Optimal execution with liquidity risk in a diffusive order book market
OOptimal execution with liquidity risk in a diffusive orderbook market
Hyoeun Lee ∗ , Kiseop Lee † April 24, 2020
Abstract
We study the optimal order placement strategy with the presence of a liquidity cost. In thisproblem, a stock trader wishes to clear her large inventory by a predetermined time horizon T . A trader uses both limit and market orders, and a large market order faces an adverse pricemovement caused by the liquidity risk. First, we study a single period model where the traderplaces a limit order and/or a market order at the beginning. We show the behavior of optimalamount of market order, m ∗ , and optimal placement of limit order, y ∗ , under different marketconditions. Next, we extend it to a multi-period model, where the trader makes sequentialdecisions of limit and market orders at multiple time points. In general, stock traders often need to handle a large order. Usually, the first step is to split a largeorder into multiple small orders before placing. This is to reduce the unfavorable price movementsto the trader caused by a large order. Selling an asset tends to move the price downward, whilebuying an asset tends to move the price upward. This effect is often called as a liquidity cost, aprice impact, or a market impact. In this paper, we are going to call this as a liquidity cost.The liquidity cost affects the optimal strategy of traders, since a large initial market order mayface a hefty liquidity cost. The optimal execution strategy under a liquidity cost has been studiedextensively.The pioneering work of Bertsimas and Lo [5] and Almgren and Chriss [3] consider a linear im-pact, such that the liquidity cost is proportional to the number shares of an order. Bank and Baum[4], C¸ etin et al. [8] , and Frey and Patie [13] investigate deeply on the liquidity cost and additionaltransient impacts. Obizhaeva and Wang [22] consider a transient (linear) impact additionally. Atransient impact is that the impact from a market order is not permanent, but only exists for ashort moment and vanishes.However, Potters and Bouchaud [23], Eisler et al. [11], and Donier [10] empirical showed thatthe liquidity cost is not linear, but rather concave. Alfonsi et al. [1] , Predoiu et al. [24], Gatheral[14] and Gu´eant [16], and many others, have proposed extensions or alternatives to Obizhaevaand Wang [22] with a nonlinear liquidity cost. ∗ Department of Statistics, University of Illinois, Email:[email protected] † Department of Statistics, Purdue University, Email: [email protected] a r X i v : . [ q -f i n . C P ] A p r urthermore, instead of the exponential decay of the transient impact, more general decay ker-nels are considered by Alfonsi et al. [2] and Gatheral et al. [15] since the exponential decay is notobserved on market data.When placing orders, the second step for investors is to decide between a market order anda limit order. A market order (MO) is an order to buy/sell an asset at the best available price.Market orders are executed immediately, while the best available price might be affected adversely(liquidity cost). Most studies so far have focused on an execution problem using solely a marketorder.A limit order (LO) is an order to buy/sell an asset at a specific price. The price of a limit order isspecified by the buyer/seller, but the execution is not guaranteed. A limit order can be cancelledby the buyer/seller before execution.A limit order book (LOB) collects all quantities and the price of limit orders. The LOB is updatedupon execution of market orders, submission of limit orders, or cancellation of pre-existing limitorders.Recently, more researchers like Jacquier and Liu [19], Alfonsi et al. [1], Guilbaud and Pham [17],Maglaras et al. [21], Cont and Kukanov [9] consider both market orders and limit orders, instead offocusing on the market order only. However, only the best bid or the best ask prices are consideredas options.Cartea et al. [7], Guo et al. [18], Figueroa-L ´opez et al. [12] have considered whether placing thelimit order deeper in the book could be preferable, which is often called as an optimal placementproblem. In [7], the optimal placement problem is studied under a continuous-time model for themid-price, a mean-reverting