Instabilities in Multi-Asset and Multi-Agent Market Impact Games
aa r X i v : . [ q -f i n . T R ] J un Instabilities in Multi-Asset and Multi-AgentMarket Impact Games
Francesco Cordoni a and Fabrizio Lillo ba Scuola Normale Superiore,Piazza dei Cavalieri, 7 - 56126 Pisa (PI), Italy.E-mail: [email protected] b Dipartimento di Matematica, Universit`a di Bologna,Piazza di Porta San Donato, 5 - 40126 Bologna (BO), Italy.E-mail: [email protected] Written: April 7, 2020; Last revised: June 5, 2020
Abstract
We consider the general problem of a set of agents trading a portfolio of assets in thepresence of transient price impact and additional quadratic transaction costs and we study,with analytical and numerical methods, the resulting Nash equilibria. Extending signifi-cantly the framework of Schied and Zhang (2018), who considered two agents and one asset,we focus our attention on the conditions on the value of transaction cost making the tradingprofile of the agents, and as a consequence the price trajectory, wildly oscillating and themarket unstable. We find that the presence of more assets, the heterogeneity of trading skills(e.g. speed or cost), and a large number of agents make the market more prone to largeoscillations and instability. When the number of assets is fixed, a more complex structure ofthe cross-impact matrix, i.e. the existence of multiple factors for liquidity, makes the marketless stable compared to the case when a single liquidity factor exists.
Keywords : Market impact; Game theory and Nash equilibria; Transaction costs; Marketmicrostructure; High Frequency Trading.
1. Introduction
Instabilities in financial markets have always attracted the attention of researchers, policy mak-ers and practitioners in the financial industry because of the role that financial crises have onthe real economy. Despite this, a clear understanding of the sources of financial instabilitiesis still missing, in part probably because several origins exist and they are different at dif-ferent time scales. The recent automation of the trading activity has raised many concerns1bout market instabilities occurring at short time scales (e.g. intraday), in part because ofthe attention triggered by the Flash Crash of May 6th, 2010 (Kirilenko et al. (2017)) and thenumerous other similar intraday instabilities observed in more recent years (Brogaard et al.(2018), Calcagnile et al. (2018), Golub et al. (2012), Johnson et al. (2013)), such as the Trea-sury bond flash crash of October 15th, 2014. The role of High Frequency Traders (HFTs), AlgoTrading, and market fragmentation in causing these events has been vigorously debated, boththeoretically and empirically (Brogaard et al. (2018), Golub et al. (2012)).One of the puzzling characteristics of market instabilities is that a large fraction of themappear to be endogenously generated, i.e. it is very difficult to find an exogenous event (e.g. anews) which can be considered at the origin of the instability (Cutler et al. (1989), Fair (2002),Joulin et al. (2008)). Liquidity plays a crucial role in explaining these events. Markets are, infact, far from being perfectly elastic and any order or trade causes prices to move, which inturn leads to a cost (termed slippage) for the investor. The relation between orders and price iscalled market impact. In order to minimize market impact cost, when executing a large volumeit is optimal for the investor to split the order in smaller parts which are executed incrementallyover the day or even across multiple days. The origin of the market impact cost is predatorytrading (Brunnermeier and Pedersen (2005), Carlin et al. (2007)): the knowledge that a traderis purchasing progressively a certain amount of assets can be used to make profit by buying atthe beginning and selling at the end of the trader’s execution. Part of the core strategy of HFTsis exactly predatory trading. Now, the combined effect on price of the trading of the predatorand of the prey can lead to large price oscillations and market instabilities. In any case, it isclear that the price dynamics is the result of the (dynamical) equilibrium between the activityof two or more agents simultaneously trading.This equilibrium can be studied by modeling the above setting as a market impact game(Carlin et al. (2007), Lachapelle et al. (2016), Moallemi et al. (2012), Schied and Zhang (2018),Sch¨oneborn (2008), Strehle (2017a,b)). In a nutshell, in a market impact game, two traderswant to trade the same asset in the same time interval. While trading, each agent modifies theprice because of market impact, thus when two (or more) traders are simultaneously present,the optimal execution schedule of a trader should take into account the simultaneous presenceof the other trader(s). As customary in these situations, the approach is to find the Nashequilibrium, which in general depends on the market impact model.Market impact games are a perfect modeling setting to study endogenously generated marketinstabilities. A major step in this direction has been recently made by Schied and Zhang (2018).By using the transient impact model of Bouchaud et al. (2009, 2004) plus a quadratic temporaryimpact cost (which can alternatively be interpreted as a quadratic transaction cost, see below),they have recently considered a simple setting with two identical agents liquidating a single assetand derived the Nash equilibrium. Interestingly, they also derived analytically the conditions onthe parameters of the impact model under which the Nash equilibrium displays huge oscillationsof the trading volume and, as a consequence, of the price, thus leading to market instabilities .Specifically, they proved the existence of a sharp transition between stable and unstable markets In their paper, Schied and Zhang interpret the large alternations of buying and selling activity observed atinstability as the ”hot potato game” among HFTs empirically observed during the Flash Crash (CFTC-SEC(2010), Kirilenko et al. (2017)).
