A market impact game under transient price impact
AA market impact game under transient price impact
Alexander Schied ∗ Tao Zhang § Abstract
We consider a Nash equilibrium between two high-frequency traders in a simple marketimpact model with transient price impact and additional quadratic transaction costs. Extendinga result by Sch¨oneborn (2008), we prove existence and uniqueness of the Nash equilibrium andshow that for small transaction costs the high-frequency traders engage in a “hot-potato game”,in which the same asset position is sold back and forth. We then identify a critical value forthe size of the transaction costs above which all oscillations disappear and strategies becomebuy-only or sell-only. Numerical simulations show that for both traders the expected costscan be lower with transaction costs than without. Moreover, the costs can increase with thetrading frequency if there are no transaction costs, but decrease with the trading frequency iftransaction costs are sufficiently high. We argue that these effects occur due to the need ofprotection against predatory trading in the regime of low transaction costs.
Keywords:
Market impact game, high-frequency trading, Nash equilibrium, transient price impact,market impact, predatory trading, M -matrix, inverse-positive matrix, Kaluza sign criterion According to the Report [10] by CFTC and SEC on the Flash Crash of May 6, 2010, the events thatlead to the Flash Crash included a large sell order of E-Mini S&P 500 contracts:. . . a large Fundamental Seller (. . . ) initiated a program to sell a total of 75,000 E-Mini contracts (valued at approximately $4.1 billion). . . . [On another] occasion it tookmore than 5 hours for this large trader to execute the first 75,000 contracts of a largesell program. However, on May 6, when markets were already under stress, the SellAlgorithm chosen by the large Fundamental Seller to only target trading volume, andnot price nor time, executed the sell program extremely rapidly in just 20 minutes.The report [10] furthermore suggests that a “hot-potato game” between high-frequency traders(HFTs) created artificial trading volume that at least contributed to the acceleration of the Funda-mental Seller’s trading algorithm: ∗ Department of Statistics and Actuarial Science, University of Waterloo, and Department of Mathematics, Univer-sity of Mannheim, Email: [email protected] § Department of Mathematics, University of Mannheim, Email: [email protected]
The authors gratefully acknowledge financial support by Deutsche Forschungsgemeinschaft (DFG) through ResearchGrants SCHI/3-1 and SCHI/3-2. a r X i v : . [ q -f i n . T R ] M a y . . HFTs began to quickly buy and then resell contracts to each other—generating a“hot-potato” volume effect as the same positions were rapidly passed back and forth.Between 2:45:13 and 2:45:27, HFTs traded over 27,000 contracts, which accounted forabout 49 percent of the total trading volume, while buying only about 200 additionalcontracts net.See also Kirilenko, Kyle, Samadi, and Tuzun [17] and Easley, L´opez da Prado, and O’Hara [11] foradditional background.Sch¨oneborn [25] observed that the equilibrium strategies of two competing economic agents, whotrade sufficiently fast in a simple market impact model with exponential decay of price impact, canexhibit strong oscillations. These oscillations have a striking similarity with the “hot-potato game”mentioned in [10] and [17]. In each trading period, one agent sells a large asset position to theother agent and buys a similar position back in the next period. The intuitive reason for this hot-potato game is to protect against possible predatory trading by the other agent. Here, predatorytrading refers to the exploitation of the drift generated by the price impact of another agent. Forinstance, if the other agent is selling assets over a certain time interval, predatory trading wouldconsist in shortening the asset at the beginning of the time interval and buying back when priceshave depreciated through the sale of the other agent. Such strategies are “predatory” in the sensethat their price impact decreases the revenues of the other agent and thus generate profit at theother agent’s expense.In this paper, we continue the investigation of the “hot-potato game”. Our first contribution isto extend the result of Sch¨oneborn [25] by identifying a unique Nash equilibrium for two competingagents within a larger class of adaptive trading strategies, for general decay kernels, and by givingan explicit formula for the equilibrium strategies. This explicit formula will be the starting pointfor our further mathematical and numerical analysis of the Nash equilibrium. Another new featureof our approach is the addition of quadratic transaction costs, which can be thought of temporaryprice impact in the sense of [6, 4] or as a transaction tax. The main goal of our paper is to studythe impact of these additional transaction costs on equilibrium strategies. Theorem 2.7, our mainresult, precisely identifies a critical threshold θ ∗ for the size θ of these transaction costs at which alloscillations disappear. That is, for transactions θ ≥ θ ∗ certain “fundamental” equilibrium strategiesconsist exclusive of all buy trades or of all sell trades. For θ < θ ∗ , the “fundamental” equilibriumstrategies will contain both buy and sell trades when the decay of price impact in between two tradesis sufficiently small.In addition, numerical simulations will exhibit some rather striking properties of equilibriumstrategies. They reveal, for instance, that the expected costs of both agents can be a decreasing function of θ ∈ [0 , θ ] when trading speed is sufficiently high. As a result, both agents can carry outtheir respective trades at a lower cost when there are transaction costs, compared to the situationwithout transaction costs. Even more interesting is the behavior of the costs as a function of thetrading frequency. We will see that, for θ = θ ∗ , a higher trading speed can decrease expected tradingcosts, whereas the costs typically increase for sufficiently small θ . In particular the latter effect issurprising, because at first glance a higher trading frequency suggests that one has greater flexibilityin the choice of a strategy and hence can become more cost efficient. So why are the costs thenincreasing in the trading frequency? We will argue that the intuitive reason for this effect is thata higher trading frequency results in greater possibilities for predatory trading by the competitorand thus requires taking additional measures of protection against predatory trading. Some of thesenumerical observations have meanwhile been derived mathematically in our follow-up paper [23],which has E. Strehle as additional coauthor.This paper builds on several research developments in the existing literature. First, there are sev-eral papers on predatory trading such as Brunnermeier and Pedersen [8], Carlin et al. [9], Sch¨oneborn2nd Schied [26], and the authors [24] dealing with Nash equilibria for several agents that are active ina market model with temporary and permanent price impact. A discrete-time market impact gamewith asymmetric information was analyzed by Moallemi et al. [20]. In contrast to these previousstudies, the transient price impact model we use here goes back to Bouchaud et al. [7] and Obizhaevaand Wang [21]. It was further developed in [1, 2, 12, 3, 22], to mention only a few related papers.As first observed in [25], the qualitative features of Nash equilibria for transient price impact differdramatically from those obtained in [9, 26, 24] for an Almgren–Chriss setting. We refer to [13, 19] forrecent surveys on the price impact literature and extended bibliographies. Among the utilized math-ematical tools are the theory of M -matrices [5], the correspondence between the inverses of triangularToeplitz matrices and reciprocals of power series [28], and Kaluza’s sign criterion for reciprocal powerseries [16, 27].The paper is organized as follows. In Section 2.1 we explain our modeling framework. Theexistence and uniqueness theorem for Nash equilibria is stated in Section 2.2. In Section 2.3 weanalyze the oscillatory behavior of equilibrium strategies. Here we will also state our main result,Theorem 2.7, on the critical threshold for the disappearance of oscillations. Our numerical results andtheir interpretation are presented in Section 2.3 and Section 2.4. Particularly, Section 2.4 contains thesimulations for the behavior of the costs as a function of transaction costs and of trading frequencyin the cases with and without transaction costs. The proofs of our results are given in Section 3. Weconclude in Section 4. We consider two financial agents, X and Y , who are active in a market impact model for one riskyasset. Market impact will be transient and modeled as in [3]; see also [7, 21, 2, 12, 22] for closelyrelated or earlier versions of this model, which is sometimes called a propagator model. When noneof the two agents is active, asset prices are described by a right-continuous martingale S = ( S t ) t ≥ on a filtered probability space (Ω , ( F t ) t ≥ , F , P ), for which F is P -trivial. The process S is oftencalled the unaffected price process. Trading takes place at the discrete trading times of a time grid T = { t , t , . . . , t N } , where 0 = t < t < · · · < t N = T . Both agents are assumed to use tradingstrategies that are admissible in the following sense. Definition 2.1.
Suppose that a time grid T = { t , t , . . . , t N } is given. An admissible trading strategy for T and Z ∈ R is a vector ζ = ( ζ , . . . , ζ N ) of random variables such that(a) each ζ i is F t i -measurable and bounded, and(b) Z = ζ + · · · + ζ N P -a.s.The set of all admissible strategies for given T and Z is denoted by X ( Z , T ).For ζ ∈ X ( Z , T ), the value of ζ i is taken as the number of shares traded at time t i , with apositive sign indicating a sell order and a negative sign indicating a purchase. Thus, the requirement(b) in the preceding definition can be interpreted by saying that Z is the inventory of the agent attime 0 = t and that by time t N = T (e.g., the end of the trading day) the agent must have a zeroinventory. The assumption that each ζ i is bounded can be made without loss of generality from aneconomic point of view. The martingale assumption is natural from an economic point of view, because we are interested here in high-frequency trading over short time intervals [0 , T ]. See also the discussions in [3, 18] for additional arguments. X and Y apply respective strategies ξ ∈ X ( X , T ) and η ∈ X ( Y , T ), theasset price is given by S ξ , η t = S t − (cid:88) t k
Suppose that T = { t , t , . . . , t N } , X and Y are given. Let furthermore ( ε i ) i =0 , ,... bean i.i.d. sequence of Bernoulli ( )-distributed random variables that are independent of σ ( (cid:83) t ≥ F t ).Then the costs of ξ ∈ X ( X , T ) given η ∈ X ( Y , T ) are defined as C T ( ξ | η ) = X S + N (cid:88) k =0 (cid:16) G (0)2 ξ k − S ξ , η t k ξ k + ε k G (0) ξ k η k + θξ k (cid:17) (3)and the costs of η given ξ are C T ( η | ξ ) = Y S + N (cid:88) k =0 (cid:16) G (0)2 η k − S ξ , η t k η k + (1 − ε k ) G (0) ξ k η k + θη k (cid:17) . X S corresponds to the book value of the position X at time t = 0. If the position X could be liquidated at book value, one would incur the expenses − X S . Therefore, the liquidationcosts as defined in (3) are the difference of the actual accumulated expenses, as represented by thesum on the right-hand side of (3), and the expenses for liquidation at book value. The followingremark provides further comments on our modeling assumptions. Remark 2.3.
