High-Frequency Trading and Limit Order Books

Transcription

1 High-Frequency Trading and Limit Order Books Part II Alexander Schied University of Mannheim Princeton RTG Summer School in Financial Mathematics Princeton University June 17 28,

2 1 Introduction What happened during the Flash Crash of May 6, 2010? From (CFTC-SEC 2010):... 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 took more than 5 hours for this large trader to execute the first 75,000 contracts of a large sell program. However, on May 6, when markets were already under stress, the Sell Algorithm chosen by the large Fundamental Seller to only target trading volume, and not price nor time, executed the sell program extremely rapidly in just 20 minutes. 2

3 Why did this algorithm execute the large trade so rapidly? In addition to the large fundamental seller there were high-frequency traders (HFTs) active as predators From (CFTC-SEC 2010, page 3):... HFTs began to quickly buy and then resell contracts to each other generating a hot-potato volume e ect as the same positions were rapidly passed back and forth. Between 2:45:13 and 2:45:27, HFTs traded over 27,000 contracts [worth approx. $1.5 billion!], which accounted for about 49 percent of the total trading volume, while buying only about 200 additional contracts net. 3

4 Why did this algorithm execute the large trade so rapidly? In addition to the large fundamental seller there were high-frequency traders (HFTs) active as predators From (CFTC-SEC 2010, page 3):... HFTs began to quickly buy and then resell contracts to each other generating a hot-potato volume e ect as the same positions were rapidly passed back and forth. Between 2:45:13 and 2:45:27, HFTs traded over 27,000 contracts [worth approx. $1.5 billion!], which accounted for about 49 percent of the total trading volume, while buying only about 200 additional contracts net. Why did this hot-potato game between HFTs happen? Two possibilities... In any case, clearly related to price impact of trades 3

5 In this part of the course: Look at some models for market microstructure / limit order books / price impact from the point of view of stability / regularity 4

6 In this part of the course: Look at some models for market microstructure / limit order books / price impact from the point of view of stability / regularity The regularity of asset pricing models has been investigated for a long time and is by now well understood. Regularity notions include no arbitrage no free lunch with vanishing risk (NFLVR) existence of equivalent (local / sigma) martingale measures no scalable arbitrage robust no-arbitrage... 4

7 The regularity of market impact models has been much less well understood. To my knowledge, the first paper is Huberman & Stanzl (2004) The following notions have so far been considered: no price manipulation no quasi-arbitrage no transaction-triggered price manipulation nonnegative expected liquidation costs These notions are perhaps not as natural as the arbitrage conditions for asset pricing. Perhaps one can do much better. We will also look at stability issues in multi-player situations. There are no formal regularity notions in this setup yet. 5

8 Overview of planned course 1. Introduction 2. Regularity of Almgren Chriss type models 2.1 The Almgren Chriss model 2.2 Other models with permanent and temporary price impact 2.3 Adding a dark pool 3. Transient price impact 3.1 Linear transient impact 3.2 Nonlinear impact, exponential resilience 3.3 Gatheral s model 4. Two-player games under transient price impact 4.1 Drift-dependence and partial equilibrium 4.2 A hot-potato game 6

9 Let us recall from last week the basic notions of a generic market impact model. Trade execution strategy: X t = asset position held at time t 2 [0, T ]. The initial position X 0 is positive for a sell strategy and negative for a buy strategy. The condition X T = 0 assures that the initial position has been unwound by time T. The path X = (X t ) t2[0,t ] will be nonincreasing for a pure sell strategy and nondecreasing for a pure buy strategy. A general strategy may consist of both buy and sell trades and hence can be described as the sum of a nonincreasing and a nondecreasing strategy. That is, X is a path of finite variation. 7

10 Market impact model: Describes the quantitative feedback of such a trade execution strategy on asset prices. Exogenously given asset price dynamics S 0 = (St 0 ) t X t = 0 for all t, the una ected price process. 0 for the case when S 0 is a semimartingale on a filtered probability space (, F, (F t ) t 0, P) and all trade execution strategies are predictable. When the strategy X is used, the price is changed from St 0 to St X, and each market impact model has a particular way of describing this change. Typically, a pure buy strategy X will lead to an increase of prices, i.e., S X t S 0 t for t 2 [0, T ], while a pure sell strategy will decrease prices. This e ect is responsible for the liquidation costs of a trade execution strategy. 8

