidity traders submit market orders to sell, N t >
0, and vice versa. If φ is large, the liquidity traders are impatient and submit large market orders to quickly reach their target.If φ is small, the liquidity traders are patient, submit smaller market orders, and only move slowly towardstheir target. One can think of − ˜ S t as the inventory the liquidity traders would want to hold if markets wereperfectly liquid. Due to illiquidity they cannot instantly acquire this inventory, and instead do so gradually.Note that since { ˜ S t } mean reverts about zero, so too does { S t } , i.e. we have E [ S T | S t , ˜ S t ] → T → ∞ almost surely ∀ t ≥ Shareholdings will also be referred to as inventories in what follows. It is actually not necessary to assume that agents can only submit affine schedules, as we will see below. For the class ofequilibria we consider, when forming an optimal response an agent can achieve any trading rate via an affine demand schedule.Thus even if agents could submit arbitrary schedules, there would still be an equilibrium where they all submit affine schedules.Of course, there may also be other equilibria where agents submit more exotic schedules. q nt , so that their inventories evolve according to dX nt = q nt dt. (33)If the market makers submit the demand schedule processes { ( α nt β nt , β nt ) } , then their trading rates q nt and thetrading price p t are determined implicitly by α nt − β nt q nt = p t ∀ n = 1 , · · · , N (34) q t + · · · + q Nt = N t . (35)Note that the price, trading rate, and inventory processes all depend on the choice of demand scheduleprocesses. This dependence is suppressed in the notation.Each market maker has cash holding M nt which evolve as a result of trading according to dM nt = − q nt p t dt .The market makers are also assumed at time t to have a common exogenous valuation of the asset as D t ,where dD t = µdt + σ D dB Dt (36)and µ, σ D > W nt = X nt D t + M nt . This is the market maker’s wealth, computed by valuingshareholdings at D t .Each market maker chooses her demand schedule to maximize the objective E (cid:20) (cid:90) ∞ e − ρt (cid:16) dW nt − γ d (cid:104) W n (cid:105) t (cid:17)(cid:21) , (37)where ρ , γ >
0. Here (cid:104) W n (cid:105) t is the quadratic variation of the market makers wealth, and so d (cid:104) W n (cid:105) t can bethought of as the variance of instantaneous wealth changes. Thus the integrand in (37) can be interpreted as amean-variance utility flow from instantaneous returns, which means that (37) embodies myopic mean-variancepreferences over returns.One can compute the objective function in (37) as E (cid:20) (cid:90) ∞ e − ρt (cid:16) − q nt ( p t − D t ) + µX nt − γσ D X nt ) (cid:17) dt (cid:21) . (38)Thus the market makers want to buy, q nt >
0, when the price is below their valuation, p t − D t <
0, and viceversa. Furthermore, they enjoy holding inventory to the extent that valuations grow on average, µ >
0, andthey are averse to holding inventory to the extent that valuations are volatile, σ D > N market makers. Indeed, the control process of each market maker is { ( α nt , β nt ) } and thecontrolled dynamics are described by (30) - (35). The coupled objectives of the market makers are given by Myopic because the utility flow comes from instantaneous returns. We take as controls the parameters of the inverse demand as opposed to the demand. This simplifies much of the algebrabelow. We will continue to refer to the controls as demand schedules.
We begin by specifying the admissibility conditions that the market makers’ demand schedules must satisfy.Firstly, we need to make sure that the system (34) - (35) can be solved uniquely to define progressivelymeasurable trading rates and prices. Secondly, prices and trading rates must be sufficiently well-behavedso that the (implicit) integrals in (33) and the double integral in (38) converge absolutely. Finally, we willwant to place some measurability restrictions on the demand schedule processes in order to reflect the typeof information that market makers have access to. This gives rise to the definition of admissible profiles ofdemand schedules.
Definition 3.1.