The market impact model we are using here has often been linked to the placementof market orders in a block-shaped limit order book, and a bid-ask spread is sometimes added to themodel so as to make this interpretation more feasible [21, 1]. For a strategy consisting exclusivelyof market orders, the bid-ask spread will lead to an additional fee that should be reflected in thecorresponding cost functional. In reality, however, most strategies will involve a variety of differentorder types and one should think of the costs (3) as the costs averaged over order types, as is oftendone in the market impact literature. For instance, while one may have to pay the spread whenplacing a market order, one essentially earns it back when a limit order is executed. Moreover,high-frequency traders often have access to a variety of more exotic order types, some of which canpay rebates when executed. It is also possible to use crossing networks or dark pools in which ordersare executed at mid price. So, for a setup of high-frequency trading, taking the bid-ask spread aszero in (1) is probably more realistic than modeling every single order as a market order and toimpose the fees. The existence of hot-potato games in real-world markets, such as the one quotedfrom [10] in Section 1, can be regarded as an empirical justification of the zero-spread assumption,because such a trading behavior could never be profitable if each trader had to pay the full spreadupon each execution of an order. See also the end of Section 2.2 for a discussion on how to replaceour quadratic transaction costs by piecewise linear ones.
We now consider agents who need to liquidate their current inventory within a given time frame andwho are aiming to minimize the expected costs over admissible strategies. The need for liquidationcan arise due to various reasons. For instance, Easley, L´opez da Prado, and O’Hara [11] argue thatthe toxicity of the order flow preceding the Flash Crash of May 6, 2010, has led the inventories ofseveral high-frequency market makers to grow beyond their risk limits, thus forcing them to unloadtheir inventories.When just a single agent is considered, the minimization of the expected execution costs is awell-studied problem; we refer to [3] for an analysis within our current modeling framework. Here weare going to investigate the optimal strategies of our two agents, X and Y , under the assumption thatboth have full knowledge of the other’s strategy and maximize the expected costs of their strategiesaccordingly. In this situation, it is natural to define optimality through the following notion of aNash equilibrium. Definition 2.4.
For given time grid T and initial values X , Y ∈ R , a Nash equilibrium is a pair( ξ ∗ , η ∗ ) of strategies in X ( X , T ) × X ( Y , T ) such that E [ C T ( ξ ∗ | η ∗ ) ] = min ξ ∈ X ( X , T ) E [ C T ( ξ | η ∗ ) ] and E [ C T ( η ∗ | ξ ∗ ) ] = min η ∈ X ( Y , T ) E [ C T ( η | ξ ∗ ) ] . To state our formula for this Nash equilibrium, we need to introduce the following notation. Fora fixed time grid T = { t , . . . , t N } , we define the ( N + 1) × ( N + 1)-matrix Γ byΓ i,j = G ( | t i − − t j − | ) , i, j = 1 , . . . , N + 1 , (4)5nd for θ ≥ θ := Γ + 2 θ Id . (5)We furthermore define the lower triangular matrix (cid:101) Γ by (cid:101) Γ ij = Γ ij if i > j , G (0) if i = j ,0 otherwise. (6)Note that Γ = (cid:101) Γ + (cid:101) Γ (cid:62) , where (cid:62) denotes the transpose of a matrix or vector. We will write forthe vector (1 , . . . , (cid:62) ∈ R N +1 . A strategy ζ = ( ζ , . . . , ζ N ) ∈ X ( Z , T ) will be identified with the( N + 1)-dimensional random vector ( ζ , . . . , ζ N ) (cid:62) . Conversely, any vector z = ( z , . . . , z N +1 ) (cid:62) ∈ R N +1 can be identified with the deterministic strategy ζ with ζ k = z k +1 . We also define the twovectors v = 1 (cid:62) (Γ θ + (cid:101) Γ) − (Γ θ + (cid:101) Γ) − w = 1 (cid:62) (Γ θ − (cid:101) Γ) − (Γ θ − (cid:101) Γ) − . (7)It will be shown in Lemma 3.2 below that the matrices Γ θ + (cid:101) Γ and Γ θ − (cid:101) Γ are indeed invertible andthat the denominators in (7) are strictly positive under our assumption (2) that G ( | · | ) is strictlypositive definite. Recall that we assume (2) throughout this paper.In the case G ( t ) = γ + λe − ρt for constants γ ≥ λ, ρ >
0, the existence of a unique Nashequilibrium in the class of deterministic strategies was established in Theorem 9.1 of [25]. Oursubsequent Theorem 2.5 extends this result in a number of ways: we allow for general positivedefinite decay kernels, include transaction costs, give an explicit form of the deterministic Nashequilibrium, and show that this Nash equilibrium is also the unique Nash equilibrium in the class ofadapted strategies. Our explicit formula for the equilibrium strategies will be the starting point forour further mathematical and numerical analysis of the Nash equilibrium. Also our proof is differentfrom the one in [25], which works only for the specific decay kernel G ( t ) = λe − ρt + γ . Theorem 2.5.
For any strictly positive definite decay kernel G , time grid T , parameter θ ≥ , andinitial values X , Y ∈ R , there exists a unique Nash equilibrium ( ξ ∗ , η ∗ ) ∈ X ( X , T ) × X ( Y , T ) .The optimal strategies ξ ∗ and η ∗ are deterministic and given by ξ ∗ = 12 ( X + Y ) v + 12 ( X − Y ) w , η ∗ = 12 ( X + Y ) v −
12 ( X − Y ) w . (8)The formula (8) shows that the vectors v and w form a basis for all possible equilibrium strategies.It follows that in analyzing the Nash equilibrium it will be sufficient to study the two cases ξ ∗ = v = η ∗ for X = 1 = Y and ξ ∗ = w = − η ∗ for X = 1 = − Y .Let us now comment on our choice of quadratic transaction costs. Such quadratic transactioncosts are often used to model “slippage” arising from temporary price impact; see [6, 4] and [12,Section 2.2]. Nevertheless, proportional transaction costs might be more realistic in many situations,and so the question arises if our results will change when the quadratic transaction costs θξ k arereplaced by (piecewise) linear transaction costs. This question is at least partially answered by the6ollowing result. It states that our quadratic transaction cost function can be replaced by proportionaltransaction costs in a neighborhood of the origin without affecting the Nash equilibrium. Since themain difference of quadratic and proportional transaction costs is their behavior at the origin, onemay therefore guess that similar results as obtained in the following sections for quadratic transactioncosts might also hold for proportional transaction costs. Proposition 2.6.
In the context of Theorem 2.5, there exists a piecewise linear, increasing, convex,and continuous transaction cost function τ with τ (0) = 0 such that ( ξ ∗ , η ∗ ) from (8) is a Nash equi-librium in X ( X , T ) × X ( Y , T ) for the the modified expected cost functional in which the quadratictransaction cost function x (cid:55)→ θx is replaced with x (cid:55)→ τ ( | x | ) . The transaction cost function τ constructed in the preceding proposition is of the form τ ( | x | ) = θ | x | + M (cid:88) k =1 θ k ( | x | − c k )1 [ c k , ∞ ) ( | x | )for certain coefficients θ k > < c < · · · c M . Transaction costs of this form canmodel a transaction tax that is subject to tax progression. With such a tax, small orders, such asthose placed by small investors, are taxed at a lower rate than large orders, which may be placedwith the intention of moving the market. We now turn toward a qualitative analysis of the equilibrium strategies. By means of numericalsimulations and the analysis of a particular example, Sch¨oneborn [25, Section 9.3] observed that theequilibrium strategies may exhibit strong oscillations if θ = 0, the time grid is equidistant, and G is of the form G ( t ) = λe − ρt + γ for constants λ, ρ > γ ≥
0. As a matter of fact, numericalsimulations, such as those presented in Figures 1 and 3, suggest that such oscillations can be observedfor a large class of decay kernels as soon as transaction costs vanish ( θ = 0) and the time grid issufficiently fine. We refer to Remark 2.10 for a possible financial interpretation of the oscillationsarising in the hot-potato game. For a single financial agent, however, optimal strategies will alwaysbe buy-only or sell-only for convex, nonincreasing decay kernels, which include those used in Figures 1and 3 (see [3, Theorem 1]). Therefore, the oscillations in our two-agent setting that are observed inthese figures must necessarily result from the interaction of both agents.It is intuitively clear that increased transaction costs will penalize oscillating strategies and thuslead to a smoothing of the equilibrium strategies. As a matter of fact, one can see in Figure 2 that for θ = 2 all oscillations have disappeared so that equilibrium strategies are then buy-only or sell-only.One can therefore wonder whether between θ = 0 and θ = 2 there might be a critical value θ ∗ atwhich all oscillations of v and w disappear, but below which oscillations are present. That is, for θ ≥ θ ∗ all equilibrium strategies should be either buy-only or sell-only, while for θ < θ ∗ equilibriumstrategies should contain both buy and sell trades (at least for certain values of N and T ). Thefollowing theorem confirms that such a critical value θ ∗ does indeed exist. We can even determineits precise value in case that we are dealing with equidistant time grids, T N := (cid:110) kTN (cid:12)(cid:12)(cid:12) k = 0 , , . . . , N (cid:111) , N ∈ N . (9)And we will be able to say even more in case G is of the form G ( t ) = λe − ρt + γ for constants λ, ρ > γ ≥
0. (10)It is well known that this class of decay kernels satisfies our assumption (2) (see, e.g., [3, Example1]), and they are clearly log-convex. 7 heorem 2.7.