11 Revenues and execution costs: Let us assume that X t is continuous in time and that St X depends continuously on the part of X that has been executed by time t. Then, at each time t, the infinitesimal amount of dx t shares is sold at price St X. Thus, the total revenues obtained from the strategy X are and the liquidation costs are R T (X) = Z T 0 S X t dx t, C T (X) = X 0 S 0 0 R T (X). When X is not continuous it may be necessary to add correction terms to these formulas. 9

12 Optimal trade execution: Maximize revenues or, equivalently, minimize costs in the class of all strategies that liquidate a given initial position of X 0 shares during a predetermined time interval [0, T ]. Optimality is usually understood in the sense that a certain risk functional is optimized. Commonly used risk functionals involve expected value as in Bertsimas & Lo (1998), Gatheral (2010) and others, mean-variance criteria as in Almgren & Chriss (1999, 2000), expected utility as in A.S., Schöneborn & Tehranchi (2010), A.S. & Schöneborn (2009) and Schöneborn (2011), or alternative risk criteria as in Forsyth, Kennedy, Tse & Windclif (2012) and Gatheral & A.S. (2011). 10

13 Defining regularity for market impact models The model must admit optimal trade execution strategies for reasonable risk criteria. Optimal strategies ought to be well-behaved. E.g., optimal execution strategy for a sell order X 0 > 0 should not involve intermediate buy orders (as long as market conditions stay reasonable) Regularity conditions should be independent of investors preferences Hence formulate conditions in terms of expected revenues or costs. One should distinguish the e ects of price impact from profitable investment strategies that can arise via trend following. Therefore, we will assume from now on that (1) S 0 is a martingale when considering the regularity or irregularity of a market impact model. Condition (1) is a common assumption in the literature even when regularity is not under investigation. 11

14 The first regularity condition was introduced by Huberman & Stanzl (2004). Definition 1 (Price manipulation). A round trip is a trade execution strategy X with X 0 = X T = 0. A price manipulation strategy is a round trip X with strictly positive expected revenues, (2) E[ R T (X) ] > 0. A price manipulation strategy allows price impact to be exploited in a favorable manner. Thus, models that admit price manipulation provide an incentive to implement such strategies (perhaps in hidden and embedded form within a more complex trading algorithm). The existence of price manipulation can often preclude the existence of optimal execution strategies for risk-neutral investors, due to the possibility of generating arbitrarily large expected revenues by adding price manipulation strategies to a given trade execution strategy. 12

15 In some models, rescaling and repeating price manipulation can lead to a weak form of arbitrage, called quasi-arbitrage. But there is also a di erence between the notions of price manipulation and arbitrage, namely price manipulation is defined as the possibility of average profits, while classical arbitrage is defined in an almost-sure sense. In a derivatives pricing model, one seeks strategies that almost surely replicate a given contingent claim. In a market impact model, one seeks trade execution strategies that are defined as minimizers of a cost functional of a risk averse investor. It is therefore completely natural to formulate regularity conditions for market impact models in terms of expected revenues or costs. 13

16 But absence of price manipulation may not be su stability of a market impact model (see below). cient to guarantee the Definition 2 (Transaction-triggered price manipulation). A market impact model admits transaction-triggered price manipulation if the expected revenues of a sell (buy) program can be increased by intermediate buy (sell) trades. Definition 3 (Negative expected liquidation costs). A market impact model admits negative expected liquidation costs if there exists T > 0 and a corresponding trade execution strategy X for which (3) E[ C T (X) ] < 0, or, equivalently, E[ R T (X) ] > X 0 S 0. 14

17 Proposition 1. (a) Any market impact model that does not admit negative expected liquidation costs does also not admit price manipulation. (b) Suppose that asset prices are decreased by sell orders and increased by buy orders. Then the absence of transaction-triggered price manipulation implies that the model does not admit negative expected liquidation costs. In particular, the absence of transaction-triggered price manipulation implies the absence of price manipulation in the usual sense. Proof: Exercise 15

18 Mean-variance optimization corresponds to maximization of a mean-variance functional of the form (4) E[ C T (X) ] + var(c T (X)), where 0 is a risk aversion parameter. Often, price manipulation strategies can become profitable for mean-variance optimizers: Suppose that " n # 0 is a sequence such that for each n there exists a price manipulation strategy X n for which C T (X n ) = " n C T (X). Then E[ C T (X n ) ] + var(c T (X n )) = " n E[ C T (X) ] + " n var(c T (X)), and the right-hand side is strictly positive for su ciently large n. 16