Given initial conditions ( (cid:126)x, ˜ s, d ) ∈ R N × R × R for ( X , · · · , X N ) , ˜ S , and D , we say thatthe profile of progressively measurable demand schedules { ( α t , β t ) } , · · · , { ( α Nt , β Nt ) } is admissible startingfrom ( (cid:126)x, ˜ s, d ) if:1. β nt > ∀ t ≥ , ∀ n = 1 , · · · , N almost surely2. (cid:82) T | q nt | dt < ∞ ∀ T ≥ , ∀ n = 1 , · · · , N almost surely3. The double integral (38) converges absolutely4. α nt , β nt ∈ σ (cid:16) { D s } ≤ s ≤ t , { p s } ≤ s Given initial conditions ( (cid:126)x, ˜ s, d ), we say that a profile of demand schedules { ( α t , β t ) } , · · · , { ( α Nt , β Nt ) } is a Nash Equilibrium starting from ( (cid:126)x, ˜ s, d ) if: 16. The profile is admissible starting from ( (cid:126)x, ˜ s, d )2. For any n = 1 , · · · , N , and for any demand schedule process { ( α t , β t ) } such that { ( α t , β t ) } , · · · , { ( α n − t , β n − t ) } , { ( α t , β t ) } , { ( α n +1 t , β n +1 t ) } , · · · , { ( α Nt , β Nt ) } (cid:1) is admissible starting from( (cid:126)x, ˜ s, d ), we have that J n (cid:0) (cid:126)x, ˜ s, d, { ( α t , β t ) } , · · · , { ( α Nt , β Nt ) } (cid:1) ≥ J n (cid:0) (cid:126)x, ˜ s, d, { ( α t , β t ) } , · · · , { ( α n − t , β n − t ) } , { ( α t , β t ) } , { ( α n +1 t , β n +1 t ) } , · · · , { ( α Nt , β Nt ) } (cid:1) . The second condition is the standard condition for a Nash equilibrium. It states that when all themarket makers but market maker n play their equilibrium demand schedules, the maximum payoff the n th market maker can earn is if she also plays her equilibrium demand schedule. In other words, when consideringthe optimal response problem against a profile of equilibrium demand schedules, each market maker finds itoptimal to also use her equilibrium demand schedule.Unfortunately, identifying all the Nash equilibria in this model is intractable and beyond the scope ofthis paper. Instead we will focus on a special class of equilibria where all the market makers’ demand scheduleshave a linear and symmetric structure. While this is fairly restrictive, the equilibria seem quite realistic andexhibit interesting dynamics. More specifically, we will only consider equilibria where all the market makersuse demand schedules with the same constant slope and with an intercept that is the same linear function ofindividual state variables. The individual state variables will be D t , X nt , and S t . The precise formulation isgiven in the next definitions. Definition 3.3. A profile of demand schedules { ( α t , β t ) } , · · · , { ( α Nt , β Nt ) } is said to be linear symmetric if ∃ a, λ, b, c, ξ ∈ R s.t. α nt = aX nt + bD t + cS t + ξ (39) β nt = λ (40) ∀ t ≥ , ∀ n = 1 , · · · , N . Definition 3.4. We say that a, λ, b, c, ξ ∈ R are a linear symmetric Nash equilibrium if the linearsymmetric profile defined by (39) and (40) is a Nash equilibrium starting from any set of initial conditions. Theorem 3.5. Fix exogenous parameters N ≥ , ρ, γ, σ D , φ, ψ, σ ˜ S > and µ ∈ R . There is a unique linearsymmetric Nash equilibrium with price p t = D t + µρ − θ γσ D N S t − γN N − N − ρσ D ( ρ + ψ )( ρ + φ ) (cid:16) δ + 1 ρ (cid:17) N t nd trading rates q nt = − κ (cid:16) X nt − S t N (cid:17) + 1 N N t , where κ := ρ ( N − ρ + ψρ + δδ := (cid:112) ρ + 2( N − ρ + ψ )( ρ + φ ) θ := ( ρ + ψ + φ ) δ − ψφ ( ρ + ψ )( ρ + φ ) δ . Before proceeding we prove the following lemma, which states the for linear symmetric profiles the admissibility conditions manifest in simple constraints on the parameters a and λ . One of the key points of the lemma isthat under a linear symmetric profile market makers can infer S t from the history of prices and valuations.The market makers are thus able to implement the demand schedules (39) and (40) given their individualinformation sets. Lemma 4.1. A linear symmetric profile is admissible if and only if λ > and aλ < ρ .Proof. Fix a linear symmetric profile given by a, λ, b, c, ξ ∈ R as in (39) and (40). We need to show that thefour conditions for admissibility in Definition 1.1 are satisfied if and only if λ > aλ < ρ . Clearly the firstcondition holds if and only if λ > 0. Next we will show that a profile satisfying (39) and (40) always satisfiesthe second and fourth conditions. Finally we will show that the third condition holds if and only if aλ < ρ ,thus completing the proof of the lemma.Note that by combining equations (39) and (40) with equations (33) - (35) we can conclude that prices,trading rates, and inventories under a linear symmetric profile must satisfy p t = (cid:16) aN + c (cid:17) S t + bD t + ξ − λN N t (41) X nt = e aλ t (cid:16) X n − S N (cid:17) + S t N (42) q nt = aλ (cid:16) X nt − S t N (cid:17) + 1 N N t (43) ∀ t ≥ ∀ n = 1 , · · · , N . These formulas imply that trading rates are