2t specific values of the market impact parameters.Although the paper of Schied and Zhang highlights an key mechanism leading to marketinstability, several important aspects are left unanswered. First, market instabilities rarelyinvolve only one asset and, as observed for example during the Flash Crash, a cascade ofinstabilities affects very rapidly a large set of assets or the entire market (CFTC-SEC (2010)).This is due to the fact that optimal execution strategies often involve a portfolio of assets ratherthan a single one (see, e.g. Tsoukalas et al. (2019)). Moreover, commonality of liquidity acrossassets (Chordia et al. (2000) and cross-impact effects (Schneider and Lillo (2019)) make thetrading on one asset trigger price changes on other assets. Thus, it is natural to ask: is a largemarket more or less prone to market instabilities? How does the structure of cross-impact andtherefore of liquidity commonality affect the market stability? A second class of open questionsregards instead the market participants. Do the presence of more agents simultaneously tradingone asset tends to stabilize the market? While the solution of Schied and Zhang considers onlytwo traders, it is important to know whether having more agents is beneficial or detrimentalto market stability. For example, regulators and exchanges could implement mechanisms tofavor or disincentive participation during turbulent periods. Answering this question requiressolving the impact game with a generic number of agents. Moreover, while in Schied andZhang the two investors are identical, real markets are characterized by huge heterogeneity intrading skills. For example, some agents (e.g. HFTs) are much faster than others, some agentsuse more sophisticated trading strategies and have smaller trading costs, etc. Is this traders’heterogeneity beneficial to market stability? Do HFTs destabilize markets?In this paper we extend considerably the setting of Schied and Zhang by answering the aboveresearch questions. Specifically, we consider (i) the case when the two agents trade multipleassets simultaneously and cross market impact is present; (ii) the case when
J > . Market Impact Games
Consider two traders who want to trade simultaneously a certain number of shares, minimizingthe trading cost. Since the trading of one agent affects the price, the other agent must take intoaccount the presence of the former in optimizing her execution. This problem is termed mar-ket impact game and has received considerable attention in recent years (Carlin et al. (2007),Lachapelle et al. (2016), Moallemi et al. (2012), Schied and Zhang (2018), Sch¨oneborn (2008),Strehle (2017a,b)). The seminal paper by Schied and Zhang, (Schied and Zhang (2018)), con-siders a market impact game between two identical agents trading the same asset in a giventime period.When none of the two agents trade, the price dynamics is described by the so called unaf-fected price process S t which is a right-continuous martingale defined on a given probabilityspace (Ω , ( F t ) t ≥ , F , P ). A trader wants to unwind a given initial position with inventory Z ,where a positive (negative) inventory means a short (long) position, during a given tradingtime grid T = { t , t , . . . , t N } , where 0 = t < t < · · · < t N = T and following an admissiblestrategy, which is defined as follows: Definition 2.1 (Admissible Strategy) . Given T and Z , an admissible trading strategy for T and Z ∈ R is a vector ζ = ( ζ , ζ , . . . , ζ N ) of random variables such that: • ζ i ∈ F t i and bounded , ∀ i = 0 , , . . . , N. • ζ + ζ + · · · + ζ N = Z .The random variable ζ k represents the order flow at trading time t k where positive (negative)flow corresponds to a sell (buy) trade of volume | ζ k | . We denote with X and Y the initialinventories of the two considered agents playing the game and with ξ = { ξ k } k ∈ T and η = { η k } k ∈ T their respective strategies.Traders are subject to fees and transaction costs and their objective is to minimize themby optimizing the execution. As customary in the literature, the costs are modeled by twocomponents. The first one is a temporary impact component modeled by a quadratic term θξ k which does not affect the price dynamics. This is sometimes called slippage and depends on theimmediate liquidity present in the order book. Notice that, as discussed in Schied and Zhang(2018), this term can also be interpreted as a quadratic transaction fee. Here we do not specifyexactly what this term represents, sticking to the mathematical modeling approach of Schiedand Zhang.The second component is related to permanent impact and affects future price dynam-ics. Following Schied and Zhang (2018), we consider the celebrated transient impact model ofBouchaud et al. (2009, 2004), which describes the price process S ξ , η t affected by the strategies ξ , η of the two traders, i.e., S ξ , η t = S t − X t k 12 ( X − Y ) w , (2)where the fundamental solutions v and w are defined as v = 1 e T (Γ θ + e Γ) − e (Γ θ + e Γ) − ew = 1 e T (Γ θ − e Γ) − e (Γ θ − e Γ) − e . e = (1 , . . . , T ∈ R N +1 . The kernel matrix Γ ∈ R ( N +1) × ( N +1) is given byΓ ij = G ( | t i − − t j − | ) , i, j = 1 , , . . . , N + 1 , and for θ ≥ θ := Γ + 2 θI , and the matrix e Γ is given by e Γ ij = Γ ij if i > j G (0) if i = j, θ and the decay kernel G . Generically, followingSchied and Zhang (2018), we say that a market is unstable if the trading strategies at the Nashequilibrium exhibit spurious oscillations, i.e., if there exists a sequence of trading times suchthat the orders are consecutively composed by buy and sell trades. In the optimal executionliterature such behavior is termed transaction triggered price manipulation , see Alfonsi et al.(2012). Figure 1 shows the simulation of the price process under the Schied and Zhang modelwhen both investors have an inventory equal to 1 for two values of θ . The unaffected priceprocess is a simple random walk with zero drift and constant volatility and the trading of thetwo agents, according to the Nash equilibrium, modifies the price path. For small θ (top panel)the affected price process exhibits wild oscillations, while when θ is large (bottom panel) theirregular behavior disappears .To clarify better our results, we introduce two definitions of market stability: Definition 2.3 (Strong Stability) . The market is strongly (uniformly) stable if ∀ θ the NashEquilibrium ( ξ ∗ , η ∗ ) ∈ X ( X , T ) × X ( Y , T ) does not exhibit spurious oscillations ∀ X , Y ∈ R . Definition 2.4 (Weak Stability) . The market is weakly stable if there exists an interval I ⊆ R such that ∀ θ ∈ I the Nash Equilibrium ( ξ ∗ , η ∗ ) ∈ X ( X , T ) × X ( Y , T ) does not exhibitspurious oscillations ∀ X , Y ∈ R .Schied and Zhang (2018) showed that when the trading time grid is equispaced, T N , andunder general assumptions on G , the market is not strongly but only weakly stable where I ,the stability region, is equal to [ θ ∗ , + ∞ ) where θ ∗ = G (0) / . Thus, they showed the existenceof a critical value of θ such that for smaller values of this threshold the equilibrium strategiesexhibit oscillations of buy and sell orders for both traders. Hence, the behavior at zero of thekernel function plays a relevant role for the equilibrium stability.As mentioned in the introduction, this result has been proved for a market with only M = 1asset, two homogeneous traders (same θ and same trading speed), and J = 2 agents. In thiswork we extend this stability result by first extending the framework of Schied and Zhang (2018) Moreover, we observe that the presence of spurious oscillations in the price dynamics may affect the consis-tency of the spot volatility estimation. Indeed, these oscillations act as a market microstructure noise, even ifthis noise is caused by the oscillations of a deterministic trend, while usually it is characterized by some additivenoise term. In particular, we find that when θ is close to zero the noise is amplified by spurious oscillations, whilefor sufficiently large θ these oscillations do not compromise the consistency of the spot volatility. 