Suppose that G is a continuous, positive definite, strictly positive, and log-convexdecay kernel and that T N denotes the equidistant time grid (9) . Then the following conditions areequivalent. (a) For every N ∈ N and T > , all components of w are nonnegative. (b) θ ≥ θ ∗ = G (0) / .If, moreover, G is of the form (10) , then conditions (a) and (b) are equivalent to: (c) For every N ∈ N and T > , all components of v are nonnegative. In the case θ < θ ∗ , one can actually obtain some stronger results on the existence of oscillations inthe vector w . These are stated in the following two propositions. First, we deal with the oscillationsof the signs of the last three trades of w , which are present as soon as θ < θ ∗ and the time gridis sufficiently fine. Recall that w completely determines the unique Nash equilibrium with initialconditions X = − Y . Proposition 2.8.
Suppose that G is a continuous and positive definite decay kernel that is nonin-creasing in a neighborhood of zero. Then for ≤ θ < θ ∗ there exists δ > such that for all timegrids T = { t , t , . . . , t N } with t N − t N − < δ and t N − − t N − < δ , the last three components of thevector w satisfy w N +1 > , w N < , and w N − > . The simulations in Figures 1 and 3 show that for θ = 0 actually all components of the vectors w and v have oscillating signs. The following propositions establishes the existence of oscillations for w in the case of an exponential decay kernel and an equidistant time grid. Proposition 2.9.
Suppose that G is of the form G ( t ) = λe − ρt for constants λ, ρ > and that T N denotes the equidistant time grid (9) for some given T > . Then there exists N ∈ N such that foreach N ≥ N there exists δ > so that for ≤ θ < δ all entries of the vector w = ( w , . . . , w N +1 ) are nonzero and have alternating signs. We refer to the right-hand panel of Figure 1 for an illustration of the oscillations of the vector w .As shown in the left-hand panel of the same figure, similar oscillations occur for the vector v andhence for equilibria with arbitrary initial conditions. The mathematical analysis for v , however, ismuch harder than for w , and at this time we are not able to prove a result that could be an analogueof Proposition 2.9 for the vector v . The existence of oscillations of w and v is also not limited toexponential decay kernels as can be seen from numerical experiments; see Figure 3 for power lawdecay and a randomly generated, non-equidistant time grid. Remark 2.10.
In this remark we will discuss a possible financial explanation for the oscillationsof equilibrium strategies observed for small values of θ . As mentioned above, the source for theseoscillations must necessarily lie in the interaction between the two agents. As observed in previousstudies on multi-agent equilibria in price impact models such as [8, 9, 26], the dominant form ofinteraction between two players is predatory trading , which consists in the exploitation of price impactgenerated by another agent. Such strategies are “predatory” in the sense that they generate profitby simultaneously decreasing the other agent’s revenues. Since predators prey on the drift createdby the price impact of a large trade, protection against predatory trading requires the cancellationof previously created price impact. Under transient price impact, the price impact of an earliertrade, say ζ , can be cancelled by placing an order ζ of the opposite side. For instance, taking ζ := − ζ G ( t − t ) will completely eliminate the price impact of ζ while the combined tradesexecute a total of ξ (1 − G ( t − t )) shares. In this sense, oscillating strategies can be understood asa protection against predatory trading by opponents (see also [25, p.150]).8 - - - Figure 1: Vectors v (left) and w (right) for the equidistant time grid T , G ( t ) = e − t , θ = 0, and T = 1. By (8), ( v , v ) is the equilibrium for X = Y = 1, and ( w , − w ) is the equilibrium for X = − Y = 1. Yet, some individual components of both v and w exceed in either direction 60% ofthe sizes of the initial positions X and Y . Figure 2: Vectors v (left) and w (right) for the equidistant time grid T , G ( t ) = e − t , θ = 2, and T = 1. - - - - Figure 3: Vectors v (left) and w (right) for power-law decay G ( t ) = 1 / √ t and a time gridgenerated from 50 independent uniformly distributed random variables on.9 emark 2.11. Alfonsi et al. [3] discovered oscillations for the trade execution strategies of a single trader under transient price impact if price impact does not decay as a convex function of time.These oscillations, however, result from an attempt to exploit the delay in market response to alarge trade, and they disappear if price impact decays as a convex function of time [3, Theorem 1].In particular, when there is just one agent active and G is convex, nonincreasing, and nonconstant(which is, e.g., the case under assumption (10)), then for each time grid T there exists a uniqueoptimal strategy, which is either buy-only or sell-only. When (10) holds and θ = 0, this strategy isknown explicitly; see [1]. Remark 2.12.
Based on numerical simulations, we believe that the statements of Theorem 2.7 andProposition 2.9 can probably be improved. Specifically, we conjecture that the equivalence betweenthe conditions (a), (b), and (c) in Theorem 2.7 remains true for all positive definite decay kernels.Our current proofs, however, cannot be extended beyond our stated conditions. Specifically, theimplication of (b) ⇒ (a) in Theorem 2.7 exploits the Toeplitz structure of the upper triangular matrixΓ θ − (cid:101) Γ, which only holds for equidistant time grids. We then use the fact that the inverse of atriangular Toeplitz matrix corresponds to the (formal) reciprocal of a power series, and we use thecelebrated Kaluza sign criterion [16, 27] to determine the signs of this reciprocal power series. Here,the log-convexity of G is essential. The proof of the implication (b) ⇒ (c) relies on the theory of M -matrices as presented in [5]. In particular, we rely on the fact that the matrix Γ − ( (cid:101) Γ + Id) isa non-singular M -matrix for G ( t ) = e − ρt (Lemma 3.11), which is no longer true, e.g., for powerlaw decay G ( t ) = 1 / (1 + t ) p with p >
0. Similarly, the proof of Proposition 2.9 exploits the factthat the upper triangular matrix Γ − (cid:101) Γ can be inverted explicitly if the time grid is equidistantand G ( t ) = e − ρt . Surprisingly, although the matrix Γ has an explicit inverse for any time grid if G ( t ) = e − ρt (see [1, Theorem 3.4]), the structure of (Γ − (cid:101) Γ) − becomes quite involved if the time gridis not equidistant. Already for equidistant time grids, the same can be said of the matrix (Γ + (cid:101) Γ) − ,which is needed to compute the vector v . Due to our explicit formulas (7) and (8), it is easy to analyze the Nash equilibrium numerically.These numerical simulations exhibit several striking effects in regards to monotonicity properties ofthe expected costs.In Figure 4 we have plotted the expected costs E [ C T N ( ξ ∗ | η ∗ ) ] = E [ C T N ( η ∗ | ξ ∗ ) ] for X = Y , G ( t ) = e − t , and T = 1 as a function of the trading frequency, N . The first observation one probablymakes when looking at this plot is the fact that for θ = 0 the expected costs exhibit a sawtooth-likepattern; they alternate between two increasing trajectories, depending on whether N is odd or even.These alternations are due to the oscillations of the optimal strategies, which also alternate with N .As can be seen from the figure, the sawtooth pattern essentially disappears already for very smallvalues of θ such as for θ = 0 . θ = 0, θ = 0 .
01, and θ = 0 . E [ C T N ( ξ ∗ | η ∗ ) ] (or alternatively E [ C T N +1 ( ξ ∗ | η ∗ ) ]) are increasing in N . This fact is surprisingbecause a higher trading frequency should normally lead to a larger class of admissible strategies.As a result, traders have greater flexibility in choosing a strategy and in turn should be able to pickmore cost efficient strategies. So why are the costs then increasing in N ? The intuitive explanation isthat a higher trading frequency increases also the possibility for the competitor to conduct predatorystrategies at the expense of the other agent (see Remark 2.10). In reaction, this other agent needs totake stronger protective measures against predatory trading. As discussed in Remark 2.10, protection10gainst predatory trading can be obtained by erasing (part of) the previously created price impactthrough placing an order of the opposite side. The result is an oscillatory strategy, whose expectedcosts increase with the number of its oscillations.Still in Figure 4, the expected costs E [ C T N ( ξ ∗ | η ∗ ) ] for the case θ = θ ∗ = 0 .
25 exhibit a verydifferent behavior. They no longer alternate in N and are decreasing as a function of the tradingfrequency. The intuitive explanation is that transaction costs of size θ ∗ = 0 .