process with jumps. [18] investigate the optimal placement problemunder a discrete-time model for the level I prices of a LOB. In [12], a problem similar to [18] isstudied under a continuous-time model, and shows that there exists an optimal placement policydifferent from the Level I-II solution of [18].While [18] and [12] studied the optimal placement behavior with a small size order without aliquidity cost, practically we cannot entirely ignore the liquidity cost. Therefore, in this paper, wesolve the optimal execution problem with liquidity cost, while considering both market ordersand limit orders as options. For the limit order, the optimal price level is also investigated as in[12]. For the price impact model, we borrow the liquidity risk introduced in [8] but we add thenecessary features of LOB.We first investigate the optimal placement problem under the single-period setting, where theinvestor can place an order only at the beginning. The investor needs to clear the inventory (ofsize M ) by a certain time horizon T . At t =
0, the investor decides the quantity of the market order m (0 ≤ m ≤ M ), and the rest M − m will be placed using the limit order. The price level for thelimit order, y , is also determined at t =
0. At time T , any unexecuted limit order is immediatelyexecuted using a market order.Next, we study an analogous problem with a multi-period setting, where the investor makesdecisions at multiple time steps, { T / n , 2 T / n , . . . T } . At each time step, the investor cancelsexisting un-executed limit orders and place new market and limit orders. At time T , similar tosingle-period setting, any remaining inventory is immediately executed using a market order.We model the asset price using the Brownian and the geometric Brownian motion, and alsomodel the liquidity cost from the investor’s market order; we use the model derived from [8], butalso include the basic feature of Limit Order book. We investigate the behavior of optimal m ∗ and y ∗ which maximize the expected cash flow from the order placement.2he rest of the paper is organized as follows. We briefly explain the liquidity risk introduced in[8] in section 2. In section 3, we discuss the optimal execution strategy in one period model. Weextend it to a multi-period case in section 4. We conclude in section 5. We recall the concepts introduced in the work of C¸ etin et al. [8]. We consider a market with a riskyasset and a money market account. The risky asset, stock, pays no dividend and we assume thatthe spot rate of interest is zero, without loss of generality. S ( t , x , ω ) represents the stock price pershare at time t ∈ [ T ] that the trader pays/receives for an order of size x ∈ R given the state ω ∈ Ω . A positive order( x >
0) represents a buy, a negative order ( x <
0) represents a sale,and the zeroth order ( x =
0) corresponds to the marginal trade. For the detailed structure of thesupply curve, we refer to Section 2 of C¸ etin et al. [8].A trading strategy (portfolio) is a triplet (( X t , Y t : t ∈ [ T ]) , τ ) where X t represents the trader’saggregate stock holding at time t (units of the stock), Y t represents the trader’s aggregate moneyaccount position at time t (units of money market account), and τ represents the liquidation timeof the stock position. Here, X t and Y t are predictable and optional processes, respectively, with X − ≡ Y − ≡
0. A self-financing strategy is a trading strategy (( X t , Y t : t ∈ [ T ]) , τ ) where X t iscadlag if ∂ S ∂ x = t , and X t is cadlag with finite quadratic variation ( [ X , X ] T < ∞ ) otherwise,and Y t = Y + X S ( X ) + (cid:90) t X u dS ( u , 0 ) − X t S ( t , 0 ) − ∑ ≤ u ≤ t ∆ X u [ S ( u , ∆ X u ) − S ( u , 0 )] − (cid:90) t ∂ S ∂ x ( u , 0 ) d [ X , X ] cu . (2.1)Therefore, it is natural to define the liquidity cost of a self-financing trading strategy ( X , Y , τ ) by L t = ∑ ≤ u ≤ t ∆ X u [ S ( u , ∆ X u ) − S ( u , 0 )] + (cid:82) t ∂ S ∂ x ( u , 0 ) d [ X , X ] cu , 0 < t ≤ T , where L − =
0, and L = X [ S ( X ) − S (