10 20 30 40 50 60 70 80 90 100 Time P r i c e =0.01 Time P r i c e =1.5 Figure 1: Blue lines exhibit the price process when both agents have inventory equals to 1. Thetop (bottom) panel shows the dynamics when θ = 0 . 01 ( θ = 1 . N + 1 = 51 points, G ( t ) = exp( − t ), the volatility of the unaffected price process is fixed to 1and S = 100. The vertical grey dotted lines delineates the trading session. The yellow linesshows the drift dynamics due to trading.in the multi-asset ( M > 1) case and solving the related Nash equilibrium. We then present anextension of the framework in a multi-agent market ( J > 3. Multi-asset market impact games As a first extension of the basic setting, we consider the case of two agents trading a portfolioof M > T and inventory Z ∈ R M , the matrix Ξ = (cid:16) ξ , ξ , . . . , ξ N (cid:17) ∈ R M × ( N +1) of random variables is an admissible strategy if (i) ξ k ∈ R M is F t k -measurable andbounded ∀ k and (ii) Z = P Nk =0 ξ k . Consistently with the previous notation, for each tradingtime ξ k and η k are M -vectors containing the order flows for the M -assets of the two agents X and Y , i.e. η k,i is the order flow of the first agent when trading asset i ar time t k . Finally, X , Y ∈ R M are the initial inventories of the two agents.The second important point is that the trading of one asset modifies also the price of theother asset(s). This effect is termed cross-impact . While self-impact may be attributed to amechanical and induced consequence of the order book, the cross-impact may be understood asan effect related to mispricing in correlated assets which are exploited by arbitrageurs bettingon a reversion to normality, see Almgren and Chriss (2001) and Schneider and Lillo (2019) forfurther details. Cross-impact has been empirically studied recently, see e.g. Mastromatteo et al.(2017), Schneider and Lillo (2019) and its role in optimal execution has been highlighted inTsoukalas et al. (2019).Mathematically cross-impact is modeled by introducing a function e Q : R M × R + → R M de-scribing how the trading of the M assets affect their prices at a certain future time. Schneider and Lillo(2019) have discussed necessary conditions for the absence of price manipulation for multi-assettransient impact models. They have shown that the cross-impact function need to be symmetricand linear in order to avoid arbitrage and manipulations. Moreover, as empirically observed byMastromatteo et al. (2017), we assume the same temporal dependence of G among the assets.Then, we assume that e Q = Q · G ( t ) where Q is linear and symmetric, i.e., Q ∈ R M × M and Q = Q T and G : R + → R . Also, we assume that Q is a nonsingular matrix. Therefore, theprice process during order execution is defined as S Ξ ,Ht = S t − X t k 2) rvs which are independent of σ ( ∪ t F t ) and mutually independent. The cost of Ξ given H is defined as C T (Ξ | H ) = h X , S i + N X k =0 (cid:18) G (0)2 h Q ξ k , ξ k i − h S Ξ ,Ht k , ξ k i + G (0) h Q ( ε k ⊗ ξ k ) , η k i + θ h ξ k , ξ k i (cid:19) , ε k ⊗ ξ k := (cid:16) ε k, · ξ k, , · · · , ε k,M · ξ k,M (cid:17) T is the Hadamard product, which means foreach trading time k and each asset i the time priority is decided by a Bernoulli game. The costof H given Ξ is defined as C T ( H | Ξ) = h Y , S i + N X k =0 (cid:18) G (0)2 h Q η k , η k i − h S Ξ ,Ht k , η k i + G (0) h Q ( ε k ⊗ ξ k ) , η k i + θ h η k , η k i (cid:19) , where ε k := (1 − ε k ).The previous definition is motivated by the following argument. When only X trades, theprices are moved from S Ξ ,Ht k to S Ξ ,Ht k + = S Ξ ,Ht k − G (0) Q ξ k . However, the order is executed at theaverage price and the player incurs in the expenses − h ( S Ξ ,Ht k + S Ξ ,Ht k + ) , ξ k i = G (0)2 h Q ξ k , ξ k i − h S Ξ ,Ht k , ξ k i . Then, Y trades immediately after X and the prices are moved linearly from S Ξ ,Ht k + to S Ξ ,Ht k + − G (0) Q η k , so the cost for Y is given by: − h ( S Ξ ,Ht k + + S Ξ ,Ht k + ) − G (0) Q η k , η k i = G (0)2 h Q η k , η k i − h S Ξ ,Ht k , η k i + G (0) h Q ξ k , η k i . We derive a compact formula for the expected costs which will be used for deriving theequation determining the Nash equilibrium. Specifically, we prove the following lemma: Lemma 3.2. E [ C T (Ξ | H )] = E [ tr (Ξ e Γ( Q Ξ) T ) + tr (Ξ e Γ( QH ) T ) + θ tr (Ξ T Ξ)]All the proofs are given in Appendix A. We now prove the existence and uniqueness of the Nash equilibrium in this multi-asset setting.This is easily achieved by using the spectral decomposition of Q to orthogonalize the assets,which we call “virtual” assets, so that the impact of the orthogonalized strategies on the virtualassets is fully characterized by the self-impact, i.e., the transformed cross impact matrix isdiagonal. Thus, the existence and uniqueness of the Nash equilibrium derives immediately byfollowing the same argument as in Schied and Zhang (2018).The notion of Nash equilibrium is generalized as follows. Definition 3.3 (Nash Equilibrium) . Given the time grid T and initial values X , Y ∈ R M ,a Nash Equilibrium is a pair of matrices (Ξ ∗ , H ∗ ) of strategies in X ( X , T ) × X ( Y , T ) suchthat: E [ C T (Ξ ∗ | H ∗ )] = min Ξ ∈ X ( X , T ) E [ C T (Ξ | H ∗ )] and E [ C T ( H ∗ | Ξ ∗ )] = min H ∈ X ( Y , T ) E [ C T ( H | Ξ ∗ )] . emark . If we suppose that Q is the identity matrix, then the multi-asset market impactgame is a straightforward generalization of the Schied and Zhang (2018) model. Indeed, eachorder of the two players for the i -th stock does not affect any other asset. So, the Nash-equilibrium can be easily found by solving M Schied and Zhang (2018) models.Let us consider the spectral decomposition of Q , i.e., QV = V D , where V is the eigenvectorsmatrix and D the diagonal matrix which contains the respective eigenvalues. Since we assumethat Q is a non singular symmetric matrix, then D is diagonal with all elements different fromzero. We define the prices of the virtual assets as P t := V T S Ξ ,Ht and we observe that P t = P t − X t k 10e remark that an Arbitrageur is a particular case of a Market Neutral agent in the limit casewhen the volume to trade in each asset is zero. Clearly in a single-asset market we have onlytwo types of the previous agents, since a Market Neutral strategy requires at least two assets. The presence of multiple assets and of cross impact can affect the trading strategy of an agentinterested in liquidating only one asset. In particular, as we show below, it might be convenientfor such an agent to trade (with zero inventory) the other asset(s) in order to reduce transactioncosts.To prove this we focus on the two-asset case, M = 2, and we analyse the Nash equilibriumwhen the kernel function has an exponential decay , G ( t ) = e − t . The first trader is a Funda-mentalist who has to liquidate the position in the first asset, i.e., X = 1, while the secondagent is an Arbitrageur, i.e., Y = 0. We set an equidistant trading time grid with 26 pointsand θ = 1 . 