25 discourage predatorytrading to a large extend, so that agents can now benefit from a higher trading frequency and pickever more cost-efficient strategies as N increases.The most surprising observation in Figure 4 is the fact that for sufficiently large N the expectedcosts for θ > θ = 0. That is, for sufficiently large tradingfrequency, adding transaction costs can decrease the expected costs of all market participants (recallthat for X = Y both agents have the same optimal strategies and, hence, the same expected costs).This fact is further illustrated in Figure 5, which exhibits a very steep initial decrease of the expectedcosts as a function of θ . After a minimum of the expected costs is reached at θ ≈ .
06, there is aslow and steady increase of the costs with an approximate slope of 0.002.The key to understanding the behavior of expected equilibrium costs as a function of tradingfrequency and transaction costs rests in the interpretation of the oscillations in equilibrium strategiesas a protection against predatory trading by the opponent (see Remark 2.10). Note that a predatorytrading strategy is necessarily a “round trip”, i.e., a strategy with zero inventory at t = 0 and T = 0(the strategy of a predatory trader with nonzero initial position would consist of a superposition of apredatory round trip and a liquidation strategy for the initial position). It therefore must consist of abuy and a sell component and is hence stronger penalized by an increase in transaction costs than abuy-only or sell-only strategy. As a result, increasing transaction costs leads to an overall reductionof the proportion of predatory trades in equilibrium. In consequence, both agents in our modelcan reduce their protection against predatory trading and therefore use more efficient strategies tocarry out their trades. They can thus fully benefit from higher trading frequencies, which leads tothe observed decrease of expected costs as a function of N if θ is sufficiently large. Moreover, forappropriate values of θ >
0, the benefit of increased efficiency outweighs the price to be paid inhigher transaction costs and so an overall reduction of costs is achieved.Let us point out that, in the case G ( t ) = e − ρt , many qualitative observations made in this sectionby means of numerical experiments have meanwhile been given rigorous mathematical proofs in ourfollow-up paper [23], which has Elias Strehle as additional coauthor. There, we investigate the limitsof equilibrium strategies and expected costs as N ↑ ∞ . We prove that, for θ = 0, both strategies andcosts oscillate indefinitely between two accumulation points, for which we provide explicit formulas.For θ >
0, however, strategies and costs converge toward limits that are independent of θ . We thenshow that the limiting strategies form a Nash equilibrium for a continuous-time version of the modelwith θ = θ ∗ , and that the corresponding expected costs coincide with the high-frequency limits ofthe discrete-time equilibrium costs. For θ (cid:54) = θ ∗ , however, continuous-time Nash equilibria do notexist unless X = Y = 0.Another interesting question is the comparison of the expected costs of the equilibrium strategieswith the expected costs that both agents would have if none of them were aware of the other’s tradingactivities. In this case, a trader with initial inventory Z will apply the strategy (cid:98) ζ Z = Z (cid:62) Γ − θ Γ − θ , which is the strategy for a single trader facing the positive definite decay G and transaction costsmeasured by the parameter θ ≥
0; this follows by taking the positive definite decay kernel G ( t ) +11 θ = 0 . θ = 0 θ = 0 . θ = θ ∗ = 0 . N Figure 4: Expected costs E [ C T N ( ξ ∗ | η ∗ ) ] = E [ C T N ( η ∗ | ξ ∗ ) ] for various values of θ as a function oftrading frequency, N , with the equidistant time grid T N , T = 1, G ( t ) = e − t , and X = Y = 1. θ Figure 5: Expected costs E [ C T ( ξ ∗ | η ∗ ) ] as a function of θ for initial values X = Y = 1 and G ( t ) = e − t . The costs decrease steeply from the value 0.7567 at θ = 0 until a minimum value ofabout 0 . θ = 0 .
06. From then on there is a moderate and almost linear increase with, e.g.,a value of 0 . θ = 0 .
5. This increase corresponds to a slope of approximately 0.002. We tookthe equidistant time grid T and ρ = 1.1 { } ( t ) in [3, Proposition 1]. We can thus define a price of anarchy in our situation by lettingPoA N ( θ, X , Y ) := E [ C T N ( (cid:98) ζ X | (cid:98) ζ Y ) ] + E [ C T N ( (cid:98) ζ Y | (cid:98) ζ X ) ] E [ C T N ( ξ ∗ | η ∗ ) ] + E [ C T N ( η ∗ | ξ ∗ ) ] , where ξ ∗ and η ∗ are the equilibrium strategies from (8). See Figure 6 for a plot. Lemma 3.1.
The expected costs of an admissible strategy ξ ∈ X ( X , T ) given another admissiblestrategy η ∈ X ( Y , T ) are E [ C T ( ξ | η ) ] = E (cid:104) ξ (cid:62) Γ θ ξ + ξ (cid:62) (cid:101) Γ η (cid:105) . (11)12 .02 0.04 0.06 0.08 0.100.0010.0020.0030.0040.0050.006 Figure 6: Price of anarchy, PoA ( θ, X , Y ), as a function of θ for X = 1 and Y = − X = 1 and Y = 1 (right) for G ( t ) = e − t . The steep increase on the right-hand panel is due tothe initial decrease of the expected costs for X = Y = 1 as shown in Figure 5. The steep of theprice of anarchy in the right-hand panel is the result of the decrease of the corresponding equilibriumstrategies as shown in Figure 5. Proof.
Without loss of generality, we may assume G (0) = 1. Since the sequence ( ε i ) i =0 , ,... is inde-pendent of σ ( (cid:83) t ≥ F t ) and the two strategies ξ and η are measurable with respect to this σ -field,we get E [ ε k ξ k η k ] = E [ ξ k η k ]. Hence, E [ C T ( ξ | η ) ] − X S = E (cid:20) N (cid:88) k =0 (cid:16) ξ k − S ξ , η t k ξ k + ε k ξ k η k + θξ k (cid:17) (cid:21) = E (cid:20) N (cid:88) k =0 (cid:18) ξ k + 12 ξ k η k − ξ k (cid:16) S t k − k − (cid:88) m =0 ( ξ m + η m ) G ( t k − t m ) (cid:17) + θξ k (cid:19) (cid:21) = E (cid:20) − N (cid:88) k =0 ξ k S t k + 12 N (cid:88) k =0 ξ k + N (cid:88) k =0 ξ k k − (cid:88) m =0 ξ m G ( t k − t m )+ N (cid:88) k =0 (cid:18) ξ k (cid:16) η k + k − (cid:88) m =0 η m G ( t k − t m ) (cid:17) + θξ k (cid:19) (cid:21) . Since each ξ k is F t k -measurable and S is a martingale, we get from condition (b) in Definition 2.1that E (cid:104) N (cid:88) k =0 ξ k S t k (cid:105) = E (cid:104) N (cid:88) k =0 ξ k S T (cid:105) = X E [ S T ] = X S . Moreover, 12 N (cid:88) k =0 ξ k + N (cid:88) k =0 ξ k k − (cid:88) m =0 ξ m G ( t k − t m ) = 12 N (cid:88) k,m =0 ξ k ξ m G ( | t k − t m | ) = 12 ξ (cid:62) Γ ξ , and N (cid:88) k =0 ξ k (cid:16) η k + k − (cid:88) m =0 η m G ( t k − t m ) (cid:17) = ξ (cid:62) (cid:101) Γ η . Putting everything together yields the assertion.We will use the convention of saying that an n × n -matrix A is positive if x (cid:62) A x > x ∈ R n , which makes sense also if A is not necessarily symmetric. Clearly, for a positive13atrix A there is no nonzero x ∈ R n for which A x = , and so A is invertible. Moreover, writing agiven nonzero x ∈ R n as x = A y for y = A − x (cid:54) = , we see that x (cid:62) A − x = y (cid:62) A (cid:62) y = y (cid:62) A y > Lemma 3.2.
The matrices Γ θ , (cid:101) Γ , Γ θ + (cid:101) Γ , Γ θ − (cid:101) Γ are positive for all θ ≥ . In particular, all termsin (7) are well-defined and the denominators in (7) are strictly positive.Proof. That Γ is positive definite, and hence positive, follows directly from (2). Therefore, for nonzero x ∈ R N +1 , 0 < x (cid:62) Γ x = x (cid:62) ( (cid:101) Γ + (cid:101) Γ (cid:62) ) x = x (cid:62) (cid:101) Γ x + x (cid:62) (cid:101) Γ (cid:62) x = 2 x (cid:62) (cid:101) Γ x , which shows that the matrix (cid:101) Γ is positive. Next, Γ − (cid:101) Γ = (cid:101) Γ (cid:62) and so this matrix is also positive.Clearly, the sum of two positive matrices is also positive, which shows that Γ θ + (cid:101) Γ = Γ + (cid:101)
Γ + 2 θ Idand Γ θ − (cid:101) Γ = Γ − (cid:101) Γ + 2 θ Id are positive for θ ≥ Lemma 3.3.