19 2 Regularity of Almgren Chriss type models 2.1 The Almgren Chriss model Bertsimas & Lo (1998), Madhavan (2000), Almgren & Chriss (1999, 2000), Konishi & Makimoto (2001), Almgren (2003) Recalling from day 2 of Rene Carmona s part: Strategies (X t ) t2[0,t ] : absolutely continuous functions of time. Price impact: for two nondecreasing functions g, h : R! R, (5) S X t = S 0 t + h(ẋt) temporary price impact R t 0 g(ẋs) ds permanent price impact Z t 0 g(ẋs) ds + h(ẋt) The una ected stock price is often taken as a Bachelier model, (6) S 0 t = S 0 + W t, where W is a standard Brownian motion. 17

20 Revenues of a strategy X: R T (X) = = Z T 0 Z T 0 S X t = X 0 S dx t S 0 t dx t Z T 0 = X 0 S0 0 C T (X) where Z T 0 X t ds 0 t Z t Ẋ t g(ẋs) ds dt 0 Z T 0 Z T 0 Z t Ẋ t g(ẋs) ds dt 0 Ẋ t h(ẋt) dt Z T 0 f(ẋt) dt (7) f(x) = xh(x). 18

21 For the particular case h = 0, the next proposition was proved first by Huberman & Stanzl (2004) in a discrete-time version of the Almgren Chriss model and by Gatheral (2010) in continuous time. Proposition 2. If an Almgren Chriss model does not admit price manipulation for all T > 0, then g must be linear, i.e., g(x) = x with a constant 0. Proof: 1st step. We show that we must have g(v) = g( v) when h = 0. Take a round trip X 0 with Ẋ 0 t = 8 < v for 0 apple t apple T/2 : v for T/2 < t apple T 19

22 Expected costs are h Z T E[ C T (X 0 ) ] = E = = = 0 Z T Ẋt 0 0 Z T/2 0 vt 2 8 X 0 t ds 0 t + Z t 0 Z T 0 g(ẋ0 s ) ds dt vtg(v) dt + Z T T/2 g(v) + g( v) Ẋ 0 t Z t 0 i g(ẋ0 s ) ds dt T ( v) 2 g(v) + (t T/2)g( v) dt When there exists x such that g(x) 6= g( x) we can get negative expected costs by by taking v = x when x(g(x) + g( x)) > 0, by taking v = x otherwise. 20

23 2nd step. Showing that nonlinear g leads to price manipulation when h = 0: Define a deterministic round trip (Xt 1 ) 0appletappleT by letting 8 < Ẋt 1 v 1 if 0 apple t apple = : if < t apple T where v 2 = T v 2 v 1 + v 2 Assuming h = 0, compute the expected costs using g(v) = g( v) from step 1 to show that one can get R T R Ẋ 1 t 0 t 0 g(ẋ1 s ) ds dt < 0, just as before. 21

24 3rd step. Suppose now that g is nonlinear and h does not vanish. For " > 0, we now define X " t = 1 " X1 "t, 0 apple t apple T " := 1 " T. Then (X " t ) 0appletappleT" is again a round trip with Ẋ" t = Ẋ1 "t with E[ C T" (X " ) ] = = Z T" Z t Ẋ t " 0 0 Z T/" Z t Ẋ"t Z t Ẋt = 1 " 2 Z T g(ẋ" s ) ds dt + g(ẋ1 "s) ds dt + Z T" 0 Z T/" 0 g(ẋ1 s ) ds dt + " f(ẋ" t ) dt Z T 0 f(ẋ1 "t) dt f(ẋ1 t ) dt. Due to step 2, when " is small enough, the term in parentheses will be strictly negative, and consequently X " will be a price manipulation strategy. 22

25 When g(x) = x for some 0, the revenues of a trade execution strategy X simplify and are given by Z T Z T R T (X) = X 0 S X t ds 0 t 2 X2 0 0 f(ẋt) dt. Proposition 3. Suppose that g(x) = x for some 0 and the function f in (7) is convex. Then for every X 0 2 R and each T > 0 the strategy (8) Xt := X 0(T t), 0 apple t apple T, T maximizes the expected revenues E[ R T (X) ] in the class of all adapted and bounded trade execution strategies (X t ) 0appletappleT. The strategy X in (8) can be regarded as a TWAP strategy (time-weighted average price). When the time parameter t does not measure physical time but volume time, it is a VWAP strategy (volume-weighted average price) 23