5. The second asset is available for trading, but let us consider as a benchmark casewhen both agents trade only the first asset. This is a Schied and Zhang (2018) game. Figure 2exhibits the Nash Equilibrium for the two players. We observe that the optimal solution for theFundamentalist is very close to the classical U-shape derived under the Transient Impact Model(TIM) , i.e. our model when only one agent is present. However, the solution is asymmetricand it is more convenient for the Fundamentalist to trade more in the last period of trading.This can be motivated by observing that at equilibrium the Arbitrageur places buy order at theend of the trading day, and thus she pushes up the price. Then, the Fundamentalist exploitsthis impact to liquidate more orders at the end of the trading session. We remark that theArbitrageur earns at equilibrium, since her expected cost is negative (see the caption).Now we examine the previous situation when the two traders solve the optimal executionproblem taking into account the possibility of trading the other asset. We define the cross im-pact matrix Q = " qq , where q = 0 . 6. In Figure 3 we report the optimal solution where theinventory of the agents are set to be X = (cid:16) (cid:17) T and Y = (cid:16) (cid:17) T . The Fundamentalistwants to liquidate only one asset, but, as clear from the Nash equilibrium, the cross-impactinfluences the optimal strategies in such a way that it is optimal for him/her to trade also theother asset. In terms of cost, for the Fundamentalist trading the two assets is worse off than inthe benchmark case (see the values of E [ C T (Ξ ∗ | H ∗ )] in captions). However, if the Fundamen-talist trades only asset 1 and Arbitrageur trades both assets, the former has a cost of 0 . All our numerical experiments are performed with exponential kernel as in (Obizhaeva and Wang (2013)).Schied and Zhang shows that the form of the kernel does not play a key role for stability, given that the conditionsgiven above are satisfied. Given the initial inventory X , the optimal strategy in the standard TIM is ξ = X e T Γ − θ e Γ − θ e , see for furtherdetails Schied and Zhang (2018). 10 20 k -0.15-0.1-0.0500.050.1 V o l u m e t r aded Fundamentalist =1.5 k -0.15-0.1-0.0500.050.1 V o l u m e t r aded Arbitrageur =1.5 Figure 2: Nash equilibrium Ξ ∗ of the Fundamentalist and H ∗ of the Arbitrageur trading onlyone asset. The trading time grid is equidistant with 26 points and θ = 1 . 5. The expected costsare equal to E [ C T (Ξ ∗ | H ∗ )] = 0 . E [ C T ( H ∗ | Ξ ∗ )] = − . F und a m e n t a li s t . , − . . , − . . , − . ( . , − . T and (0 0) T , respectively. We have highlighted in red the Nash Equilibriumassociated with this payoff matrix .Fundamentalist and Arbitrageur and it is clear that both agents prefer to trade both assets.Actually, the state where both agents trade two assets is the Nash equilibrium of the gamewhere each agent can choose how many assets to trade.The solution presented above is generic, but an important role is played by the transactioncost modeled by the temporary impact. When the temporary impact parameter θ increases,the benefit of the cross-impact vanishes, and the optimal strategy of the Fundamentalist tendsto the solution provided by the simple TIM with one asset and no other agent. We find thatthe difference between these expected costs is negative, i.e. it is always optimal to trade alsothe second asset, but converges to zero for large θ . 4. Market instability and cross impact structure In this Section we study whether the increase of the number of assets and the structure of crossimpact matrix help avoiding oscillations and market instability at equilibrium according to the12 10 20 k -0.1-0.0500.050.1 V o l u m e t r a d e d on asse t Fundamentalist k -0.1-0.0500.050.1 Arbitrageur k -0.1-0.0500.050.1 V o l u m e t r a d e d on asse t k -0.1-0.0500.050.1 Figure 3: Optimal strategies for a Fundamentalist (Ξ ∗ ) and an Arbitrageur ( H ∗ ), where theirinventories are equal to (1 0) T and (0 0) T , respectively. Q = (cid:20) . . (cid:21) , and the trading timegrid is an equidistant time grid with 26 points. The expected costs are equal E [ C T (Ξ ∗ | H ∗ )] =0 . E [ C T ( H ∗ | Ξ ∗ )] = − . θ = 1 . Q describing the complexity of the market for what concerns commonality in liquidity.We first show that instabilities are generically observed also in the multi-asset case and thatactually more assets make the market less stable. For simplicity let us consider M = 2 assetsand a game between a Fundamentalist and an Arbitrageur (similar results hold for differentcombinations of agents). We choose G ( t ) = e − t , the cross impact equal to Q = " . . , andwe consider θ = 0 . 25; remember that for the one asset case the market is stable for this value of θ . Figure 4 shows that for this value of θ the strategies are oscillating and therefore the marketis not strongly stable. More surprisingly, the fact that oscillations are observed for θ = 0 . θ to ensure stability. In the following weprove that this is the case and we determine the threshold value.Figure 4 shows also the case θ = 0. Notably, in this case the oscillations in the secondasset disappear. This is due to the fact that, since Γ , (Γ ), the Γ matrix associated with thefirst (second) virtual asset is equal to (1 + q )Γ, ((1 − q )Γ), the combination of “fundamental”solutions v and w are the same for the two virtual assets. Thus, at the equilibrium the twosolutions for the second asset are exactly zero.13 20 40 k -0.6-0.4-0.200.20.4 V o l u m e t r a d e d on asse t Fundamentalist k -0.6-0.4-0.200.20.4 Arbitrageur k -0.15-0.1-0.0500.050.1 V o l u m e t r a d e d on asse t k -0.15-0.1-0.0500.050.1 Figure 4: Nash Equilibrium for a Fundamentalist and an Arbitrageur, where their inventoriesare equal to (1 0) T and (0 0) T respectively. The blue lines are the optimal solution when θ = 0and the red lines when θ = 0 . 25. The trading time has 51 points and Q = (cid:20) . . (cid:21) .Finally, it is worth noting that, if S = P k | ξ k, | denotes the total absolute volume tradedby the Fundamentalist on the second asset, then lim θ → S = 0 and lim θ →∞ S = 0 as exhibitedfrom Figure 5. This means, that when the cost of trades increases it is not anymore convenientfor both traders to try to exploit the cross impact effect as observed in Section 3. C u m u l a t i v e V o l u m e Log( ) C u m u l a t i v e V o l u m e Figure 5: Cumulative traded volume of the second asset by the Fundamentalist when playingagainst an Arbitrageur as a function of θ . The inset shows the same curve in semi-log scale.14e have shown in a simple setting that having more than one available asset does not helpimproving the strong stability of the market and increases the threshold value between stableand unstable markets. We show that when the number of assets tends to infinity the marketdoes not satisfy the weak stability condition.In the one asset setting, if we choose a sufficiently large θ the instability vanishes. Therefore,this raises the question whether the equilibrium instability is still present when the number ofassets increases. To this end we introduce the definition of asymptotic stability. Definition 4.1 (Asymptotically weakly stable) . The market is asymptotically weakly stable ifit is weakly stable when M → ∞ . Given this definition, we prove the following: Theorem 4.2 (Asymptotic Instability) . Suppose that G is a continuous, positive definite,strictly