0, 0 )] .A practically important problem of a trader is how to execute a large order in the market withliquidity risk. Guo et al. [18] studied an optimal placement problem under a discrete time setting.Figueroa-Lopez et al. [12] studied an optimal placement problem in continuous time setting, butboth studies did not consider the liquidity cost. We plan to study how to place an order usingboth market and limit orders in a market with liquidity risk. As the first step, let us consider a simple case where the marginal price S ( t , 0 ) is a diffusion process.The goal is to sell M orders by time horizon T . We consider two cases. In a single period case,we will study the optimal placement problem in a single step (decision is made once when t = t = t , . . . , t N = T ), and at each time step the traderplace market/limit order. 3or the supply curve, we use S ( t , x ) = S ( t , 0 ) − d − β ( K + x ) − , β > x < S ( t , 0 ) = S + µ t + σ W t , or S ( t , 0 ) = S e ( µ − σ ) t + σ W t ,where d ( > ) is the half of the bid and ask spread, and K is the initial market depth at the bestbid price. Therefore, the price does not move up to the first K shares, then moves down linearlyafterward. [6], [8] suggest linear supply curve: For liquid stocks, the supply curve is linear in x and for highly illiquid commodities, [6] suggests a jump-linear supply curve. In our work, weapply a linear curve, and just use one β since we focus on only one side (sell). Also, we considerthe necessary limit order book feature, the market depth K .First, let us consider a single period case. The placement is made only once at time 0. At time0, the trader makes a decision. If a trader sells M orders using only a market order at time 0, thetrader’s cash flow becomes M ∗ ( S ( − M ) − f ) where f is the fee per an executed market order.A more general case is when a trader sells m orders using a market order at time 0, and places a M − m sell limit order at the price level S (
0, 0 ) + y , y ≥ d . Any remaining unexecuted orders attime T are converted to a market order and are executed immediately with paying the liquiditycost.Let τ be the first time that the trader’s limit order becomes the best ask. In other word, τ is thefirst time S (
0, 0 ) + y becomes the best ask. Let L be the number of executed shares at the pricelevel S (
0, 0 ) + y and denote L = ( M − m ) ρ where ρ is the proportion of the execution.Notice that even when the trader’s limit order becomes the best ask, there is no guarantee of ex-ecution, since there will be orders at the exact same level from other traders too. The trader’s cashflow from the initial market order is m ∗ ( S ( − m ) − f ) . If τ > T , no limit order will be executedand all will be put as a market order. Therefore, the expected cash flow from the remaining M − m shares will be ( M − m ) E ( S ( T , − ( M − m )) − f | τ > T ) . When τ < T , the expected cash flow fromthe remaining M − m shares becomes E ( L ( S (
0, 0 ) + y + r )) + E (( M − m − L )( S ( T , − ( M − m )) − f ) | τ < T ) , where r is the rebate per an executed limit order.Our goal is to find the optimal ( m , y ) which maximizes the expected cash flow m ( S ( − m ) − f ) + ( M − m ) E ( S ( T , − ( M − m )) − f | τ > T ) P ( τ > T )+ { E ( L ( S (
0, 0 ) + y + r )) + E (( M − m − L )( S ( T , − ( M − m − L )) − f ) | τ < T ) } P ( τ < T ) . (3.1)To do this, we need to calculate the conditional expectation E ( S ( T , 0 ) | τ < T ) . Since S ( t , 0 ) is afunction of a Brownian motion, the distribution of τ is related to the hitting time of a Brownianmotion. It is well known that it is obtained by the reflection property of a Brownian motion andits running maximum. That is a common problem especially in a barrier option. It is also knownthat when τ a is the hitting time of a for a standard Brownian motion B t , 0 < b ≤ a , and M t is itsrunning maximum, we have P ( B T < b | τ a < T ) = P ( B T > a + ( a − b ) | τ a < T ) ,and P ( M T ≥ a , B T < b ) = P ( M T ≥ a , B T > a − b ) .We will apply this joint distribution to calculate the conditional expectation E ( S ( T , 0 ) | τ < T ) . Itbecomes a function of y only. 4ext two lemmas give us useful equations about expected stock prices and the probability of thelimit order execution. Lemma 3.1 describes the expected stock price, which follows a Brownianmotion, in two different cases: when the stock price hits a certain price before t and when hittingdoes not happen until t . Lemma 3.2 shows analogous expected stock prices when the stock pricefollows a geometric Brownian motion. Remark 1.