26 By means of Proposition 1, the next result follows. Corollary 1. Suppose that g(x) = x for some 0 and the function f in (7) is convex. Then the Almgren Chriss model is free of transaction-triggered price manipulation, negative expected liquidation costs, and price manipulation. The assumptions that g is linear and that f is convex is not inconsistent with empirical observation; see Almgren, Thum, Hauptmann & Li (2005), where it was argued that f(x) is well approximated by a multiple of the power law function x 1+ with

27 The Almgren Chriss model is highly tractable and can easily be generalized to multi-asset situations; see, e.g., Konishi & Makimoto (2001) or Schöneborn (2011). Mean-variance optimization corresponds to maximization of a mean-variance functional of the form (9) E[ R T (X) ] var(r T (X)), where 0 is a risk aversion parameter. This problem was studied by Almgren & Chriss (1999, 2000), Almgren (2003), and Lorenz & Almgren (2011). The first three papers solve the problem for deterministic trade execution strategies, while the latter one gives results on mean-variance optimization over adaptive strategies. This latter problem is much more di cult than the former, mainly due to the time inconsistency of the mean-variance functional. When forcing time consistency, one obtains a connection with Dawson Watanabe superprocesses (A.S. 2013). Konishi & Makimoto (2001) study the closely related problem of maximizing the functional for which variance is replaced by standard deviation, i.e., by the square root of the variance. 25

28 Expected-utility maximization corresponds to the maximization of (10) E[ u(r T (X)) ], where u : R! R is a concave and increasing utility function. In contrast to the mean-variance functional, expected utility is time consistent, which facilitates the use of stochastic control techniques. 26

29 2.2 Other models with permanent and temporary price impact The Bertsimas Lo model The Bertsimas Lo model was introduced in Bertsimas & Lo (1998) to remedy the possible occurrence of negative prices in the Almgren Chriss model. In the following continuous-time variant of the Bertsimas Lo model, the price impact of an absolutely continuous trade execution strategy X acts in a multiplicative manner on una ected asset prices: Z (11) St X = St 0 t exp g(ẋs) ds + h(ẋt), 0 for two nondecreasing functions g, h : R! R that describe the respective permanent and temporary impact components. The una ected price process S 0 is often taken as (risk-neutral) geometric Brownian motion: St 0 2 = exp W t 2 t, where W is a standard Brownian motion and 27 is a nonzero volatility

30 parameter. The following result was proved by Forsyth et al. (2012). Proposition 4. When g(x) = x for some 0, the Bertsimas Lo model does not admit price manipulation in the class of bounded trade execution strategies. The computation of optimal trade execution strategies is more complicated in this model than in the Almgren Chriss model. We refer to Bertsimas & Lo (1998) for a dynamic programming approach to the maximization of the expected revenues in the discrete-time version of the model. Forsyth et al. (2012) use Hamilton Jacobi Bellman equations to analyze trade execution strategies that optimize a risk functional consisting of the expected revenues and the expected quadratic variation of the portfolio value process. 28

31 Further models An early market impact model described in the academic literature is the one by Frey & Stremme (1997). In this model, price impact is obtained through a microeconomic equilibrium analysis. As a result of this analysis, permanent price impact of the following form is obtained: (12) S X t = F (t, X t, W t ) for a function F and a standard Brownian motion W. This form of permanent price impact has been further generalized by Baum (2001) and Bank & Baum (2004) by assuming a smooth family (S t (x)) x2r of continuous semimartingales. The process (S t (x)) t 0 is interpreted as the asset price when the investor holds the constant amount of x shares. The asset price resulting from a strategy (X t ) 0appletappleT is then given as S X t = S t (X t ). The dynamics of such an asset price can be computed via the Itô-Wentzell formula. This analysis reveals that continuous trade execution strategies of bounded variation do not create any liquidation costs (Bank & Baum 2004, 29