For given time grid T and initial values X and Y , there exists at most one Nashequilibrium in the class X ( X , T ) × X ( Y , T ) .Proof. We assume by way of contradiction that there exist two distinct Nash equilibria ( ξ , η ) and( ξ , η ) in X ( X , T ) × X ( Y , T ). Here, the fact that the two Nash equilibria are distinct means thatthey are not P -a.s. equal. Then we define for α ∈ [0 , ξ α := α ξ + (1 − α ) ξ and η α := α η + (1 − α ) η . We furthermore let f ( α ) := E (cid:104) C T ( ξ α | η ) + C T ( η α | ξ ) + C T ( ξ − α | η ) + C T ( η − α | ξ ) (cid:105) . Since according to (2) the matrix Γ θ is positive definite, the functional ξ (cid:55)−→ E [ C T ( ξ | η ) ] = E (cid:104) ξ (cid:62) Γ θ ξ + ξ (cid:62) (cid:101) Γ η (cid:105) is strictly convex with respect to ξ . Since the two Nash equilibria ( ξ , η ) and ( ξ , η ) are distinct, f ( α ) must also be strictly convex in α and have its unique minimum in α = 0. That is, f ( α ) > f (0) for α > . It follows that lim h ↓ f ( h ) − f (0) h = df ( α ) dα (cid:12)(cid:12)(cid:12) α =0+ ≥ . (12)Next, by the symmetry of Γ θ , E [ C T ( ξ α | η ) ] = E (cid:20) α ( ξ ) (cid:62) Γ θ ξ + α (1 − α )( ξ ) (cid:62) Γ θ ξ + 12 (1 − α ) ( ξ ) (cid:62) Γ θ ξ + α ( ξ ) (cid:62) (cid:101) Γ η + (1 − α )( ξ ) (cid:62) (cid:101) Γ η (cid:21) . Therefore, ddα (cid:12)(cid:12)(cid:12) α =0+ E [ C T ( ξ α | η ) ] = E (cid:104) ( ξ − ξ ) (cid:62) Γ θ ξ + ( ξ − ξ ) (cid:62) (cid:101) Γ η (cid:105) . ddα (cid:12)(cid:12)(cid:12) α =0+ f ( α )= E (cid:20) ( ξ − ξ ) (cid:62) Γ θ ξ + ( ξ − ξ ) (cid:62) (cid:101) Γ η + ( ξ − ξ ) (cid:62) Γ θ ξ + ( ξ − ξ ) (cid:62) (cid:101) Γ η +( η − η ) (cid:62) Γ θ η + ( η − η ) (cid:62) (cid:101) Γ ξ + ( η − η ) (cid:62) Γ θ η + ( η − η ) (cid:62) (cid:101) Γ ξ (cid:21) = − E (cid:20) ( ξ − ξ ) (cid:62) Γ θ ( ξ − ξ ) + ( η − η ) (cid:62) Γ θ ( η − η ) (cid:21) + E (cid:20) ( ξ − ξ ) (cid:62) (cid:101) Γ( η − η ) + ( ξ − ξ ) (cid:62) (cid:101) Γ (cid:62) ( η − η ) (cid:21) = − E (cid:20) ( ξ − ξ ) (cid:62) Γ θ ( ξ − ξ ) + ( η − η ) (cid:62) Γ θ ( η − η ) (cid:21) − E (cid:20) ( ξ − ξ ) (cid:62) Γ( η − η ) (cid:21) . Now, ( ξ − ξ ) (cid:62) Γ( η − η ) + 12 (cid:16) ( ξ − ξ ) (cid:62) Γ θ ( ξ − ξ ) + ( η − η ) (cid:62) Γ θ ( η − η ) (cid:17) ≥ (cid:16) ( ξ − ξ + η − η ) (cid:62) Γ( ξ − ξ + η − η ) (cid:17) ≥ . Thus, and because the two Nash equilibria ( ξ , η ) and ( ξ , η ) are distinct, we have ddα (cid:12)(cid:12)(cid:12) α =0+ f ( α ) ≤ − E (cid:20) ( ξ − ξ ) (cid:62) Γ( ξ − ξ ) + ( η − η ) (cid:62) Γ( η − η ) (cid:21) < , which contradicts (12). Therefore, there can exist at most one Nash equilibrium in the class X ( X , T ) × X ( Y , T ).Now let us introduce the class X det ( Z , T ) := (cid:110) ζ ∈ X ( Z , T ) (cid:12)(cid:12)(cid:12) ζ is deterministic (cid:111) of deterministic strategies in X ( Z , T ). A Nash equilibrium in the class X det ( X , T ) × X det ( Y , T )is defined in the same way as in Definition 2.4. Lemma 3.4.
A Nash equilibrium in the class X det ( X , T ) × X det ( Y , T ) of deterministic strategiesis also a Nash equilibrium in the class X ( X , T ) × X ( Y , T ) of adapted strategies.Proof. Assume that ( ξ ∗ , η ∗ ) is a Nash equilibrium in the class X det ( X , T ) × X det ( Y , T ) of determin-istic strategies. We need to show that ξ ∗ minimizes E [ C T ( ξ | η ∗ ) ] and η ∗ minimizes E [ C T ( η | ξ ∗ ) ] inthe respective classes X ( X , T ) and X ( Y , T ) of adapted strategies. To this end, let ξ ∈ X ( X , T )be given. We define ξ ∈ X det ( X , T ) by ξ k = E [ ξ k ] for k = 0 , , . . . , N .Applying Jensen’s inequality to the convex function R N +1 (cid:51) x (cid:55)→ x (cid:62) Γ θ x , we obtain E [ C T ( ξ | η ∗ ) ] = E (cid:104) ξ (cid:62) Γ θ ξ + ξ (cid:62) (cid:101) Γ η ∗ (cid:105) = E (cid:104) ξ (cid:62) Γ θ ξ (cid:105) + ξ (cid:62) (cid:101) Γ η ∗ ≥ ξ (cid:62) Γ θ ξ + ξ (cid:62) (cid:101) Γ η ∗ = E [ C T ( ξ | η ∗ ) ] ≥ E [ C T ( ξ ∗ | η ∗ ) ] . This shows that ξ ∗ minimizes E [ C T ( ξ | η ∗ ) ] over ξ ∈ X ( X , T ). One can show analogously that η ∗ minimizes E [ C T ( η | ξ ∗ ) ] over η ∈ X ( Y , T ), which completes the proof.15 emark 3.5. Before proving Theorem 2.5, we briefly explain how to derive heuristically the explicitform (8) of the equilibrium strategies. By Lemma 3.1 and the method of Lagrange multipliers, anecessary condition for ( ξ ∗ , η ∗ ) to be a Nash equilibrium in X det ( X , T ) × X det ( Y , T ) is the existenceof α, β ∈ R , such that (cid:40) Γ θ ξ ∗ + (cid:101) Γ η ∗ = α ;Γ θ η ∗ + (cid:101) Γ ξ ∗ = β . (13)By adding the equations in (13) we obtain(Γ θ + (cid:101) Γ)( ξ ∗ + η ∗ ) = ( α + β ) . (14)By Lemma 3.2, the matrix Γ θ + (cid:101) Γ is positive and hence invertible, so that (14) can be solved for ξ ∗ + η ∗ . Since we must also have (cid:62) ( ξ ∗ + η ∗ ) = X + Y , we obtain ξ ∗ + η ∗ = ( X + Y ) (cid:62) (Γ θ + (cid:101) Γ) − (Γ θ + (cid:101) Γ) − = ( X + Y ) v . Similarly, by subtracting the two equations in (13) yields(Γ θ − (cid:101) Γ)( ξ ∗ − η ∗ ) = ( α − β ) . It follows again from Lemma 3.2 that (Γ θ − (cid:101) Γ) is invertible, and so we have ξ ∗ − η ∗ = ( X − Y ) T (Γ θ − (cid:101) Γ) − (Γ θ − (cid:101) Γ) − = ( X − Y ) w . Thus, ξ ∗ and η ∗ ought to be given by (8). Proof of Theorem 2.5.
By Lemmas 3.3 and 3.4 all we need to show is that (8) defines a Nash equi-librium in the class X det ( X , T ) × X det ( Y , T ) of deterministic strategies. For ( ξ , η ) ∈ X det ( X , T ) × X det ( Y , T ) we have E [ C T ( ξ | η ) ] = 12 ξ (cid:62) Γ θ ξ + ξ (cid:62) (cid:101) Γ η . (15)Therefore minimizing E [ C T ( ξ | η ) ] over ξ ∈ X det ( X , T ) is equivalent to the minimization of thequadratic form on the right-hand side of (15) over ξ ∈ R N +1 under the constraint (cid:62) ξ = X .Now we prove that the strategies ξ ∗ and η ∗ given by (8) are indeed optimal. We haveΓ θ ξ ∗ + (cid:101) Γ η ∗ = 12 ( X + Y )(Γ θ + (cid:101) Γ) v + 12 ( X − Y )(Γ θ − (cid:101) Γ) w = µ , where µ = ( X + Y )2 (cid:62) (Γ θ + (cid:101) Γ) + ( X − Y )2 (cid:62) (Γ θ − (cid:101) Γ) . Now let ξ ∈ X det ( X , T ) be arbitrary and define ζ := ξ − ξ ∗ . Then we have ζ (cid:62) = 0. Hence, by thesymmetry of Γ θ ,12 ξ (cid:62) Γ θ ξ + ξ (cid:62) (cid:101) Γ η ∗ = 12 ( ξ ∗ ) (cid:62) Γ θ ξ ∗ + 12 ζ (cid:62) Γ θ ζ + ζ (cid:62) Γ θ ξ ∗ + ( ξ ∗ ) (cid:62) (cid:101) Γ η ∗ + ζ (cid:62) (cid:101) Γ η ∗ = 12 ( ξ ∗ ) (cid:62) Γ θ ξ ∗ + ( ξ ∗ ) (cid:62) (cid:101) Γ η ∗ + 12 ζ (cid:62) Γ θ ζ + µ ζ (cid:62) ≥
12 ( ξ ∗ ) (cid:62) Γ θ ξ ∗ + ( ξ ∗ ) (cid:62) (cid:101) Γ η ∗ , θ is positive definite and that ζ (cid:62) = 0. Therefore ξ ∗ minimizes (15) in the class X det ( X , T ) for η = η ∗ . In the same way, one shows that η ∗ minimizes E [ C T ( η | ξ ∗ ) ] over η ∈ X det ( X , T ). Proof of Proposition 2.6.