For the rest of the paper, the pdf, cdf, and survival or tail distribution of a standard normal r.v.Z are denoted by φ ( z ) = e − z /2 / √ π , N ( z ) = (cid:82) z − ∞ φ ( x ) dx. B t is a standard Brownian motion. Lemma 3.1.
Let us assume that the price process follows a Brownian motion with drift, S ( t , 0 ) = S (
0, 0 ) + µ t + σ B t . Let τ y : = inf u { S ( u , 0 ) + d = S (
0, 0 ) + y } . ThenP ( τ y > t ) = N (cid:18) ( y − d ) − µ t σ √ t (cid:19) − e ( y − d ) µσ N (cid:18) − ( y − d ) − µ t σ √ t (cid:19) , E [( S ( t , 0 ) − S (
0, 0 )) I ( τ y > t )] = µ tN (cid:18) y − d − µ t σ √ t (cid:19) + e ( y − d ) µσ ( − ( y − d ) − µ t ) N (cid:18) − ( y − d ) − µ t σ √ t (cid:19) , E [( S ( t , 0 ) − S (
0, 0 )) I ( τ y < t )] = µ tN (cid:18) − ( y − d ) + µ t σ √ t (cid:19) + e ( y − d ) µσ ( ( y − d ) + µ t ) N (cid:18) − ( y − d ) − µ t σ √ t (cid:19) . Proof.
Let M t : = max u ≤ t S ( u , 0 ) − S (
0, 0 ) . From the definition of τ y , { τ y > t } = { M t ≤ y − d } . FromJeanblanc et al. [20], note that P (( S ( t , 0 ) − S (
0, 0 )) ∈ dz , M t ≤ a ) = φ (cid:18) z − µ t σ √ t (cid:19) σ √ t − e µ a φ (cid:18) z − a − µ t σ √ t (cid:19) .Then P ( τ y > t ) = P ( M t ≤ y − d ) = (cid:90) ( y − d ) − ∞ φ (cid:18) z − µ t σ √ t (cid:19) σ √ t − e µ ( y − d ) φ (cid:18) z − ( y − d ) − µ t σ √ t (cid:19) dz = N (cid:18) ( y − d ) − µ t σ √ t (cid:19) − e ( y − d ) µσ N (cid:18) − ( y − d ) − µ t σ √ t (cid:19) .Next, E [( S ( t , 0 ) − S (
0, 0 )) I ( τ y > t )] = E [( S ( t , 0 ) − S (
0, 0 )) I ( M t ≤ ( y − d ))]= (cid:90) ( y − d ) − ∞ zP (( S ( t , 0 ) − S (
0, 0 )) ∈ dz , M t ≤ ( y − d ))= (cid:90) ( y − d ) − ∞ z φ (cid:18) z − µ t σ √ t (cid:19) σ √ t − e µ ( y − d ) φ (cid:18) z − ( y − d ) − µ t σ √ t (cid:19) dz = µ tN (cid:18) y − d − µ t σ √ t (cid:19) + e ( y − d ) µσ ( − ( y − d ) − µ t ) N (cid:18) − ( y − d ) − µ t σ √ t (cid:19) .Finally, E [( S ( t , 0 ) − S (
0, 0 )) I ( τ y < t )] = E [( S ( t , 0 ) − S (
0, 0 ))] − E [( S ( t , 0 ) − S (
0, 0 )) I ( τ y > t )]= µ t − E [( S ( t , 0 ) − S (
0, 0 )) I ( τ y > t )]= µ tN (cid:18) − ( y − d ) + µ t σ √ t (cid:19) + e ( y − d ) µσ ( ( y − d ) + µ t ) N (cid:18) − ( y − d ) − µ t σ √ t (cid:19) .5he next lemma gives us similar results when the price process is a geometric Brownian motion. Lemma 3.2.