32 Lemma 3.2). Since any reasonable trading strategy can be approximated by such strategies (Bank & Baum 2004, Theorem 4.4), it follows that, at least asymptotically, the e ects of price impact can always be avoided in this model. A related model for temporary price impact was introduced by Çetin, Jarrow & Protter (2004). Here, a similar class (S t (x)) x2r of processes is used, but the interpretation of x 7! S t (x) is now that of a supply curve for shares available at time t. Informally, the infinitesimal order dx t is then executed at price S t (dx t ). Also in this model, continuous trade execution strategies of bounded variation do not create any liquidation costs (Çetin et al. 2004, Lemma 2.1). The model has been extended by Roch (2011) so as to allow for additional price impact components. We also refer to the survey paper Gökay, Roch & Soner (2011) for an overview for further developments and applications of this model class and for other, related models. 30

33 2.3 Adding a dark pool to an Almgren Chriss model Section based on Klöck, A.S. & Sun (2011). See also Kratz & Schöneborn (2010, 2012) Execution of orders at an exchange creates market impact, which has an adverse e ect on prices and which can provide a signal to predatory trading algorithms. To avoid these e ects, alternative trading venues such as dark pools have been created. In a dark pool, orders are invisible to other market participants and trade execution does not generate price impact (although there may be indirect e ects). Hence, large orders can be executed at lower cost and without information leakage 31

34 Typically, dark pools have no price finding mechanism. Orders are executed, for instance, at the mid price quoted at the exchange. Possible questions: Do dark pools make markets more e cient? Do they stabilize the system or do they create instabilities? 32

35 There can be an incentive for price manipulation strategies. E.g., - place sell order in dark pool - simultaneously buy at exchange to drive up price Specifically, there are fishing strategies in which predatory trading algorithms place small orders in a dark pool so as to detect hidden liquidity. Once hidden liquidity is found, the price at the exchange is manipulated. Then a dark-pool order is placed. See Mittal (2008). 33

36 There can be an incentive for price manipulation strategies. E.g., - place sell order in dark pool - simultaneously buy at exchange to drive up price Specifically, there are fishing strategies in which predatory trading algorithms place small orders in a dark pool so as to detect hidden liquidity. Once hidden liquidity is found, the price at the exchange is manipulated. Then a dark-pool order is placed. See Mittal (2008). Here, we investigate what happens if we add a dark pool to an Almgren Chriss model. 33

37 An Almgren Chriss model is defined in terms of the parameters We assume: (, h, S 0 ) (S 0 t ) t 0 is a càdlàg martingale on (, F, (F t ), P), where F 0 is P-trivial strategies (X t ) are adapted to (F t ) > 0 h is continuous, strictly increasing, and satisfies h(0) = 0 and h(x)! 1 for x! 1 the function f(x) := xh(x) is convex As seen above, these assumptions exclude all notions of price manipulation 34

38 Adding a dark pool In addition to trading at the exchange, investors can place an order of bx shares into the dark pool at time t = 0. This order will be matched with incoming orders of the opposite side. These orders arrive at random times and sizes 0 < 1 < 2 <... and Ỹ 1, Ỹ2,... > 0 We consider only orders that are a possible match: the Ỹi will describe sell orders when b X > 0 is a buy order and buy orders when b X < 0 is a sell order. By Y i := 8 >< sgn ( X)Ỹi, b if P i j=1 Ỹj apple b X, bx sgn ( X) b P i 1 j=1 >: Y j, if P i 1 j=1 Ỹj apple X b and P i j=1 Ỹj > X, b 0, if P i 1 j=1 Ỹj > X, b we describe the part of the incoming order that is actually executed against the remainder of b X. 35

39 The amount of shares that have been executed in the dark pool until time t is Z t := XN t i=1 where N t counts the number of fills until time t: Y i N t := inf{k 2 N k apple t} Assumption 1. We assume the following conditions: (13) 0 < 1 < 1 P-a.s. and 0 := inf 0< apple1 1 P[ 1 apple ] > 0; (14) there exists x 0 > 0 such that 1 := inf >0 P[ Ỹ1 x 0 1 apple ] > 0. We furthermore assume that S 0 is a martingale also under the filtration (G t ) generated by (F t ) and Z. 36

40 To execute an order of X 0 shares, the investor can first place an order of bx 2 R shares in the dark pool and then choose a liquidation strategy of Almgren Chriss-type for the execution of the remaining assets at the exchange. This latter strategy will be described by a process ( t ) that parameterizes the speed by which shares are sold on the exchange. Until fully executed, the remaining part of the order X b can be cancelled at a stopping time < T. Hence, the number of shares held by the investor at time t is X t := X 0 + Z t 0 s ds + Z t All such strategies must satisfy the liquidation constraint X T = X 0 + Z T 0 t dt + Z = 0 37