positive, log-convex decay kernel and that the time grid is equidistant. Let ( λ i ) i =1 ,..,M bethe spectrum of the cross-impact matrix Q . The market is unstable if θ < θ ∗ where θ ∗ = max i =1 , ,...,M G (0) · λ i . (3) Moreover, if the largest eigenvalue of the cross-impact matrix diverges for M → ∞ , i.e. lim M → + ∞ λ max = + ∞ , then the market is not asymptotically weakly stable. The theorem tells that the instability of the market is related to the spectral decompositionof the cross-impact matrix, i.e. to the liquidity factors. We analyze some realistic cross-impactmatrices and their implications for the stability of the Nash equilibrium. Schneider and Lillo(2019) have derived constraints on the structure of the cross-impact for the absence of dynamicarbitrage. They showed that the symmetry of the cross-impact matrix is one of these conditions.Mastromatteo et al. (2017) estimated the cross-impact matrix on 150 US stocks showing that itis roughly symmetric and has a block structure with blocks related to economic sectors. There-fore, we focus on symmetric positive definite cross-impact matrix. The positive definiteness isrequired in order to have a positive kernel on the virtual assets, recall Subsection 3.1 where itis shown that for each virtual assets i the decay kernel is given by G ( t − t k ) · λ i , for all tradingtimes t k , where λ i is the i -th eigenvalue of Q . We consider the following cases: • One Factor Matrix . We say that Q is a one factor matrix if Q = (1 − q ) I + q · ee T , where e = (1 , . . . , T ∈ R M and q ∈ (0 , q guarantee the positive definitenessof the cross-impact matrix. Then it holds: Corollary 4.3. If the cross-impact matrix is a one factor matrix, then the market is notasymptotically weakly stable. This implies that when M increases the transactions cost θ must raise in order to preventmarket instability, since θ ∗ = G (0) λ max / ∼ G (0) qM/ 4, because λ max = 1 + q ( M − θ = 1 . q = 0 . M = 2000. The inventory of the Fundamentalist is 1 for the first 1000 assetsand zero for the others. The solutions clearly show spurious oscillations of buy and sell15igure 6: Nash equilibrium when θ = 1 . X = (1 , . . . , , , . . . , T ∈ R M and Y = (0 , . . . , T ∈ R M , where M = 2 , , 000 assets, while the green ones arethose for any of the last 1 , 000 assets.orders. Notice that in the one asset case this value of θ gives a stable market. We observethat the eigenvector corresponding to λ max is given by e , which represents an equallyweighted portfolio. As a consequence, if we consider a Market Neutral agent against anArbitrageur the solution becomes stable ∀ θ > (1 − q ) / 4, since both traders have zeroinventory on the first virtual asset. Thus, oscillations might disappear when the inventoryof the agents in the first virtual asset is zero.A generalization of the above model considers Q as a rank-one modification matrix, i.e. Q = D + ββ T , where D = diag(1 − β , . . . , − β M ) and β ∈ R M is a fixed vector. In thisway the cross impact is not the same across all pairs of stocks. We find again that themarket is not asymptotically stable because the threshold increases with M . Differentlyfrom the previous case this is observed also in the case of a Market Neutral against anArbitrageur. • Block Matrix . We now assume that the cross impact matrix has a block structure in sucha way that cross impact between two stocks in the same block i is q i , while when the twostocks are in different blocks the cross impact is q . As mentioned above, this is consistentwith the empirical evidence in Mastromatteo et al. (2017).Let us denote with M i the number of stocks in block i , ( i = 1 , . . . K ), and let Q i =(1 − q i ) I + q i · e i e Ti ∈ R M i × R M i with q i ∈ (0 , 1) and e i = (1 , . . . , T ∈ R M i , where K is16he number of blocks. We define the cross impact matrix as: Q := Q q e e T · · · q e e TK q e e T Q · · · q e e TK ... . . . ... q e K e T · · · q e K e TK − Q K , If the average number of stocks of a cluster tends to infinity when M goes to infinity, weprove an analogue result as for the one factor matrix case: Corollary 4.4. If Q is a block matrix, where each block is a one factor matrix, if lim M → + ∞ MK → + ∞ , then the market is not asymptotically weakly stable. k -1-0.8-0.6-0.4-0.200.20.40.60.81 V o l u m e t r aded Market Neutral k -1-0.8-0.6-0.4-0.200.20.40.60.81 V o l u m e t r aded Arbitrageur Figure 7: Nash equilibrium when θ = 1 . X =(1 , . . . , , − , . . . , − T ∈ R M and for the Arbitrageur Y = (0 , . . . , T ∈ R M , where M = 2000.The cross impact matrix is a block matrix with K = 10. The figure exhibits the equilibriarelated to one (the first) asset for each block. The trading time grid is an equidistant time gridwith 26 points. Each block has a cross-impact q i equal to 0 . , . , . . . , . i = 1 , , . . . , . 95 for the last one.As an example, we consider K = 10 equally sized blocks from an universe M = 2 , 000 assetsand set q = 0 . 1. With this kind of cross impact matrix, we have K large eigenvalues whoseeigenvectors correspond to virtual assets displaying oscillations. The optimal trading strategiesfor stocks belonging to the same block are the same. Thus in Figure 7 we show the Nashequilibrium for the first asset in each of the 10 blocks when the two agents are a Market Neutraland an Arbitrageur. The oscillations are evident, as expected, in all traded assets.We now study how the critical value θ ∗ varies when the number of assets increases fordifferent structures of the cross impact matrix and therefore of the liquidity factors. Comparing17ifferent matrix structures is not straightforward since the critical value depends on the valuesof the matrix elements. To this end we consider the set of symmetric cross impact matrices of M assets having one on the diagonal and fixed sum of the off diagonal elements. More preciselylet h ∈ R , then we introduce for each M the set A Mh := { A ∈ R M × M | A T = A, X i>j a ij = h, a ii = 1 } , One important element of this set is the cross impact matrix Q fac ∈ R M × M of a one factormodel (see above) with off-diagonal elements equal to 2 h/M ( M − Theorem 4.5. For a fixed h ∈ R , let us consider the related one-factor matrix Q fac ∈ A Mh ,then λ ( Q ) ≥ λ ( Q fac ) , ∀ Q ∈ A Mh , i.e. among all the matrices with one in the diagonal and constant sum of the off-diagonal terms,the one-factor matrix (i.e. where all the off-diagonal elements are equal) is one with the smallestlargest eigenvalue. Moreover, we prove in the last part of Appendix A that the previous is not a strict inequality,by showing that both a diagonal block matrix, with identical blocks, and the one-factor matrixhave the same maximum eigenvalue.This theorem implies that among all the cross impact matrices belonging to A Mh , the onefactor case is among the most stable cross-impact matrices. For example, it is direct to constructan example of a block diagonal cross impact matrix with non-zero off block elements (i.e. similarto what observed empirically) and to prove that its critical θ ∗ is larger than the critical valuefor the one factor matrix having the same value h of total cross-impact. 