almost surely continuous and thus thesecond condition holds.To prove the fourth condition it suffices to prove that S t ∈ σ (cid:16) { D s } ≤ s ≤ t , { p s } ≤ s 0. Furthermore the ODEabove implies that S t = e Nλ (cid:16) aN + c (cid:17) t S + (cid:90) t e Nλ (cid:16) aN + c (cid:17) ( t − s ) A s ds from which it follows that S t ∈ σ (cid:16) { D s } ≤ s ≤ t , { p s } ≤ s 20s admissible, and by (46) and (47) it gives the remaining market maker the trading rate process { q t } = { ˜ q t } .To prove (48) it suffices to prove that { S u } ≤ u ≤ t ∈ σ (cid:16) { D s } ≤ s ≤ t , { p s } ≤ s 0. Indeed (50) follows simply by plugging ( α t , β t ) = ( aX t + bD t + cS t + ξ, λ ) in (46) - (47) and solvingfor q t .This third point shows that if the remaining market maker follows the linear symmetric profile, thenher trading rate is the maximizer in (44). The first two points imply that solving the optimal responseproblem against a linear symmetric profile is the same as solving the constrained optimization problem in theproposition. Hence the proposition follows.This proposition associates to each linear symmetric profile a, λ, b, c, ξ ∈ R an optimization problem,as well as a candidate solution of the problem. The parameters provide an equilibrium if and only if thecandidate is truly a solution. Below we will use the theory of stochastic control to derive first order conditionsfor the optimization problem. Using these we will demonstrate that there is a unique profile for which thecandidate is a true solution, and thus there is a unique linear symmetric equilibrium.Before proceeding, we establish some notation relevant to the proposition and it’s use below. Given a setof parameters a, λ, b, c, ξ ∈ R , the optimization in the proposition is a standard stochastic control problem onan infinite horizon, with state space ( x, d, ˜ s, s ) ∈ R and control space q ∈ R . Denote the covariables for theproblem by y = ( y x , y d , y ˜ s , y s ) ∈ R and z = z xx z xd z x ˜ s z xs z dx z dd z d ˜ s z ds z ˜ sx z ˜ sd z ˜ s ˜ s z ˜ ss z sx z sd z s ˜ s z ss ∈ R × . H : R × R × R × × R → R given by H ( x, d, ˜ s, s, y, z, q ) = qy x + µy d − ψ ˜ sy ˜ s − φ ( s − ˜ s ) y s + σ D z dd + σ S z ˜ s ˜ s + µx − γσ D x − q (cid:0) P ( x, d, ˜ s, s, q ) − d (cid:1) . where P : R × R → R is the function specifying prices as a function of state and control from (45) above, i.e. P ( x, d, ˜ s, s, q ) = (cid:16) aN − c + λφN − (cid:17) s + bd + ξ − λφN − s − aN − x + λN − q. Also denote by Q : R → R the mapping corresponding to the Markov control in (44), i.e. Q ( x, d, ˜ s, s ) = aλ (cid:0) x − sN ) − φN ( s − ˜ s ) . (51) H and P describe the optimization problem, and Q is the candidate solution. The functions H, P , and Q alldepend on the parameters a, λ , b, c, and ξ , but this dependence is suppressed in the notation.We will prove that a, λ, b, c, ξ ∈ R are a linear symmetric Nash equilibrium if and only if a = − N − δ γσ D (52) λ = N − N − ργσ D ( ρ + ψ )( ρ + φ ) (cid:16) δ + 1 ρ (cid:17) (53) b = 1 (54) c = − ρ ( ρ + ψ + φ )( ρ + ψ )( ρ + φ ) γσ D N (cid:16) δ + 1 ρ (cid:17) + γσ D δ (55) ξ = µρ . (56)prices and trading rates are given by (41) - (43), and plugging in these values gives the formulas in thestatement of the theorem.We begin by proving the only if part of the statement. To this end, suppose the parameters a, λ, b, c, ξ ∈ R are a linear symmetric Nash equilibrium. Denote by V ( x, d, ˜ s, s ) the value function of the optimization problemin Proposition 4.2, i.e. the value of the supremum in (44) when the initial conditions are ( X , D , ˜ S , S ) =( x, d, ˜ s, s ).Note that the value function is smooth. Indeed because the parameters provide an equilibrium, (44)must hold, and therefore V ( x, d, ˜ s, s ) can be computed by evaluating the expectation in (44) along the controlprocess aλ (cid:16) X t − S t N (cid:17) + φN ( S t − ˜ S t ). This provides us with an explicit expression for V , and by direct inspectionit follows that V is smooth.It follows that V satisfies the following HJB equation: ρV ( x, d, ˜ s, s ) = sup q ∈ R H ( x, d, ˜ s, s, ∇ V, ∇ V, q ) ∀ ( x, d, ˜ s, s ) ∈ R . (57)Furthermore, since Q is an optimal Markov control, it also follows that Q ( x, d, ˜ s, s ) ∈ arg max q ∈ R H ( x, d, ˜ s, s, ∇ V, ∇ V, q ) ∀ ( x, d, ˜ s, s ) ∈ R . (58)2257) is the classical result that if the value function is smooth then it satisfies the HJB equation [Tou13]. Whenan optimal Markov control is known to exist, one way to prove (57) is to first prove (58) [Car16]. A lemmaexplicitly proving (58) is included in the appendix to this paper for completeness.The first order condition for (58) is V x ( x, d, ˜ s, s ) = P (cid:0) x, d, ˜ s, s, Q ( x, d, ˜ s, s ) (cid:1) − d + λN − Q ( x, d, ˜ s, s ) ∀ ( x, d, ˜ s, s ) ∈ R . (59)Anti-differentiating (59) it follows that ∃ a smooth function w : R → R such that V ( x, d, ˜ s, s ) = 12 aN − x + ( b − xd − (cid:16) N − N ( N − λ (cid:17) x ˜ s + (cid:16) N − N ( N − a + c (cid:17) xs + ξx + w ( d, ˜ s, s ) (60) ∀ ( x, d, ˜ s, s ) ∈ R .In summary, we’ve shown that if a, λ, b, c, ξ ∈ R are a linear symmetric Nash equilibrium then (57) - (60)hold. Combining these equations, we conclude that ∃ a smooth function w : R → R such that ρ aN − x + ρ ( b − xd − ρ N − N ( N − λx ˜ s + ρ (cid:16) N − N ( N − a + c (cid:17) xs + ρξx + ρw ( d, ˜ s, s )= (cid:16) a λ ( N − − γσ D (cid:17) x + (cid:16) aN − N − N ( N − ψλ + c (cid:17) x ˜ s − a N ( N − λ xs + µbx + µw d − ψ ˜ sw ˜ s + ˜ sw s + σ D w dd + σ N w ˜ s ˜ s + λ ( N − N (cid:16) ˜ s − aλ s (cid:17) (61) ∀ ( x, d, ˜ s, s ) ∈ R .It remains to be shown that (61) implies (52) - (56). Note that since the function w is independent of x ,the coefficients of x , xd, x ˜ s, xs , and x must be equal on the left and right hand sides of (32). Equatingthe coefficients of xd and x immediately gives (54) and (56). Equating the remaining coefficients yields thealgebraic system ρ aN − a λ ( N − − γσ D − ρ N − N ( N − λ = aN − N − N ( N − 1) ( ψ + φ ) λ + c (63) ρ (cid:16) N − N ( N − 1) ( a + λφ ) + c (cid:17) = − a N ( N − λ − N − N ( N − φ λ − φ (cid:16) aN − c (cid:17) . (64)Equation (63) gives c in terms of a and λ , from which it follow that if (52) and (53) hold then so too does(55). Next we plug this expression for c into (64) and solve for λ in terms of a to get λ = − ρ + ψ (cid:16) aN − − N − N − γσ D ρ (cid:17) .From this equation it follows that if (52) holds then so too does (53). Hence it only remains to prove that (52)holds. Plugging this expression for λ into (62), we see that a must satisfy a = ( N − γ σ D δ . Now, becausethe parameters provide an equilibrium, the constraint aλ < ρ must hold. The positive root for a violates theconstraint and the negative root satisfies it, so it follows that (52) holds.We now prove the if part of the statement. To this end, suppose that a, b, c, ξ , and λ are given byequations (52) - (56). We need to show (44) holds, i.e. that the mapping Q in (51) provides an optimal Markovcontrol for the stochastic control problem in Proposition 4.2. This will be done by by using the verificationtheorem for the HJB equation [Pha09]. 23iven a set of initial conditions, denote by { ˆ q t } and { ˆ X t } the trading rate and inventory processesarising from using the Markov control given by Q , i.e. d ˆ X t = Q ( ˆ X t , D t , ˜ S t , S t ) dt and ˆ q t = Q ( ˆ X t , D t , ˜ S t , S t ).These processes depend on the choice of initial conditions, but this is suppressed in the notation. We need toshow that ∃ a smooth function V ( x, d, ˜ s, s ) such that1. (57) holds2. (58) holds3. e − ρt E [ V ( ˆ X t , D t , ˜ S t , S t )] → t → ∞ for any choice of initial conditions4. { ˆ q t } satisfies the the constraints in Proposition 4.2 for any choice of initial conditions.If this can be done then it follows by the verification theorem that Q is an optimal Markov control for thestochastic control problem in Proposition 4.2.Note that we can find a second order polynomial w ( d, ˜ s, s ) that satisfies the equation ρw = µw d − ψ ˜ sw ˜ s − φ ( s − ˜ s ) w s + σ D w dd + σ S w ˜ s ˜ s + (cid:32) φN (cid:114) λN − s − ˜ s ) + aN (cid:112) λ ( N − s (cid:33) on all of R . Now define the function V ( x, d, ˜ s, s ) by equation (60). Then the function V is smooth and byconstruction equations (59) and (61) hold. Notice that as a function of q the Hamiltonian is a quadraticpolynomial with leading coefficient − λN − . Since λ > V is a second order polynomial, in order to prove the third condition it suffices to show that E [ e − ρt ˆ X t ], E [ e − ρt D t ], E [ e − ρt ˜ S t ], and E [ e − ρt S t ] converge to 0 as t → ∞ for any set of initial conditions. Thisfollows from the fact that aλ < ρ and ψ, φ > { ˆ q t } satisfies the constraints in Proposition 3.1. The third constraint istrivial since ˆ q t = Q ( ˆ X t , D t , ˜ S t , S t ). This formula also implies { ˆ q t } is almost surely continuous, and thus thefirst condition holds. Lastly, the second condition holds because aλ < ρ and ψ, φ > Consider the equilibrium price process given in Theorem 3.5. The four terms admit intuitive economicinterpretations. The first term D t is simply the market makers’ current valuation for the asset. The secondterm µρ is a premium for expected valuation growth. µ is the drift of valuations, so on average valuationsincrease by µ ( T − t ) over a time interval [ t, T ]. As this is common knowledge among the market makers, thismust be reflected in the price at time t . Otherwise, a profitable deviation from equilibrium would be to buy24he asset at time t and sell it at time T . Since market makers discount payoffs from time T to time t by e − ρ ( T − t ) , the appropriate premium in the price to prevent this deviation is µρ . Said another way, we have µρ = E t (cid:20) (cid:90) ∞ t e − ρ ( T − t ) dD T (cid:21) . Thus the second term is the (risk-neutral) present value of expected future changes in the asset’s value.Since the market makers are not risk neutral, they also require risk compensations to take exposures tothe asset. This is the role of the third term in the price − θ γσ D N S t . When the market makers are in aggregatelong the asset, so S t is positive, this term is negative and thus the asset is trading at a relatively low price.Because the asset is trading at a low price, market makers don’t find it profitable to deviate from equilibriumby selling the asset to reduce their exposures. Similarly, when the market makers are in aggregate short theasset, this term causes the asset to trade at a high price, and thus market makers don’t find it profitable tobuy the asset to reduce their exposures.The magnitude of the compensation per unit of exposure is given by θ γσ D N . σ D is the volatility ofvaluations, so this is the amount of risk per unit of exposure to the asset. γN is the aggregate risk aversion ofthe market makers, so this is the dollar compensation they require to hold a unit of risk. Thus γσ D N is thedollar compensation the market makers require to hold a unit of exposure to the asset. The intuitive role ofthe parameter θ is to take in to account fluctuations in future risk exposures and to discount them to thepresent. This will be made clearer in Proposition 5.4 below.The final term in the price process is − γN N − N − ρσ D ( ρ + ψ )( ρ + φ ) (cid:16) δ + ρ (cid:17) N t , which is the price impact theliquidity traders face on their trades, or equivalently the slope of the supply curve they face when trading.The liquidity traders submit market orders to trade at rate −N t , so they are buying when N t < N t > 0. When the liquidity traders are buying, price impact causes the trading price to be high, andwhen the liquidity traders are selling, price impact causes the trading price to be low. This is the model’sanalogue of liquidity traders’ market orders walking the book. As suggested by [Kyl89], we define liquidity inthe model as the reciprocal of price impact and study it’s comparative statics. Definition 5.1. P rice Impact := γN N − N − ρσ D ( ρ + ψ )( ρ + φ ) (cid:16) δ + ρ (cid:17) Definition 5.2. Liquidity := P rice Impact Proposition 5.3. ∂∂γ Liquidity < .Liquidity is decreasing in market makers’ risk aversion.2. ∂∂σ D Liquidity < .Liquidity is decreasing in fundamental volatility.3. If γN is held fixed then ∂∂N Liquidity > .Liquidity is increasing in market maker competition. The notation here and below is E t [ · ] := E [ ·|F t ]. . ∂∂ψ Liquidity > .Liquidity is decreasing in order flow uncertainty. To understand the fourth point, recall that the liquidity traders’ order flow is driven by { ˜ S t } , whichis an Ornstein-Uhlenbeck process whose stationary distribution has variance σ N ψ . Thus as ψ increases, theliquidity traders’ orders arrive with less uncertainty.These comparative statics agree with real word intuition about liquidity, and thus they justify thetheoretical definition of liquidity. Put simply, liquidity should reflect market makers’ willingness to absorbtemporary flow. Market makers are less willing to absorb temporary flow when they are more risk averse,when the asset is riskier, when there are not many of them, or when order flow is more uncertain. By theproposition, liquidity is also lower in the model in these situations.Next we study what happens to price impact in the competitive limit of the model. This is the limitwhen N → ∞ and γN → γ . Essentially one considers the sequence of models with N market makers eachhaving risk aversion γ N := N γ . Thus for any model in the sequence, the aggregate risk aversion of themarket makers is γ N N = γ . Hence the sequence considers increasingly competitive market making sectorsthat in aggregate have the same risk bearing capacity. Taking the limit of the sequence provides a perfectlycompetitive benchmark for the model. Proposition 5.4. In the competitive limit, we have that P rice Impact → γ σ D ( ρ + φ )( ρ + ψ ) and p t → D t + µρ − E t (cid:20) (cid:90) ∞ t e − ρ ( T − t ) γ σ D S T dT (cid:21) ∀ ( t, ω ) ∈ [0 , ∞ ) × ΩThat price impact does not vanish in the competitive limit is somewhat surprising. Naive intuition wouldsuggest that market makers exercise their market power by charging price impact. So in the competitive limit,when individual market makers no longer have market power, price impact should vanish. The reason thisintuition does not hold is given by the limiting value of the equilibrium