41 (Virtual) price at exchange Z t S t = St s ds + Z t + h( t ) The parameter describes the permanent impact of a dark pool execution on the exchange price. Why should this impact exist? Without the dark pool, the matching order would have been executed at the exchange and generated permanent price impact in a favorable direction. Thus, the term Z t can be understood in terms of a deficiency in opposite price impact. 38

42 (Virtual) price of i th incoming order in dark pool Ŝ i = S 0 i + Z i 0 s ds + Z i + Y i + appleh( i ) Orders executed at the exchange have full permanent impact, but their possible temporary impact di ers from temporary impact at exchange by a constant apple 0. The parameter 0 describes additional slippage related to the dark-pool execution, which will result in transaction costs of the size Yi 2. In addition to ordinary transaction costs and taxes, there may be hidden costs related to using dark pools such as those arising from the phenomena of adverse selection or fishing. In addition, data may be sparse so that there will be a high degree of model uncertainty. A high value of can thus also serve as a penalization in view of possible model misspecification. 39

43 (Virtual) price of i th incoming order in dark pool Ŝ i = S 0 i + Z i 0 s ds + Z i + Y i + appleh( i ) With these definitions, the revenues of a strategy ( b X,, ) are R T = Z T 0 s S s ds N X T ^ i=1 Y i Ŝ i 39

44 We assume: 2 [0, 1] 0 apple 0 Definition 4. The dark-pool extension of a given Almgren Chriss model is defined in terms of (,, apple, ( i ), (Ỹi)) 40

45 General regularity results Theorem 1. For given (,, apple, ( i ), (Ỹi)), the following are equivalent. (a) For any Almgren Chriss model, the dark-pool extension does not admit negative expected liquidation costs for every time horizon T > 0. (b) For any Almgren Chriss model, the dark-pool extension does not admit price manipulation for every time horizon T > 0. (c) We have = 1, 1 2 and apple = 0. 41

46 General regularity results Theorem 1. For given (,, apple, ( i ), (Ỹi)), the following are equivalent. (a) For any Almgren Chriss model, the dark-pool extension does not admit negative expected liquidation costs for every time horizon T > 0. (b) For any Almgren Chriss model, the dark-pool extension does not admit price manipulation for every time horizon T > 0. (c) We have = 1, 1 2 and apple = 0. Remark 1. = 1 means that an execution of a dark-pool order must generate the same permanent impact as a similar order executed at the exchange. 1 2 means that the execution of a dark-pool order of size Y i needs to generate slippage of at least 2 Yi 2. This latter amount is just equal to the costs from permanent impact one would have incurred by executing the order at the exchange. apple = 0 means that temporary impact from trades executed at the exchange must not a ect the price at which dark-pool orders are executed. 41

47 General regularity results Theorem 1. For given (,, apple, ( i ), (Ỹi)), the following are equivalent. (a) For any Almgren Chriss model, the dark-pool extension does not admit negative expected liquidation costs for every time horizon T > 0. (b) For any Almgren Chriss model, the dark-pool extension does not admit price manipulation for every time horizon T > 0. (c) We have = 1, 1 2 and apple = 0. 1 Parameter values = 1, 2 pools. and apple = 0 won t be found for real-world dark 41

48 In Theorem 1, it is crucial that we may vary at least one of the parameters or h of the underlying Almgren-Chriss model. If both parameters are fixed, we can no longer conclude that = 1 and only obtain the following implication instead of an equivalent characterization of regular models. Theorem 2. Suppose an Almgren Chriss model with parameters (, h, P 0 ) has been fixed. When a dark-pool extension (,, apple, ( i ), (Ỹi)) of this model does not admit price manipulation for all T > 0, then (15) If, in addition, there is equality in (15) then apple = 0 and =

49 Special model characteristics We will make the following simplifying but natural assumption on the dark-pool extension defined through (,, apple, ( i ), (Ỹi)). Assumption 2. From now on, we assume the following conditions. (a) Slippage is zero: = 0. (b) The process (N t ) is a standard Poisson process with parameter > 0 and (Ỹi) are i.i.d. random variables with common distribution µ on (0, 1]. We also assume that the stochastic processes (S 0 t ), (N t ), and (Ỹi) are independent. (c) There is a fixed Almgren Chriss model with linear temporary impact, i.e., h(x) = x 43