5. Instability in market impact games with many agents We now consider J > M = 1) in order to study howthe stability of the market is influenced by an increasing competition. This model representsanother important generalization of Schied and Zhang (2018) which considered only the case J = 2.The unaffected price process S t is always assumed to be a right continuous martingale in asuitable probability space. We denote with ξ jk the order flow of agent j at time t k , so that theaffected price process is defined as S ξ t := S t − X t k Lemma 5.1. E [ C T ( ξ j | ξ − j )] = E [ 12 ( ξ j ) T Γ θ ξ j ] + N X k =0 ξ jk J X i =2 X l = j ( i − J · ( J − ξ lk + k − X m =0 G ( t m − t k ) X l = j ξ lm . We now analyse numerically the Nash equilibrium emerging from the interaction of differenttypes of agent. Unless otherwise specified, we fix θ = 10 and N = 15. Moreover we introducea scaling parameter β ≥ J increases, i.e., G ( t ) = J − β e − t . Thecase when β = 0 corresponds to the additive case, while for β = 1 the total instantaneousimpact does not depend on the number of agents. Hence, when β = 1 the aggregate impact ofall the agents is independent from J and we want to test whether the trading patterns are alsoindependent from it. Recent empirical evidences (Bucci et al. (2020)) show that market impactdepends on the aggregate net order flow of the optimal executions simultaneously present,without individually distinguishing them.To this end, we first study an homogeneous market with J identical Fundamentalist sellerswith the same initial inventory. In Figure 8 we report the Nash equilibria for the agents when β = 0 , / , J . We remark that for each β the initial inventory X j ,19 k V o l u m e t r aded =0 J=2J=3J=5J=10 k V o l u m e t r aded =0.5 J=2J=3J=5J=10 k V o l u m e t r aded =1.0 J=2J=3J=5J=10 Figure 8: Optimal solution of a fundamental seller for different number J of agents and when the scaling parameter β is set to 0 , / , 1. The timegrid contains 16 points. Number of Agents * =0=0.5=1 Figure 9: Numerical estimates of the critical value of θ as a function of the number of agentswhen the time grid is fixed with 7 points for some scaling parameters. All agents are identicalFundamentalist sellers.for all agent j , is rescaled by J − β , and then, for improve readability, the displayed solutions arenormalized such that the sum of orders is equal to 1.When the competition in the market increases, regardless of β , the optimal solution tendsto place more orders at the beginning of the trading sessions, instead of waiting the last tradingtime. This synchronized trading of all the agents at the beginning of the trading period, asresulting from the Nash equilibrium, could be at the origin of the market instability. On theother hand, when β increases the equilibrium seems to be approximated by a flat straight linewith a constant trading volume over time. In any case, it is important to notice that when β = 1the Nash equilibrium is not independent from the number of agents J . Thus, for example, theequilibrium of J = 2 agents is not the same as the equilibrium of J = 4 agents with half theimpact and half the initial inventory .We now consider, in the same setting as above, how the stability of the market depends onthe number of agents and on the scaling parameter β . Specifically, we compute numerically thecritical value of θ after which the market is not stable. We choose a time grid equal to N = 6and we study the three cases when β = 0 , / , θ ∗ (see Figure 9). For β = 0 the points lieapproximately on a line and the addition of an agent increases θ ∗ by approximately 0 . 4. On theother hand, the instability seems to be relaxed when the scaling parameter increases, where therelation with J becomes flatter. Notice however that when β = 1, i.e. the total impact of agents This analysis assumes that the initial inventory is the same for the agents. In Appendix B we explore thecase when this assumption is relaxed. k -0.1-0.0500.050.10.150.2 V o l u m e t r aded Fund-SellerFund-SellerArbitr. k -0.2-0.15-0.1-0.0500.050.10.150.2 V o l u m e t r aded Fund-SellerFund-BuyerArbitr. k -202 V o l u m e t r aded -4 Figure 10: Blue and red lines are the Nash equilibrium for the Fundamentalist traders. Theyellow line refers to the equilibrium of the Arbitrageur. The scaling parameter β is set to 0.becomes approximately independent from J , θ ∗ → . J increases, and this asymptoticvalue is larger than the one for two agents which is θ ∗ = 0 . / 2. We conclude that the increaseof the competition in a market may deteriorates its stability and, less surprisingly, it increasesthe expected costs of a strategy.Now we consider the cases when the agent are of different types. In particular, we focus onthe role of an Arbitrageur when two other agents are either identical or of opposite type.First, we consider the case of two identical Fundamentalist sellers and one Arbitrageur( J = 3) and we fix θ = 1 . 5. Figure 10 (left) displays the equilibrium solution for β = 0, to becompared with the two agent case of Fig. 2. While the trading pattern of the Arbitrageur isqualitatively similar to the one of the two agent case, the Fundamentalists trade significantly lesstoward the end of the day. This is likely due to the fact that it might be costly to trade for oneFundamentalist given the presence of the other. The expected costs for the two Fundamentalistsis equal to 0 . − . . 13 and − . · − for the twoFundamentalists and the Arbitrageur, respectively. This indicates that the two Fundamentalistsare able to reduce significantly their costs with respect to the previous case, increasing theirprotection against predatory trading strategies and that the Arbitrageur is not able to act as amarket maker. On the other hand, we observe (see the inset of Fig. 10) that the solution forthe Arbitrageur starts to exhibit some oscillations of purchase and sale, although his/her ordersremains of negligible size with respect to that of Fundamentalists. 6. Instability and heterogeneity of agents’ trading skills Here we study the case of two agents trading one asset, but, differently from Schied and Zhang(2018), we assume that the two agents are different in their trading skills. We consider separatelythree sources of heterogeneity: different permanent impact, different temporary impact (ortrading fees), and different trading speed.In the first setting, we study the case when the two agents affect with a different impact22he price process. This might correspond to more sophisticated traders who are able to trademore efficiently, for example using better algorithms for posting orders in the market. There isan alternative interpretation, more in line with traditional market microstructure. Permanentimpact is considered a measure of the informativeness of trades, thus an agent with a very smallpermanent impact might be interpreted as a noise trader. In our model the two interpretationsare indistinguishable. For analytical convenience, we assume the kernel G ( t ) is the same for thetwo agents and we introduce two scaling parameters β i ( i = 1 , 2) such that the price processdynamic affected by trading becomes S ξ , η t = S t − N X k =0 G ( t − t k )( J − β ξ k + J − β η k ) , (4)where J = 2. We observe that when two agents have different β ’s this does not compromise themarket stability. Indeed, the previous equation can be rewritten as S ξ , η t = S t − N X k =0 G ( t − t k )( e ξ k + e η k ) , (5)where e ξ k , e η k are the strategies associated with the transformed inventories e X = J − β X , e Y = J − β Y . Therefore, the Nash equilibrium of equation 4 is related to the equilibrium ofequation 5, i.e., the are equal up to constants, and in particular the stability is not compromised.We just mention that if, for instance, β = 0 and β = 1, i.e. the impact of the second agentis half the one of the first agent, and