Let us assume that the price process follows a geometric Brownian motion, dS ( t , 0 ) = µ S ( t , 0 ) dt + σ S ( t , 0 ) dB t . Also, let’s denote τ y : = inf u { S ( u , 0 ) = S (
0, 0 ) + y − d } . ThenP ( τ y > t ) = N (cid:18) a − µ t + σ /2 σ √ t (cid:19) − e a µσ − a N (cid:18) − a − µ t + σ t /2 σ √ t (cid:19) , E [ S ( t , 0 ) I ( τ y > t )] = S (
0, 0 ) (cid:18) e µ t N (cid:18) a − µ t + σ t /2 σ √ t (cid:19) − e a µσ + µ t + a N (cid:18) − a − µ t + σ t /2 σ √ t (cid:19)(cid:19) , E [ S ( t , 0 ) I ( τ y < t )] = S (
0, 0 ) (cid:18) e µ t N (cid:18) − a + µ t − σ t /2 σ √ t (cid:19) + e a µσ + µ t + a N (cid:18) − a − µ t + σ t /2 σ √ t (cid:19)(cid:19) . where a : = ln (cid:18) ( S (
0, 0 ) + y − d ) S (
0, 0 ) (cid:19) . Proof.
Let X t = ln ( S ( t , 0 ) / S (
0, 0 )) . Then X t = ( µ − σ /2 ) t + σ B t . Let’s denote M t : = max u ≤ t X u .From the definition of τ y , note that { τ y > t } = { M t ≤ ln (cid:16) ( S ( )+ y − d ) S ( ) (cid:17) } . Then, we may apply theproof of Lemma 3.1 by using a = ln (cid:16) ( S ( )+ y − d ) S ( ) (cid:17) and P ( X t ∈ dz , M t ≤ a ) = φ (cid:18) z − ( µ − σ ) t σ √ t (cid:19) σ √ t − e ( µ − σ ) a φ (cid:18) z − a − ( µ − σ ) t σ √ t (cid:19) .First, P ( τ y > t ) = P ( M t ≤ a ) = (cid:90) a − ∞ φ (cid:32) z − ( µ − σ ) t σ √ t (cid:33) σ √ t − e ( µ − σ ) a φ (cid:32) z − a − ( µ − σ ) t σ √ t (cid:33) dz = N (cid:18) a − µ t + σ /2 σ √ t (cid:19) − e a µσ − a N (cid:18) − a − µ t + σ t /2 σ √ t (cid:19) Next, E (cid:20) S ( t , 0 ) S (
0, 0 ) I ( τ y > t ) (cid:21) = E [ e X t I ( M t ≤ a )] = (cid:90) a − ∞ e z P ( X t ∈ dz , M t ≤ a )= (cid:18) e µ t N (cid:18) a − µ t + σ t /2 σ √ t (cid:19) − e a µσ + µ t + a N (cid:18) − a − µ t + σ t /2 σ √ t (cid:19)(cid:19) , E (cid:20) S ( t , 0 ) S (
0, 0 ) I ( τ y < t ) (cid:21) = E [ e X t ] − E [ e X t I ( M t ≤ a )]= e µ t − (cid:18) e µ t N (cid:18) a − µ t + σ t /2 σ √ t (cid:19) − e a µσ + µ t + a N (cid:18) − a − µ t + σ t /2 σ √ t (cid:19)(cid:19) = (cid:18) e µ t N (cid:18) − a + µ t − σ t /2 σ √ t (cid:19) + e a µσ + µ t + a N (cid:18) − a − µ t + σ t /2 σ √ t (cid:19)(cid:19) .Using Lemma 3.1 , we can calculate the expected cash flow (ECF) for a Brownian motion model.Similarly, the ECF for a geometric Brownian model is explicitly shown in Lemma 3.4 using resultsfrom Lemma 3.2. 6 emark 2. For the rest of the paper, we use the following notations: α t = y − d + µ t σ √ t , β t = y − d − µ t σ √ t , ˜ N ( − α t ) = e ( y − d ) µ / σ N ( − α t ) , (cid:101) = (cid:0) N ( β T ) − ˜ N ( − α T ) (cid:1) . Lemma 3.3.