Following Lemma 3.1, the expected cost functional with x (cid:55)→ τ ( | x | ) replac-ing x (cid:55)→ θx is given by E [ C T ( ξ | η ) ] := E (cid:20) ξ (cid:62) Γ ξ + ξ (cid:62) (cid:101) Γ η + N (cid:88) k =0 τ ( | ξ k | ) (cid:21) , ξ ∈ X ( X , T ) , η ∈ X ( Y , T ) . Now let ξ ∗ and η ∗ be as in Theorem 2.5. Since both ξ ∗ and η ∗ are deterministic, | ξ ∗ k | and | η ∗ k | takejust finitely many values as k ranges from 0 to N . After adding the value 0 to this list and arrangingit in increasing order, the values from that list correspond to numbers 0 = c < c < c < · · · < c M − .Then we take c M := c M − + 1 and let τ : [0 , ∞ ) → [0 , ∞ ) be the linear interpolation of the function x (cid:55)→ θx with respect to the grid c , c , . . . , c M and with linear continuation beyond [ c M − , c M ]. Then τ ( | ξ ∗ k | ) = θ ( ξ ∗ k ) and τ ( | η ∗ k | ) = θ ( η ∗ k ) holds for all k , and it follows that E [ C T ( ξ ∗ | η ∗ ) ] = E [ C T ( ξ ∗ | η ∗ ) ].Let us now suppose by way of contradiction that ( ξ ∗ , η ∗ ) is not a Nash equilibrium in X ( X , T ) × X ( Y , T ). Then there exist ξ ∈ X ( X , T ) or η ∈ X ( Y , T ) such that E [ C T ( ξ | η ∗ ) ] < E [ C T ( ξ ∗ | η ∗ ) ]or E [ C T ( η | ξ ∗ ) ] < E [ C T ( η ∗ | ξ ∗ ) ]. By symmetry, it is sufficient to consider only the first possibility.For α ∈ [0 , ξ α := (1 − α ) ξ ∗ + α ξ . By the convexity of the expected cost functional, we have E [ C T ( ξ α | η ∗ ) ] < E [ C T ( ξ ∗ | η ∗ ) ] for all α ∈ (0 , ε ∈ (0 ,
1] such that | ξ εk | ≤ c M for k = 0 , . . . , N P -a.s. Thus, the convexityof x (cid:55)→ θx implies that τ ( | ξ εk | ) ≥ θ ( ξ εk ) P -a.s. for k = 0 , . . . , N . Hence, E [ C T ( ξ ε | η ∗ ) ] ≥ E [ C T ( ξ ε | η ∗ ) ] > E [ C T ( ξ ∗ | η ∗ ) ] = E [ C T ( ξ ∗ | η ∗ ) ] , which is the desired contradiction. Proof of Proposition 2.8.
According to (7) and Lemma 3.2, the vector w is a positive multiple of(Γ θ − (cid:101) Γ) − . The matrix Γ θ − (cid:101) Γ is an invertible upper triangular matrix, whose diagonal entries areall equal to ν := G (0) / θ . We may assume without loss of generality that ν = 1; otherwise wedivide G by ν . Then we will have G (0) >
1, and there exists δ > G ( δ ) >
1. Nowwe take δ ≤ δ such that G is nonincreasing in [0 , δ ]. The off-diagonal elements of Γ θ − (cid:101) Γ are equalto Γ i,j = G ( t j − − t i − ) for i < j and they vanish for i > j . Let u = ( u , . . . , u N +1 ) (cid:62) = (Γ θ − (cid:101) Γ) − .A straightforward computation shows that u N +1 = 1 , u N = 1 − Γ N,N +1 , and u N − = 1 − Γ N − ,N +1 + Γ N − ,N (Γ N,N +1 − . Clearly, u N +1 > δ , we have Γ i − ,i = G ( t i − t i − ) > i = N, N + 1. In particular, u N < u N − > − Γ N − ,N +1 + (Γ N,N +1 −
1) = G ( t N − t N − ) − G ( t N − t N − ) ≥ , where the latter inequality follows from the assumption that G is nonincreasing in [0 , δ ].17 roof of Proposition 2.9. Recall that here G ( t ) = λe − ρt for constants λ, ρ >
0. We need to computethe inverse of the matrix Γ θ − (cid:101) Γ. Setting κ := 2 θ/λ + and a := e − ρT , we haveΓ θ − (cid:101) Γ = λ κ a N a N · · · a N − N a κ a N · · · a N − N a N − N κ a N · · · · · · · · · κ . It is easy to verify that the inverse of this matrix is given byΠ N := 1 λ κ − a N κ − a N ( κ − κ · · · − a N − N ( κ − N − κ N − a NN ( κ − N − κ N +1 κ − a N κ · · · − a N − N ( κ − N − κ N − − a N − N ( κ − N − κ N κ − a N κ · · · · · · · · · κ . Let us denote by u = ( u , u , . . . , u N +1 ) ∈ R N +1 the vector λ Π N . Then we have u N +1 = κ and, for n = 1 , . . . , N , u n = u n +1 − a ( N +1 − n ) /N ( κ − N − n /κ N +2 − n . That is, u n = 1 κ − a N κ N (cid:88) m = n (cid:16) a N ( κ − κ (cid:17) N − m = 1 κ − a N κ N − n (cid:88) k =0 (cid:16) a N ( κ − κ (cid:17) k = 1 κ (cid:20) − a N κ (1 − a N ) + a N + ( − N +1 − n a N κ (1 − a N ) + a N (cid:16) a N (1 − κ ) κ (cid:17) N +1 − n (cid:21) . (16)If θ = 0, we have u n = 2 (cid:20) − a N a N + ( − N +1 − n a N +2 − nN a N (cid:21) . Since a <
1, we have 0 ≤ − a N a N < − a N −→ N ↑ ∞ .On the other hand, we have2 a N +2 − nN a N ≥ a N +2 − nN ≥ a N +1 N −→ a as N ↑ ∞ .Therefore, the signs of u n will alternate as soon as N is large enough to have 1 − a N < a N +1 N . Thisproves part (a). As for part (b), since the expression (16) is continuous in κ , the signs of u n willstill alternate if, for fixed N ≥ N , we take κ slightly larger than 1 /
2. (Note however that the term(1 − κ ) N /κ N tends to zero faster than 1 − a N , so we cannot get this result uniformly in N ).18 .3 Proof of Theorem 2.7 Proof of (a) ⇒ (b) in Theorem 2.7. It is well known and easy to see that G (0) ≥ G ( t ) for all t ≥ G ( | · | ) is positive definite. The log-convexity of G thereforeimplies that G must be nonincreasing in a neighborhood of zero. Therefore, Proposition 2.8 isapplicable. It implies that w must have some components with negative sign if θ < θ ∗ . This yieldsthe assertion.The proof of the implication (b) ⇒ (a) in Theorem 2.7 relies on the following classical result on thesigns of power series, which is due to Kaluza [16] and Szeg˝o [27]. Here we state it in the formulationof Jurkat [15, Theorem 3]. Theorem 3.6 (Kaluza sign criterion) . For n ≥ , let a n > be coefficients in the power series f ( x ) = (cid:80) ∞ n =0 a n x n satisfying the condition that a n +1 /a n is nondecreasing in n ≥ . Then thecoefficients b n of the formal reciprocal power series f ( x ) = ∞ (cid:88) n =0 b n x n satisfy b = 1 /a > and b n ≤ for n ≥ . If, moreover, the power series for f is convergent for | x | < , then it follows that lim x ↑ f ( x ) = (cid:80) ∞ n =0 b n exists and is nonnegative. This result is connected with our situation as follows. Let ( a n ) n ≥ be a sequence of numbers suchthat a > A = ( (cid:101) a i,j ) i,j =1 ,...,N with coefficients (cid:101) a i,j = a j − i if i ≤ j and (cid:101) a i,j = 0 otherwise. The inverse B = A − is then also an upper triangularToeplitz matrix. It is generated by the sequence ( b n ) n ≥ that satisfies b = 1 /a and is otherwisedetermined recursively through the convolution identities m (cid:88) k =0 a k b m − k = 0 , m ≥ . But these conditions also determine the coefficients ( b n ) n ≥ of the (formal) reciprocal of the powerseries (cid:80) ∞ n =0 a n x n , so that there is a one-to-one correspondence between the inversion of triangularToeplitz matrices and the formal development of reciprocal power series; see [28]. Proof of (b) ⇒ (a) in Theorem 2.7. Let a = G (0) / θ and a n = G ( nT /N ) for n ≥
1. Then thematrix Γ θ − (cid:101) Γ is equal to the upper triangular Toeplitz matrix constructed as above from the sequence( a n ) n ≥ . Clearly, we have a n > n , and the fact that G is log-convex implies that a n +1 /a n is nondecreasing in n ≥
1. If θ ≥ θ ∗ , then we will also have a /a ≤ a /a . Moreover, the fact that G is positive definite implies once again that G ( t ) ≤ G (0) for all t so that the sequence ( a n ) n ∈ N isbounded and the power series (cid:80) ∞ n =0 a n x n converges for | x | <
1. It follows that we may apply allparts of Theorem 3.6. It yields that the coefficients ( b n ) n ≥ satisfy b > b n ≤ n ≥
1, andthat (cid:80) ∞ n =0 b n exists and is nonnegative. Therefore, we must have (cid:80) kn =0 b n ≥ k ≥
0. Butthese sums coincide with the components of the vector (Γ θ − (cid:101) Γ) − , which is in turn proportional to w . Proof of (c) ⇒ (b) in Theorem 2.7. We consider the case N = 1. By definition, v is proportional tothe vector 2 det(Γ θ + (cid:101) Γ)(Γ θ + (cid:101) Γ) − = (cid:18) λ (3 − a ) + γ + 4 θλ (3 − a ) − γ + 4 θ (cid:19) . a ∈ (0 ,
1) and θ ≥
0. By sending a ↑ θ < θ ∗ and a sufficiently close to 1.Thus, we cannot have v ≥ ⇒ (c) in Theorem 2.7. It relies on resultsfor so-called M -matrices stated in the book [5] by Berman and Plemmons. We first introduce somenotations. If A is a matrix or vector, we will write(a) A ≥ A is nonnegative;(b) A > A ≥ A (cid:29) A is strictly positive. Definition 3.7 (Definition 1.2 in Chapter 6 of [5]) . A matrix A ∈ R n × n is called a nonsingular M -matrix if it is of the form A = s Id − B , where the matrix B ∈ R n × n satisfies B ≥ s > B .Also recall that a matrix A ∈ R n × n is called a Z -matrix if all its off-diagonal elements arenonpositive. Berman and Plemmons [5] give 50 equivalent characterizations of the fact that a given Z -matrix is a nonsingular M -matrix. We will need three of them here and summarize them in thefollowing statement. Theorem 3.8 (From Theorem 2.3 in Chapter 6 of [5]) . For a Z -matrix A ∈ R n × n , the followingconditions are equivalent. (a) A is a nonsingular M -matrix; (b) All the leading principal minors of A are positive. (c) A is inverse-positive; that is, A − exists and A − ≥ . (d) A + α Id is nonsingular for all α ≥ . We start with the following auxiliary lemma.