50 What can one say about the role played by T in the existence of price manipulation. Proposition 5. Suppose that = 1. If apple = 0 then there is no price manipulation if and only if (16) T apple 2. 44

51 We now assume that (17) = 0, apple = 0, µ = 1 Proposition 6. For X 0 2 R and < 2, (18) lim T "1 sup 2X (X 0,T ) E[R T ] = X 0 P X Equation (18) is remarkable, because it implies that the condition of positive expected liquidation costs is violated, while there is no price manipulation and T = 1. Corollary 2. There is no price manipulation for every T > 0 if and only if apple 2 But there is always transaction-triggered price manipulation for su ciently large T. 45

52 3 Transient price impact 46

53 Limit order book before market order buyers bid offers sellers ask offers bid-ask spread best bid price best ask price prices 47

54 Limit order book before market order volume of sell market order buyers bid offers sellers ask offers bid-ask spread best bid price best ask price 48

55 Limit order book after market order volume of sell market order buyers bid offers sellers ask offers new bid-ask spread new best bid price best ask price 49

56 Resilience of the limit order book after market order buyers bid offers sellers ask offers new best bid price new best ask price 50

57 There are some first models in which limit order books are modeled as stochastic queuing or particle systems. See, e.g., Avellaneda & Stoikov (2008), Bovier, Černý & Hryniv (2006), Cont & de Larrard (2013, 2010), Cont, Kukanov & Stoikov (2010) and other papers mentioned in Rene Carmona s part. Here, we consider an idealized limit order book with continuous, flat distribution of limit orders introduced by Obizhaeva & Wang (2013). Models with nonconstant or stochastic resilience and/or limit order distribution were recently analyzed in Alfonsi & Infante Acevedo (2012), Klöck (2012), Fruth, Schöneborn & Urusov (2011) 51

58 Limit order book model without large trader buyers bid offers sellers ask offers unaffected best bid price, is martingale unaffected best ask price 52

59 Limit order book model after large trades actual best bid price actual best ask price 53

60 Limit order book model at large trade t = q(b t B t+ ) sell order executed at average price B t B t+ xq dx similarly for buy orders q( 54

61 Limit order book model immediately after large trade 55

62 Resilience of the limit order book t G( t) + decay of previous trades B t+ t 56

63 Model with collapsed bid-ask spread buyers bid offers sellers ask offers 57

64 3.1 Linear transient impact Obizhaeva & Wang (2013), Alfonsi et al. (2008) used exponential decay and Gatheral (2010), Alfonsi et al. (2012), and Gatheral et al. (2012) use general decay as proposed by Bouchaud et al. (2004) and Gatheral (2010) Section based on Alfonsi et al. (2012), and Gatheral et al. (2012) Una ected price process: martingale S 0 Admissible trategy: predictable process X = (X t ) that describes the number of shares held by the trader t! X t is leftcontinuous with finite total variation the signed measure dx t has compact support w.l.o.g. X t = 0 for large enough t. Note: These strategies are of bounded variation. When they are also continuous there will be no liquidation costs in models such as those from Bank & Baum (2004), Çetin et al. (2004) 58

65 Impacted price process: S X t = S 0 t + Z {s<t} G(t s) dx s, where G : (0, 1)! [0, 1) is the decay kernel. It describes the empirical resilience of price impact between trades; see, e.g., Bouchaud et al. (2004), Moro, Vicente, Moyano, Gerig, Farmer, Vaglica, Lillo & Mantegna (2009) We first assume (19) G is bounded and G(0) := lim t#0 G(t) exists. 59

66 Costs of a strategy X: When X is continuous at t, then the infinitesimal order dx t is executed at price St X, so St X dx t is the cost increment. Thus, the total expenses of a continuous strategy are Z Z Z Z St X dx t = St 0 dx t + G(t s) dx s dx t. {s<t} When X has a jump X t, then the price is moved from S X t to S X t+ = S X t + X t G(0) This linear price impact corresponds to a constant supply curve for which G(0) 1 dy buy or sell orders are available at each price y. The trade X t is thus carried out at the following cost, Z S X t+ S X t yg(0) 1 dy = 1 2G(0) (SX t+) 2 (S X t ) 2 = G(0) 2 ( X t) 2 + X t S X t. 60