both agents are Fundamentalist sellers, the sophisticated(or uninformed) agent Y has the advantage as if he/she should trade half of his volume. Figure11 exhibits the Nash equilibrium for both traders. As expected, the optimal trading patternis different for the two traders. More interestingly, the trading profile of the agent with smallimpact resembles the one of an Arbitrageur (see for comparison Fig. 2). Thus, from the pointof view of the trading profile, an Arbitrageur with zero inventory or a Fundamental trader withsmall impact behaves in a similar way at equilibrium. Remember that the latter can be seeneither as a skilled trader or as a noise trader. Moreover, the expected cost for the β = 0 and β = 1 sellers are equal to 0 . . β = 0, thecosts are equal to 0 . θ . This might correspond to the again to the case where one of the two agents has moresophisticated trading strategy or to a market where different class of traders are allowed to paydifferent fees. This is indeed quite standard for example in ATS where liquidity providers havelower fees than liquidity takers. In particular, we want to test whether only one agent with θ below threshold is sufficient to destabilize the market or if both agents must have θ < θ ∗ .We denote with θ X and θ Y the temporary impact parameter for the two agents. We studythe Nash equilibrium by solving numerically the Schied and Zhang (2018) game in the setting Note that this is equivalent to saying that the G ( t ) of the two agents are proportional one to each other, i.e.the two agents differ in the value of G (0). 10 20 k -0.0500.050.10.150.20.25 V o l u m e t r aded =0 k -0.0500.050.10.150.20.25 V o l u m e t r aded =1 Figure 11: Nash equilibrium for two Fundamentalist sellers when θ = 1 . 5. The left (right) figureis referred to the agent with β = 0 ( β = 1). The trading time grid is an equidistant grid with26 points and the decay kernel is G ( t ) = exp( − t ).where both agents are Fundamentalist seller with X = Y = 1. When both agents are abovethreshold ( θ ∗ < θ Y < θ X = 2 θ Y , see Figure 12), we observe that the more significant differencebetween the two solutions is in the trading schedule of low θ agent, who trades more at thebeginning of the trading session. It is interesting to consider the changes in cost. When θ = 1 forboth agents, the individual cost is 0 . θ Y is reduced to 0 . Y is reduced (0 . X who sees her cost soar to 0 . θ for a class of agents impacts negatively the other market participants.When we examine the case when one of the agents has a θ Y smaller than the critical value θ ∗ = 0 . 25, we find numerical evidences that the equilibrium becomes unstable also for the Fun-damentalist agent X for which θ X > θ ∗ . This suggests that in a market with few sophisticatedtraders with small temporary impact (HFTs) or agents allowed to pay lower fees, the instabilitycomes up for all the agents in the market, regardless for their own θ s. In other words, marketinstability is driven by the agent with the smallest θ and having even few sophisticated traders(in terms of costs or fees) destabilizes the market.The third and last setting assumes that the two traders are different because one of the twois faster than the other. A fast trader could represent, for example, an HFT. In market impactgames one has to decides who arrives first in the market at each time step, because the laggardis going to pay the impact of the trade of the leader. In the standard setting, this is decided bya fair Bernoulli trial, which means that on average each trader is the leader half of the times.We modify this and for each trading time t k the Bernoulli game ε k deciding the trading priority,24 k V o l u m e t r aded Fundamentalist X =1=1 k V o l u m e t r aded Fundamentalist Y =0.5=1 Figure 12: Red lines are the Nash equilibrium when both traders have θ = 1. Blue lines arethe solution when X has θ = 1 and Y has θ = 0 . 5. The trading time grid is an equidistant gridwith 16 points and the decay kernel is G ( t ) = exp( − t ).is no longer fair, i.e., ε k = , with probability p , with probability 1 − p p ∈ [0 , ε k = 1 means that Y places the k -th order before X , so that X has to take into accountthe impact of Y in her cost. Let us denote with ξ ( η ) the Nash equilibrium for X ( Y ), then weobserve that E [ C T ( ξ | η )] = E [ ξ T Γ θ ξ + ξ T Γ p η ] and E [ C T ( η | ξ )] = E [ η T Γ θ η + η T Γ − p ξ ], since N X k =0 ξ k pη k + k − X m =0 η m G ( t k − t m ) ! = ξ T Γ p η where (Γ p ) ij = e Γ ij , i = jp · G (0) , i = j . E [ C T ( ξ | η )] E [ C T ( η | ξ )] p X and Y as a function of the trading speed p (probability of arriving first on the market) of Y . When p = 1 the agent Y places orders alwaysbefore of X . 25 k V o l u m e t r aded Fundamentalist X p=1/2p=2/3p=3/4p=4/5p=1 k V o l u m e t r aded Fundamentalist Y p=1/2p=2/3p=3/4p=4/5p=1 Figure 13: Nash equilibrium in function of the trading speed p (probability of arriving first onthe market) of Y . The red dotted lines are the Nash equilibrium when both traders have thesame trading speed, i.e., p = 0 . 5. The time grid has 16 points, θ = 1 and G ( t ) = exp( − t ).Figure 13 exhibits the Nash equilibrium of the two traders for several values of p and Table2 reports the corresponding expected costs. We observe that when p increases, the optimalsolution for the HFT Y is to liquidate slightly faster in the first period and then sell theremaining part of the inventory with a lower intensity, with respect to the solution with p = 0 . X the behavior is exactly opposite to theone of the fast trader. However the differences between Nash equilibria for different p are quitesmall, while the greatest benefit for the HFT is the smaller trading costs (see Table 2).Finally, we ask whether the presence of a fast trader modifies the critical value of θ whenthe instability starts. Figure 14 shows how θ ∗ varies as a function of the trading speed p fora games between two Fundamentalist sellers . Since we compute the critical value using anumerical method there are some small oscillations. We observe a small but significant trendof θ ∗ as a function of p . This result indicates that the presence of an HFT makes the marketmore prone to oscillations and instabilities. However the effect is relatively small compared tothe other possible sources of instabilities studied in the previous sections. 7. Conclusions In this paper we used market impact games to investigate several potential determinants ofmarket instabilities driven by finite liquidity and simultaneous trade execution of more agents.Specifically, we extended the results of Schied and Zhang (2018) in several directions. We firstconsidered a multi-asset market where we introduced the cross-impact effect among assets.We solve the Nash equilibrium, we analysed the optimal solution provided by the equilibrium,and we studied the impact of transaction costs on liquidation strategies. Secondly, we studied We also repeat the same experiment in the presence of an Arbitrageur against a Fundamentalist and we findanalogous results. .55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 p * Figure 14: Numerical estimates of the critical value θ ∗ as a function of the trading speed p (probability of arriving first on the market) of the HFT. We set N = 6 for the time grid and G ( t ) = exp( − t ). Both agents are Fundamentalist sellers.the stability of the market when the number of assets increases and we found that for mostrealistic cross-impact structures the market is intrinsically unstable. Even if asymptotically theinstability arises in all cases, we found that when the structure of the cross-impact matrix iscomplex, for example it has a block or multi-factor structure, the instability transition occursfor higher values of the impact parameter. Thus, all else being equal, the temporary impact (orthe transaction fees) must be larger in order to observe stability. 