Lemma 3.9.
A triangular Z -matrix A ∈ R n × n with positive diagonal is an M -matrix.Proof. Let A = a a · · · a n a · · · a n ... . . . . . . ...0 · · · · · · a nn be an upper triangular Z -matrix with positive diagonal. Then all of its leading principle minors arepositive: A [ k ] = k (cid:89) i =1 a ii > , for k ∈ , , . . . , N . By Theorem 3.8 (b), A is an M -matrix. 20t will be convenient to define the matricesΦ ij := e − ρ | t i − − t j − | and Ψ ij := 1for i, j = 1 , . . . , N + 1. Recalling that G ( t ) = λe − ρt + γ , we then have Γ = λ Φ + γ Ψ and Γ θ = λ Φ + γ Ψ + 2 θ Id. Moreover, for any matrix A we let (cid:101) A ij := A ij if i > j , A ij if i = j ,0 otherwise.Note that this notation is consistent with (6), and we get (cid:101) Γ = λ (cid:101) Φ + γ (cid:101) Φ. We finally define (cid:98)
Φ := (cid:101)
Φ + 12 Id . Lemma 3.10.
For α ≥ , the inverse of the matrix (cid:98) Φ + α Φ is given by β − a N µβ − a N µβ · · · − a N − N µβ N − aα α β N − a N β (1 + (1 − a N ) α ) β − a N µνβ · · · − a N − N µνβ N − a N − N µβ N − a N β (1 + (1 − a N ) α ) β · · · − a N − N µνβ N − − a N − N µβ N − . . . . . . ... ...... . . . . . . − a N β (1 + (1 − a N ) α ) β − a N µβ · · · · · · − a N β β , where β = (cid:0) − a N ) α (cid:1) − , µ = (1 − a N ) α, ν = (1 − a N )(1 + α ) . Proof.
Let the matrix in the statement be denoted by P . We rewrite P as P ij = β, if i = j = 1 or i = j = N + 1;(1 + (1 − a N ) α ) β , if i = j ∈ { , . . . , N } ; − a N β, if i − j = 1; − a j − iN β j − i +2 µν, if j − i ∈ { , . . . , N − } and i (cid:54) = 1 and j (cid:54) = N + 1; − a j − iN β j − i +1 µ, if j − i ∈ { , . . . , N − } and either i = 1 or j = N + 1; − aα α β N , if i = 1 and j = N + 1 , , if i ≥ j + 2.On the other hand, the matrix (cid:98) Φ + α Φ can be written as( (cid:98)
Φ + α Φ) ij = α, if i = j ; αa j − iN , if i < j ;(1 + α ) a i − jN , if i > j. Checking N +1 (cid:88) k =1 P ik ( (cid:98) Φ + α Φ) kj = N +1 (cid:88) k =1 ( (cid:98) Φ + α Φ) ik P kj = δ ij for all i and j completes the proof. 21et (cid:98) Ψ := (cid:101) Ψ (cid:62) −
12 Id . Lemma 3.11.
The matrix Φ − (cid:0)(cid:98) Φ − γλ (cid:98) Ψ (cid:1) is a Z -matrix and a nonsingular M -matrix.Proof. It was shown in [1, Theorem 3.4] thatΦ − = 11 − a N − a N · · · · · · − a N a N − a N · · ·
00 . . . . . . . . . . . . ...... . . . . . . . . . . . . ...... . . . . . . − a N a N − a N · · · · · · − a N . (17)The matrix (cid:98) Φ − γλ (cid:98) Ψ is equal to − γλ − γλ · · · · · · − γλ a N − γλ · · · · · · − γλ a N . . . . . . . . . . . . ...... . . . . . . . . . . . . ...... . . . . . . a N − γλ a · · · · · · a N a N . A straightforward computation now yields that the matrix (1 − a N )Φ − (cid:0)(cid:98) Φ − γλ (cid:98) Ψ (cid:1) is equal to − a N − a N − γλ − (1 − a N ) γλ − (1 − a N ) γλ · · · − (1 − a N ) γλ a N γλ − a N − (1 − a N + a N ) γλ − (1 − a N ) γλ · · · − (1 − a N ) γλ a N γλ − a N − (1 − a N + a N ) γλ · · · − (1 − a N ) γλ − (1 − a N ) γλ ... ... ... 0 1 + a N γλ − a N − (1 − a N + a N ) γλ · · · · · · a N γλ , which is an upper triangular Z -matrix with positive diagonal. By Lemma 3.9, Φ − (cid:0)(cid:98) Φ − γλ (cid:98) Ψ (cid:1) is hencea nonsingular M -matrix. Lemma 3.12.
For δ ≥ the matrix Λ δ := Φ − (cid:0)(cid:98) Φ − γλ (cid:98) Ψ (cid:1) + δ Φ − is a nonsingular M -matrix.Proof. For δ = 0 the result follows from Lemma 3.11. So let us assume henceforth that δ >
0. Notefirst that Λ δ is a Z -matrix since both Φ − (cid:0)(cid:98) Φ − γλ (cid:98) Ψ (cid:1) and Φ − are Z matrices by Lemma 3.11 and(17), respectively. Hence condition (d) of Theorem 3.8 will imply that Λ δ is a nonsingular M -matrixas soon as we can show that Λ δ + α Id is invertible for all α ≥ γ = 0 in Lemma 3.11 yields that Φ − (cid:98) Φ is a nonsingular M -matrix. Hence ( α Id + Φ − (cid:98) Φ) − ≥ α ≥
0. It follows that (cid:0)(cid:98)
Φ + α Φ (cid:1) − = (cid:0) Id + ( α Φ) − (cid:98) Φ (cid:1) − ( α Φ) − = (cid:16) α Id + Φ − (cid:98) Φ (cid:17) − Φ − > , − = 11 + a N (cid:16) , − a N , . . . , − a N , (cid:17) T (cid:29) . (18)Since moreover ( (cid:98) Φ + α Φ) − is a Z -matrix by Lemma 3.10, it follows that ( (cid:98) Φ + α Φ) − is a diagonallydominant Z -matrix for all α ≥ Q := ( (cid:98) Φ + α Φ) − (cid:16) δ Id − γλ (cid:98) Ψ (cid:17) is a Z -matrix. Denoting again P := ( (cid:98) Φ + α Φ) − , we get Q ij = δP ij − γλ j − (cid:88) k =1 P ik , with the convention that (cid:80) k =1 a k = 0. It follows that Q ii ≥ i , because P ii ≥ γλ (cid:80) i − k =1 P ik ≤ P is a Z -matrix. Since P is diagonally dominant, we have (cid:80) j − k =1 P ik ≥ j > i and hence Q ij = δP ij − γλ (cid:80) j − k =1 P ik ≤ j > i . Using the fact that P ik = 0 for k ≤ i −
1, we get that for j < i Q ij = δP ij − γλ j − (cid:88) k =1 P ik = δP ij ≤ . This shows that Q is a Z -matrix.We show next that Q is a nonsingular M -matrix. To this end, we note first that the triangularmatrix (cid:16) δ Id − γλ (cid:98) Ψ (cid:17) is invertible under our assumption δ >
0. As a matter of fact, an easy calculationverifies that its inverse is given by1 δ σ σ (1 + σ ) · · · σ (1 + σ ) N − σ (1 + σ ) N − σ σ (1 + σ ) · · · σ (1 + σ ) N − σ · · · σ (1 + σ ) N − σ · · · · · · · · · ≥ , where σ := γλδ >
0. Hence, Q − = (cid:16) δ Id − γλ (cid:98) Ψ (cid:17) − ( (cid:98) Φ + α Φ) ≥ . So Theorem 3.8 (c) shows that Q is a nonsingular M -matrix.For the final step, we note first that Theorem 3.8 (d) implies that Id + Q is a nonsingular M -matrix. In particular, ( Id + Q ) − exists, and so we can define the matrix( Id + Q ) − ( (cid:98) Φ + α Φ) − Φ = (cid:16) δ Id + (cid:98)
Φ + α Φ − γλ (cid:98) Ψ (cid:17) − Φ= (cid:16) Id + ( δ Φ − ) − (cid:16) Φ − (cid:16)(cid:98) Φ − γλ (cid:98) Ψ (cid:17) + α Id (cid:17)(cid:17) − ( δ Φ − ) − = (cid:16) δ Φ − + Φ − (cid:16)(cid:98) Φ − γλ (cid:98) Ψ (cid:17) + α Id (cid:17) − = (Λ δ + α Id) − . This proves that Λ δ + α Id is invertible and the proof is complete.23 emma 3.13.