67 Hence, the total costs of an arbitrary admissible strategy X are given by Z St X dx t + G(0) X ( Xt ) 2 2 Z Z Z = St 0 dx t + G(t s) dx s dx t + G(0) X ( Xt ) 2 {s<t} 2 Z = St 0 dx t + 1 Z Z G( t s ) dx s dx t. 2 It therefore follows from the martingale property of S 0 that the expected costs of an admissible strategy are h Z i X 0 S0 0 + E St 0 dx t + 1 E[ C(X) ], 2 where C(X) := Z Z G( t s ) dx s dx t. 61

68 Next if T is such that X T = 0, then Z S 0 t dx t = X 0 S 0 0 Hence, X T S 0 T h Z i E St 0 dx t Z T 0 = X 0 S 0 0, X t ds 0 t. and the expected costs are 1 E[ C(X) ]. 2 62

69 Remark: Instead of this simple market impact model, one can consider models for (block-shaped) electronic limit order books with nonzero bid-ask spread. In these models one can then show that Expected costs 1 E[ C(X) ] 2 with equality for monotone strategies X; see Alfonsi et al. (2008), Alfonsi & A.S. (2010). 63

70 Two questions: Can there be model irregularities? Existence, uniqueness, and structure of strategies minimizing the expected costs? Recall: Definition 5. A price manipulation strategy is a round trip with strictly negative expected costs. Clearly, there is no price manipulation when C(X) 0 for all strategies X. Such a situation was analyzed by Bochner (1932): 64

71 Proposition 7 (Straightforward extension of Bochner s thm). C(X) 0 for all strategies X () G( ) can be represented as the Fourier transform of a positive finite Borel measure µ on R, i.e., Z G( x ) = e ixz µ(dz); (G is positive definite). If, in addition, the support of µ is not discrete, then C(X) > 0 for every nonzero admissible strategy X (G is strictly positive definite). Remark 2. Suppose that X is a step function with jumps at times t 0,..., t N, i.e., X X t = X 0 i. t i <t Then C(X) = X i j G( t i t j ) 65

72 Proof of Proposition 7: Suppose first that C(X) 0 for all strategies X. When considering strategies with discrete support we are in the context of Bochner s theorem, and so G( ) must be the Fourier transform of a positive finite Borel measure µ on R. Conversely, suppose that G( x ) = R R eixz µ(dz). When X is an admissible strategy, then Z Z Z C(X) = e iz(t s) µ(dz) dx s dx t = Z Z e izt dx t Z e izs dx s µ(dz) = Z b X(z) 2 µ(dz) 0, where b X(z) = R e itz dx t is the Fourier transform of X. It is well-defined due to our assumption that X has compact support. 66

73 Let us finally show that C is even positive definite when the support of µ is not discrete. Since X has compact support, the function X(z) b has a continuation to an entire analytic function on the complex plane. Indeed, one easily uses Lebesgue s theorem to see that Z bx(z) = e itz dx t is finite and di erentiable as a function of z 2 C. Hence, for X 6= 0, the zero set of X b must be a discrete set. Thus, for the integral Z C(X) = X(z) b 2 µ(dz) to vanish, the measure µ needs to have discrete support. 67

74 Optimal trade execution problem: Minimizing expected costs, 1 E[ C(X) ] 2 for strategies that liquidate a given long or short position of y shares within a given time frame. Time constraint: compact set T [0, 1). Boils down to minimizing C( ) over n o X (y, T) := X deterministic strategy with X 0 = y and support in T. 68

75 Suppose first that T is discrete, i.e., T = {t 0,..., t N }. Then the problem is equivalent to minimize NX i,j=0 x i x j G( t i t j ) over x 2 R with x > 1 = y where 1 = (1,..., 1) > Minimizers always exist when G is positive definite. When G is strictly positive definite, the optimal x is proportional to the solution of Mx = 1, i.e., to M 1 1 where M ij = G( t i t j ) Existence of minimizers not clear when T is not discrete. 69

76 Proposition 8. When G is strictly positive definite then there exists at most one optimal strategy for given y and T. Proof: Let C(X, Y ) = 1 2 C(X + Y ) C(X) C(Y ) = Z Z G( t s ) dx s dy t First, X 6= Y implies that 0 < C(X Y ) = C(X) + C(Y ) 2C(X, Y ). Therefore, 1 C 2 X Y = 1 4 C(X) C(Y ) C(X, Y ) < 1 2 C(X) + 1 C(Y ), 2 which implies the uniqueness of optimal trade execution strategies when they exist. 70