price.In the competitive limit, market makers have an aggregate risk aversion of γ > 0, so the equilibriumprice should consist of a risk neutral present value as well as a discount based on the market makers’ aggregateexposure. The first two terms in the limiting price, D t + µρ , are a risk neutral present value as discussed above.Thus the third term − E t (cid:20) (cid:82) ∞ t e − ρ ( T − t ) γ σ D S T dT (cid:21) should be the appropriate risk discount.As discussed above, S T is the aggregate exposure of the market makers at time T , and − γ σ D S T isthe dollar compensation they require in order to maintain this exposure. However, because S T evolves with T , this is only the exposure over the infinitesimal time interval [ T, T + dt ]. Furthermore, standing at time t ,market makers require a compensation for the entire path of exposures they will be taking over [ t, ∞ ). Theappropriate compensation for the exposure over [ T, T + dt ] is γ σ D S T , and market makers discount payoffsfrom time T to time t by e − ρ ( T − t ) , so the appropriate compensation for the entire path of exposures over26 t, ∞ ) is − E t (cid:20) (cid:82) ∞ t e − ρ ( T − t ) γ σ D S T dT (cid:21) . The presence of the parameter θ in Theorem 3.5 ensures that thisexpected present value arises in the limit. Thus θ should be thought of as inflating/deflating the standard riskpremium so as to account for the expected future path of flow.Said another way, the third term is exactly what is needed to prevent market makers deviating fromequilibrium by pursuing a strategy that buys/sells based on the current level of the aggregate exposure. If S T does not evolve with T and has a constant value equal to S , then this term reads − ρ γ σ D S . This is exactlythe risk discount a representative CARA investor with risk aversion γ and time discount rate ρ would requireto hold S shares of an asset with volatility σ D .Based on this analysis, we can conclude, as suggested previously, that − ρ γσ D N S t is not the full riskcompensation component of the equilibrium price, but instead that − E t (cid:20) (cid:82) ∞ t e − ρ ( T − t ) γ σ D S T dT (cid:21) is. Thuspart of the price impact component of prices compensates market makers for risk, and so it does not vanish inthe competitive limit. The rest of the price impact component of prices is a manifestation of market makers’market power, and so it does vanish in the competitive limit. The interpretation is that market makerscharge price impact for (at least) two reasons. The first reason is simply because they can, since they havemarket power, and doing so is profitable. The second reason is that incoming trades change the entire path ofexposures that market makers will be taking going forward. In order for the market makers to find it utilitymaximizing to clear the incoming trade, this must be reflected in the trading price. Next consider the equilibrium trading rates given in Theorem 3.5. The two terms encode two importantproperties of the market makers’ equilibrium trading behavior. The first is that in aggregate the marketmakers must buy at rate N t , or equivalently the average trading rate of the market makers must be N N t .This is simply a consequence of market clearing, since the liquidity traders are selling at rate N t . That this isindeed the case is guaranteed by the second term in the formula for trading rates; the first terms all cancel outwhen aggregating/averaging.The first term in the trading rates dictates which market makers buy more/less than the average. Notethat κ > 0, so the market makers with inventories larger than S t N buy less, and vice versa. This brings us tothe second important property of the market makers’ trading behavior: they are continually moving towards aPareto optimal allocation amongst themselves. The market makers must in aggregate hold S t shares, simplyby market clearing. Since they are all identical, it would be Pareto optimal for everyone to hold S t N shares.However, this certainly can’t hold at time 0, as initial inventories are exogenous and arbitrary. Beyond time 0efficiency of the allocations depends on endogenous trading behavior.Using the formula for trading rates, we can compute that in equilibrium the trajectory of each marketmaker’s inventory is X nt = e − κt ( X n − S N ) + S t N . Thus each market maker deviates from the efficient allocation by the first term. This term is non-zero if and27nly if X n (cid:54) = S N , and in this case it converges monotonically towards 0 over time. Thus the market makerstake as given the inefficiency in their initial allocations, and then trade amongst themselves to make allocationsmore efficient. Allocations are inefficient at time t is because they were inefficient at time 0; the endogenoustrading behavior