Let A be an invertible matrix and suppose that α ∈ R is such that A + α Ψ is invertible.Then the vector A − is proportional to ( A + α Ψ) − .Proof. Note that Ψ x is proportional to for any vector x . Hence,( A + α Ψ) A − = ( Id + α Ψ A − ) = (1 + β ) for some constant β . Applying ( A + α Ψ) − to both sides of this equation yields the result.We are now ready to prove the remaining implication of Theorem 2.7. Proof of (b) ⇒ (c) in Theorem 2.7. We need to show that v has only nonnegative components for θ ≥ λ + γ . The vector v is proportional to (Γ θ + (cid:101) Γ) − . When setting δ := 4 θ − ( λ + γ )2 λ ≥ , we find thatΓ θ + (cid:101) Γ − γ Ψ = λ Φ + γ Ψ + 2 θ Id + λ (cid:101) Φ + γ (cid:101) Ψ − γ Ψ= λ Φ + 2 θ Id + λ (cid:101) Φ − γ (cid:101) Ψ (cid:62) = λ Φ + λ (cid:16)(cid:98) Φ − γλ (cid:98) Ψ + δ Id (cid:17) = λ Φ (cid:0) Λ δ + Id (cid:1) , and we know from Lemma 3.12 that the latter matrix is invertible. It therefore follows fromLemma 3.13 that v is proportional to (cid:0) Φ(Λ δ + Id) (cid:1) − = (Λ δ + Id) − Φ − . As noted in (18), we have Φ − (cid:29)
0. Moreover, Λ δ , and hence Λ δ + Id, are nonsingular M -matrices byLemma 3.12 and Theorem 3.8 (d). Via Theorem 3.8 (c), these facts imply that (cid:0) Φ(Λ δ + Id) (cid:1) − ≥ v ≥ We have considered a Nash equilibrium for two competing agents in a market impact model withgeneral transient price impact. We have seen that without transaction costs both agents engagein a “hot-potato game”, which has some similarities to certain events during the Flash Crash thathave been reported in [10, 17]. We have then analyzed the behavior of equilibrium strategies asfunctions of transaction costs, θ , and trading frequency, N . In Theorem 2.7 we have determined thecritical value of transaction costs at which the equilibrium strategies v and w become buy-only orsell-only. In Section 2.4, numerical simulations have shown that expected costs can be increasingin the trading frequency for small θ , while they generally decrease for sufficiently large θ . We havealso seen that the expected costs of both agents can be lower with additional transaction costs thanwithout. These observations provide some support for the common claim that additional transactioncosts can, at least under certain circumstances such as during a fire sale, have a calming effect onfinancial markets. Acknowledgement:
The authors thank Ria Grindel, Elias Strehle, and an anonymous referee forcomments that helped to improve previous versions of the manuscript.24 eferences [1] A. Alfonsi, A. Fruth, and A. Schied. Constrained portfolio liquidation in a limit order bookmodel.
Banach Center Publications , 83:9–25, 2008.[2] A. Alfonsi, A. Fruth, and A. Schied. Optimal execution strategies in limit order books withgeneral shape functions.
Quant. Finance , 10:143–157, 2010.[3] A. Alfonsi, A. Schied, and A. Slynko. Order book resilience, price manipulation, and the positiveportfolio problem.
SIAM J. Financial Math. , 3:511–533, 2012.[4] R. Almgren and N. Chriss. Optimal execution of portfolio transactions.
Journal of Risk , 3:5–39,2000.[5] A. Berman and R. J. Plemmons.
Nonnegative matrices in the mathematical sciences , volume 9of
Classics in Applied Mathematics . Society for Industrial and Applied Mathematics (SIAM),Philadelphia, PA, 1994. Revised reprint of the 1979 original.[6] D. Bertsimas and A. Lo. Optimal control of execution costs.
Journal of Financial Markets ,1:1–50, 1998.[7] J.-P. Bouchaud, Y. Gefen, M. Potters, and M. Wyart. Fluctuations and response in financialmarkets: the subtle nature of ‘random’ price changes.
Quant. Finance , 4:176–190, 2004.[8] M. K. Brunnermeier and L. H. Pedersen. Predatory trading.
Journal of Finance , 60(4):1825–1863, August 2005.[9] B. I. Carlin, M. S. Lobo, and S. Viswanathan. Episodic liquidity crises: cooperative and preda-tory trading.
Journal of Finance , 65:2235–2274, 2007.[10] CFTC-SEC. Findings regarding the market events of May 6, 2010. Report, 2010.[11] D. Easley, M. L´opez de Prado, and M. O’Hara. The microstructure of the flash crash: Flow toxi-city, liquidity crashes and the probability of informed trading.
Journal of Portfolio Management ,37(2):118–128, 2011.[12] J. Gatheral. No-dynamic-arbitrage and market impact.
Quant. Finance , 10:749–759, 2010.[13] J. Gatheral and A. Schied. Dynamical models of market impact and algorithms for orderexecution. In J.-P. Fouque and J. Langsam, editors,
Handbook on Systemic Risk , pages 579–602.Cambridge University Press, 2013.[14] G. Huberman and W. Stanzl. Price manipulation and quasi-arbitrage.
Econometrica ,72(4):1247–1275, 07 2004.[15] W. B. Jurkat. Questions of signs in power series.
Proc. Amer. Math. Soc. , 5:964–970, 1954.[16] T. Kaluza. ¨Uber die Koeffizienten reziproker Potenzreihen.
Math. Z. , 28:161–170, 1928.[17] A. A. Kirilenko, A. S. Kyle, M. Samadi, and T. Tuzun. The flash crash: The impact of highfrequency trading on an electronic market.
Preprint, available at SSRN 1686004 , 2010.[18] F. Kl¨ock, A. Schied, and Y. Sun. Price manipulation in a market impact model with dark pool.
Preprint , 2011. 2519] C.-A. Lehalle. Market microstructure knowledge needed to control an intra-day trading process.In J.-P. Fouque and J. Langsam, editors,
Handbook on Systemic Risk , pages 549–578. CambridgeUniversity Press, 2013.[20] C. C. Moallemi, B. Park, and B. Van Roy. Strategic execution in the presence of an uninformedarbitrageur.
Journal of Financial Markets , 15(4):361 – 391, 2012.[21] A. Obizhaeva and J. Wang. Optimal trading strategy and supply/demand dynamics.
Journalof Financial Markets , 16:1–32, 2013.[22] S. Predoiu, G. Shaikhet, and S. Shreve. Optimal execution in a general one-sided limit-orderbook.
SIAM J. Financial Math. , 2:183–212, 2011.[23] A. Schied, E. Strehle, and T. Zhang. High-frequency limit of nash equilibria in a market impactgame with transient price impact. arXiv:1509.08281 , 2015.[24] A. Schied and T. Zhang. A state-constrained differential game arising in optimal portfolioliquidation.
To appear in Mathematical Finance , 2017.[25] T. Sch¨oneborn. Trade execution in illiquid markets. Optimal stochastic control and multi-agentequilibria. Doctoral dissertation, TU Berlin, 2008.[26] T. Sch¨oneborn and A. Schied. Liquidation in the face of adversity: stealth vs. sunshine trading.SSRN Preprint 1007014, 2009.[27] G. Szeg˝o. Bemerkungen zu einer Arbeit von Herrn Fej´er ¨uber die Legendreschen Polynome.
Math. Z. , 25:172–187, 1926.[28] W. F. Trench. Inverses of lower triangular toeplitz matrices.