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E [ C T (Ξ | H )] − h X , S i == E (cid:20) N X k =0 (cid:18) h Q ξ k , ξ k i − h S Ξ ,Ht k , ξ k i + h Q ( ε k ⊗ ξ k ) , η k i + θ h ξ k , ξ k i (cid:19) (cid:21) = E (cid:20) N X k =0 (cid:18) h Q ξ k , ξ k i + 12 h Q ξ k , η k i + − h S t k − k − X m =0 G ( t k − t m ) Q ( ξ m + η m ) , ξ k i + θ h ξ k , ξ k i (cid:19)(cid:21) = E (cid:20) − N X k =0 h ξ k , S t k i + 1 / N X k =0 h Q ξ k , ξ k i + N X k =0 h k − X m =0 G ( t k − t m ) Q ξ m , ξ k i ++ N X k =0 h / · Q η k + k − X m =0 G ( t k − t m ) Q η m , ξ k i + N X k =0 θ h ξ k , ξ k i (cid:21) , but E (cid:20) N X k =0 h ξ k , S t k i (cid:21) = E (cid:20) h N X k =0 ξ k , S T i (cid:21) = h X , S i , N X k =0 θ h ξ k , ξ k i = θ tr(Ξ T Ξ) , / N X k =0 h Q ξ k , ξ k i + N X k =0 h k − X m =0 G ( t k − t m ) Q ξ m , ξ k i = tr(Ξ e Γ( Q Ξ) T ) , N X k =0 h / · Q η k + k − X m =0 G ( t k − t m ) Q η m , ξ k i = tr(Ξ e Γ( QH ) T ) , and we conclude that E [ C T (Ξ | H )] = E [tr(Ξ e Γ( Q Ξ) T ) + tr(Ξ e Γ( QH ) T ) + θ tr(Ξ T Ξ)]. Proof of Theorem 4.2. We first observe that when we transform the real assets to the virtualones also the inventories of the traders are transformed. Indeed, if P t denotes the virtual assets, P t = P t − X t k The eigenvalues of Q are λ = 1 − q + qM and λ M = 1 − q , where v = e , the vector with all 1, is the virtual asset associated with λ . Then, when M → ∞ thefirst eigenvalue diverges so for Theorem 4.2 we conclude. Proof of Corollary 4.4. We first note that by Theorem 4.2 it is sufficient to prove that thereexists a cluster which is unbounded. Indeed, we observe that Q = b Q + q e e ... e K h e e · · · e K i where b Q = Q − q e e T · · · Q − q e e T · · · · · · Q K − q e K e TK . Then by Theorem 8.1.8 pag.443 of Golub and Van Loan (2013) λ ( Q ) ≥ λ ( b Q ) where λ i ( Q )denotes the i -th largest eigenvalue of Q and respectively of b Q . However, the eigenvalues of b Q are given by the eigenvalues of Q i − q e i e T i for i = 1 , , . . . , K. For each i , λ ( Q i − q e i e T i ) =1 − q i + M i ( q i − q ) and the rests M i − − q i . So, if there exists acluster such that M i is unbounded for any value of θ , then λ ( Q i − q e i e Ti ) is unbounded andalso the respective eigenvalue of Q , so by Theorem 4.2 we conclude that there is no a finitevalue for θ such that the market is weakly stable.So, let us first start by fixing the number of cluster to K < ∞ . Then, when M tendsto infinity at least one of the cluster will increase to infinity, which means that there exists acluster such that λ ( Q i − q e i e T i ) → ∞ and also the respective eigenvalue of Q goes to infinity.Therefore, we conclude for Theorem 4.2. 31or the general case we conclude by contradiction. If K ( M ) is the number of cluster for afixed M , and K ( M ) → ∞ when M → ∞ then the set { M i : i ∈ N } is unbounded. Indeed, ifsup i ∈ N M i = S < ∞ , then the average number of stocks in a cluster is P K ( M ) i =1 M i K ( M ) ≤ S for all M and this is in contradiction with the assumptions that lim M → + ∞ MK ( M ) → + ∞ . So since { M i : i ∈ N } is unbounded we conclude that there is no finite value of θ such that it is greaterthan all the eigenvalues of Q when M → ∞ . Note that the bound is not strict since the largest eigenvalue of a block diagonal matrix withidentical blocks is also 1 + hM . Indeed, Let consider the block diagonal matrix with K identicalclusters Q := Q ( ρ ) 0 · · · Q ( ρ ) · · · · · · Q ( ρ ) ∈ R M × M , where Q ( ρ ) ∈ R M c is a one-factor matrix and M c · K = M . We observe that Q ∈ A Mh if andonly if ρ = h ( M c − M , therefore λ ( Q ) = 1 + ( M c − ρ = 1 + 2 hM . Proof of Lemma 5.1. WLOG G (0) = 1. We observe that E [ { σ k ( l )
12 ( ξ jk ) + θ ( ξ jk ) + J X i =2 J X l =1 ,l = j { σ k ( l )
12 ( ξ jk ) + θ ( ξ jk ) + J X i =2 J X l =1 ,l = j { σ k ( l )
12 ( ξ j ) T Γ θ ξ j (cid:21) + N X k =0 ξ jk (cid:18) J X i =2 X l = j ( i − J · ( J − ξ lk + k − X m =0 G ( t m − t k ) X l = j ξ lm (cid:19) Appendix B. The role of initial inventories in multi-agent mar-ket impact games Here we analyse how the equilibrium solution is affected by the inventories of the agents. Weconsider J = 4 Fundamentalist sellers having different initial inventory. In all cases we fixthe total initial inventory to X + X + X + X = 4. As a benchmark case we consider thecase of equal inventories X i = 1 (see top left panel of Figure 15). When the inventory of oneseller is 1 / / / 10 incompetition with other three sellers with inventory equal to 13 / 10, the behavior of the formerbecomes more similar to that of an Arbitrageur (see bottom left panel of Fig. 15). In fact, theoptimal solution is to place all positive orders at the beginning and wait the end of the tradingsession to liquidate the excess volume, exploiting the position of the big Fundamentalists. Alsowe observe that in contrast to the J = 2 case the shape of the solution for big sellers is notaffected by the presence of the small one.To get some intuition on this result, we remind that in the two agents case Schied and Zhang(2018) showed that the solution is characterized by the sum and difference of the inventories,see Equation (1) and (2). Thus, if the two traders have the same inventory the solution isfully characterized by the vector v , which is scaled by the sum of the inventories. Figure 2 ofSchied and Zhang (2018) shows the shapes of v and w , and we note that the former is a vector33 k V o l u m e t r aded Fund-Seller X =1Fund-Seller X =1Fund-Seller X =1Fund-Seller X =1 k V o l u m e t r aded Fund-Seller X =1/2Fund-Seller X =7/6Fund-Seller X =7/6Fund-Seller X =7/6 k -0.04-0.0200.020.040.060.080.10.120.14 V o l u m e t r aded Fund-Seller X =1/10Fund-Seller X =13/10Fund-Seller X =13/10Fund-Seller X =13/10 k -0.0500.050.10.150.20.250.30.350.4 V o l u m e t r aded Fund-Seller X =37/10Fund-Seller X =1/10Fund-Seller X =1/10Fund-Seller X =1/10 Figure 15: Nash equilibrium for J = 4 Fundamentalist sellers for four different sets of initialinventory such that the total volume traded is 4. The time grid has 16 points, G ( t ) = exp( − t ),and θ = 10 . which assigns more trades at the beginning, while w tends to concentrate orders at the endof trading session. Let us suppose that X < X , then the solution for trader 1 is a linearcombination of v and w with opposite signs (thus positive at the beginning and negative atthe end), while for the second trader the solution is approximately given by a positive linearcombination of v and w (an asymmetric U-shape), so that its equilibrium’s shape converges toa U-shape. In particular, in the case of X << X , at the equilibrium, the first seller will placepositive orders at the beginning while at the end he/she will place opposite sign orders.Let us go back to the J = 4 case. We have observed how the presence of a single small sellerdoes not affect the shapes of the big ones. However, this may be related to market dominanceof the big sellers. Indeed, if we analyse the complementary case when there is only one bigseller with inventory equal to 37 / 10 against three small sellers whose inventories are all equal to1 /