of the market makers does not in any way create allocational inefficiencies.But the market makers’ trading behavior also does not perfectly remove the initial inefficiency inallocations. Indeed, inventories converge to efficiency at an exponential rate of κ , and this rate is not infinite.Based on the formula for κ , we see that the two main drivers of this rate are ψ , governing the uncertainty inorder flow, and N , governing the degree of competition. As ψ → ∞ , so order flow uncertainty vanishes, or as N → ∞ , so the market is perfectly competitive, this rate of convergence goes to infinity.The story here is essentially one of individual market makers behaving strategically in an attempt to earnprofits. Note that the price in the absence of liquidity trades ( N t = 0), interpreted as the , is D t + µρ − θ γN σ D S t .Any market maker holding more than S t N shares should find this price high relative to her exposure and shouldwant to sell, and vice versa. However, market makers take into account price impact, so they know they can’tactually trade to S t N at this price. The market makers also know that at any moment a random liquiditytrade might come in and move their exposure in the right direction, but now without price impact . Thusinstead of trading all the way to S t N , market makers only move partially and take the chance that the liquiditytraders’ flow will move them the rest of the way. In the absence of order flow uncertainty, or under perfectcompetition, the market makers no longer go through this calculation and simply trade all the way to S t N ,converging instantly to efficiency (i.e. at an infinite rate). This paper presents a mathematical framework to model market making and to thereby generate an endogenousprice impact function. The key insight is that price impact is the mechanism through which market makersearn profits while matching their books. The paper presents this idea in a stylized static model as well as in arich continuous time model. The theory provides insights on the connection between market making and priceimpact, and as such is provides practical guidance on how to interpret measures of execution costs. References [AC01] Robert Almgren and Neil Chriss. “Optimal Execution of Portfolio Transactions”. The Journal ofRisk More specifically, price impact favors the market maker in this situation, as her limit order is getting hit by an incomingliquidity trader market order. If the market maker was insistent on moving to S t N , then she would have to place a market order,so price impact would go against her. Asset Pricing and Portfolio Choice Theory . Second edition. Financial ManagementAssociation Survey and Synthesis Series. Oxford ; New York: Oxford University Press, 2017. 722 pp. isbn : 978-0-19-024114-8.[Bar+15] Nicholas Barberis et al. “X-CAPM: An Extrapolative Capital Asset Pricing Model”. Journal ofFinancial Economics Finance and Stochastics Lectures on BSDEs, Stochastic Control, and Stochastic Differential Games withFinancial Applications . Philadelphia: Society for Industrial and Applied Mathematics, 2016. 265 pp. isbn : 978-1-61197-423-2.[CJP15] ´Alvaro Cartea, Sebastian Jaimungal, and Jos´e Penalva. Algorithmic and High-Frequency Trading .Cambridge, United Kingdom: Cambridge University Press, 2015. 343 pp. isbn : 978-1-107-09114-6.[CK93] John Y. Campbell and Albert S. Kyle. “Smart Money, Noise Trading and Stock Price Behaviour”. The Review of Economic Studies Asset Pricing . Rev. ed. Princeton, N.J: Princeton University Press, 2005. 533 pp. isbn : 978-0-691-12137-6.[FPR13] Thierry Foucault, Marco Pagano, and Ailsa R¨oell. Market Liquidity: Theory, Evidence, and Policy .Oxford ; New York: Oxford University Press, 2013. 424 pp. isbn : 978-0-19-993624-3.[GM88] Sanford J. Grossman and Merton H. Miller. “Liquidity and Market Structure”. The Journal ofFinance Journal ofEconomic Theory 165 (Sept. 1, 2016), pp. 487–516.[Gu´e16] Olivier Gu´eant. The Financial Mathematics of Market Liquidity: From Optimal Execution toMarket Making . Chapman & Hall/CRC Financial Mathematics Series. Boca Raton: CRC Press,Taylor & Francis Group, 2016. 278 pp. isbn : 978-1-4987-2547-7.[Kyl89] Albert S. Kyle. “Informed Speculation with Imperfect Competition”. The Review of EconomicStudies Market Microstructure Theory . repr. Malden, Mass.: Blackwell, 2011. 290 pp. isbn : 978-0-631-20761-0.[Ped15] Lasse Heje Pedersen. Efficiently Inefficient: How Smart Money Invests and Market Prices AreDetermined . Princeton: Princeton University Press, 2015. 348 pp. isbn : 978-0-691-16619-3.[Pha09] Huyˆen Pham. Continuous-Time Stochastic Control and Optimization with Financial Applications .Stochastic Modelling and Applied Probability 61. Berlin: Springer, 2009. 232 pp. isbn : 978-3-540-89499-5. 29Tou13] Nizar Touzi. Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE . FieldsInstitute Monographs v. 29. New York: Springer, 2013. 214 pp. isbn : 978-1-4614-4286-8.[Wil79] Robert Wilson. “Auctions of Shares”. The Quarterly Journal of Economics Fix a filtered probability space (Ω , F , {F t } , P ) equipped with a d-dimensional Brownian motion { B t } andsatisfying the usual conditions. Consider a standard infinite horizon stochastic control problem with state space R S , control space R A , and randomness coming from { B t } . Thus we take as given mappings b : R S × R A → R S , σ : R S × R A → R S × d , and f : R S × R A → R . We assume that b and σ are uniformly Lipschitz in their firstvariables and that f is Borel measurable. Also fix a constant β > A the set of progressively measurable R A valued processes α = { α t } such that E (cid:20) (cid:90) T | b (0 , α t ) + | σ (0 , α t ) | dt (cid:21) ∀ T > . Given x ∈ R S and α = { α t } ∈ A , denote by X x,α = { X x,αt } the unique strong solution to the SDE dX t = b ( X t , α t ) dt + σ ( X t , α t ) dB t X = x. For each x ∈ R S denote by A ( x ) the subset of α = { α t } ∈ A such that E (cid:20) (cid:90) ∞ e − βt | f ( X x,αt , α t ) | dt (cid:21) < ∞ and assume that A ( x ) is nonempty ∀ x ∈ R S .For x ∈ R S and α = { α t } ∈ A ( x ) define the reward functional by J ( x, α ) = E (cid:20) (cid:90) ∞ e − βt f ( X x,αt , α t ) dt (cid:21) . The value function is defined for x ∈ R S by V ( x ) = sup α ∈A ( x ) J ( x, α ) . We also define the Hamiltonian H : R S × R S × R S × S × R A → R by H ( x, y, z, a ) = b ( x, a ) · y + 12 trace (cid:0) σ ( x, a )) σ T ( x, a ) z (cid:1) + f ( x, a ) . The significance of the Hamiltonian is that it describes how functions of a controlled state process evolveover time. That is, by Ito’s formula, for any α = { α t } ∈ A , h > g ∈ C , (cid:0) [0 , ∞ ) × R S (cid:1) we have that g ( t + h, X x,αt + h ) − g ( t, X x,αt ) = (cid:90) t + ht ∂g∂t ( s, X x,αs ) + H (cid:0) X x,αs , ∇ g ( s, X x,αs ) , ∇ g ( s, X x,αs ) , α s (cid:1) − f ( X x,αs , α s ) ds + M artingale. The assumptions on b and σ and the definition of A were made precisely so that this equation admits a unique strong solution[Pha09]. efinition 7.1. We say that a : R S → R A is an optimal Markov control if ∀ x ∈ R S , ∃ ˆ α x = { ˆ α xt } ∈ arg max α ∈A ( x ) J ( x, α ) such that ˆ α xt = a (cid:0) X x, ˆ α x t (cid:1) ∀ t ≥ Lemma 7.2. Suppose that V ∈ C ( R S ) and H is continuous. If a : R S → R A is continuous and provides anoptimal Markov control then a ( x ) ∈ arg max ˜ a ∈ R A H ( x, ∇ V ( x ) , ∇ V ( x ) , ˜ a ) ∀ x ∈ R S . Proof. We proceed by contradiction. Suppose that the conclusion of the theorem does not hold. Then ∃ x ∈ R S , ˜ a ∈ R A and (cid:15) > H ( x , ∇ V ( x ) , ∇ V ( x ) , a ( x )) < H ( x , ∇ V ( x ) , ∇ V ( x ) , ˜ a ) − (cid:15). By continuity of a , H and V it follows that ∃ a neighborhood U of x such that H ( x, ∇ V ( x ) , ∇ V ( x ) , a ( x )) < H ( x, ∇ V ( x ) , ∇ V ( x ) , ˜ a ) − (cid:15) ∀ x ∈ U . (65)Let { X t } be the unique strong solution to the SDE dX t = b ( X t , a ( X t )) dt + σ ( X t , a ( X t )) dBtX = x and let { ˜ X t } be the unique strong solution to the SDE d ˜ X t = b ( ˜ X t , ˜ a ) dt + σ ( ˜ X t , ˜ a ) dBt ˜ X = x . Note that by smoothness of V and continuity of H we can find neighborhoods U and U of x such that | V ( x ) − V ( y ) | < (cid:15)β ∀ x, y ∈ U (66) | H ( x, ∇ V ( x ) , ∇ V ( x ) , ˜ a ) − H ( y, ∇ V ( y ) , ∇ V ( y ) , ˜ a ) | < (cid:15) ∀ x, y ∈ U . (67)Define the stopping time τ = inf { t ≥ X t , ˜ X t ) (cid:54)∈ U ∩ U ∩ U × U ∩ U ∩ U } . Since the processes { X t } and { ˜ X t } have a.s. continuous paths it follows that τ > V ( x ) = E (cid:20) (cid:90) τ e − βt f ( X t , a ( X t )) dt + e − βτ V ( X τ ) (cid:21) = V ( x ) + E (cid:20) (cid:90) τ e − βt (cid:16) H (cid:0) X t , ∇ V ( X t ) , ∇ V ( X t ) , a ( X t ) (cid:1) − βV ( X t ) (cid:17) dt (cid:21) ≤ V ( x ) + E (cid:20) (cid:90) τ e − βt (cid:16) H (cid:0) X t , ∇ V ( X t ) , ∇ V ( X t ) , ˜ a (cid:1) − (cid:15) − βV ( X t ) (cid:17) dt (cid:21) ≤ V ( x ) + E (cid:20) (cid:90) τ e − βt (cid:16) H (cid:0) ˜ X t , ∇ V ( ˜ X t ) , ∇ V ( ˜ X t ) , ˜ a (cid:1) + (cid:15) − (cid:15) − βV ( ˜ X t ) + (cid:15) (cid:17) dt (cid:21) = E (cid:20) (cid:90) τ e − βt f ( ˜ X t , ˜ a ) dt + e − βτ V ( ˜ X τ ) (cid:21) − (cid:15)β E [1 − e − βτ ] ≤ V ( x ) − (cid:15)β E [1 − e − βτ ] . The first equality uses the dynamic programing principle [Pha09] and the optimality of the Markov controlgiven by a . The second and last equality use Ito’s lemma. The last inequality uses the dynamic programmingprinciple. The inequalities in the middle follow from the definition of τ and inequalities (36) - (38). Since (cid:15)β > 0, it follows from this estimate that E [ e − βτ ] ≥≥