Let us assume that the price process follows a Brownian motion, S ( t , 0 ) = S (
0, 0 ) + µ t + σ B t .Then ECF ( y , m ) can be summarized as follows:ECF ( y , m ) = MS (
0, 0 ) + m ( − d − β ( K − m ) − − f )+ ( M − m ) (cid:8) µ TN ( β T ) − ( ( y − d ) + µ T ) ˜ N ( − α T ) + (cid:101) ( − d − β ( K − M + m ) − − f ) (cid:9) + ( − (cid:101) ) (cid:0) E [ L ]( y + r ) − E [( M − m − L )( d + β ( K − M + m + L ) − + f )] (cid:1) + ( M − m − E [ L ]) (cid:0) µ TN ( − β T ) + ( ( y − d ) + µ T ) ˜ N ( − α T ) (cid:1) . (3.2) Proof.
By definition of S ( t , − m ) , ECF ( y , m ) can be rewritten as follows: ECF ( y , m ) = m ( S − d − β ( K − m ) − − f )+ ( M − m ) (cid:8) E [ S ( T , 0 ) I ( τ > T )] + P ( τ > T )( − d − β ( K − M + m ) − − f ) (cid:9) + E [ L ]( S (
0, 0 ) + y + r ) P ( τ < T ) − E [( M − m − L )( d + β ( K − M + m + L ) − + f )] P ( τ < T )+ E [( M − m − L )] E [( S ( T , 0 ) I ( τ < T )] . (3.3)From Lemma 3.1, note that E [ S ( t , 0 ) I ( τ > t )] = S (
0, 0 ) P ( τ > t ) + µ tN ( β t ) − ( ( y − d ) + µ t ) ˜ N ( − α t ) , E [ S ( t , 0 ) I ( τ < t )] = S (
0, 0 ) P ( τ < t ) + µ tN ( − β t ) + ( ( y − d ) + µ t ) ˜ N ( − α t ) , P ( τ > t ) = N ( β t ) − ˜ N ( − α t ) . (3.4)Then, by plugging expressions in (3.4) into (3.3), we have the final expression as in (3.2).Similarly, using Lemma 2, we can calculate the expected cash flow for a geometric Brownianmotion model. Lemma 3.4.
Let us assume that the price process follows geometric Brownian motion, dS t = µ S t dt + σ S t dB t . Then ECF ( y , m ) can be summarized as follows:ECF ( y , m ) = m ( S − d − β ( K − m ) − − f )+ ( M − m ) (cid:110) S (cid:16) e µ T N (cid:0) ˜ β T (cid:1) − ˜ N ( − ˜ α T ) (cid:17) + ( N (cid:0) ˜ β T (cid:1) − ˜ N ( − ˜ α T ))( − d − β ( K − M + m ) − − f ) (cid:111) + E [ L ]( S (
0, 0 ) + y + r ) (cid:0) N (cid:0) − ˜ β T (cid:1) + ˜ N ( − ˜ α T ) (cid:1) − E [( M − m − L )( d + β ( K − M + m + L ) − + f )] (cid:0) N (cid:0) − ˜ β T (cid:1) + ˜ N ( − ˜ α T ) (cid:1) + E [( M − m − L )] S (cid:16) e µ T N (cid:0) − ˜ β T (cid:1) + ˜ N ( − ˜ α T ) (cid:17) , (3.5) wherea : = ln (cid:18) ( S + y − d ) S (cid:19) , ˜ α t = a + µ t − σ t /2 σ √ t , ˜ β t = a − µ t + σ t /2 σ √ t , ˜ N ( − ˜ α t ) = e a µσ + µ t + a N ( − ˜ α t ) .7 roof. By definition of S ( t , − m ) , ECF ( y , m ) can be rewritten as follows: ECF ( y , m ) = m ( S − d − β ( K − m ) − − f )+ ( M − m ) (cid:8) E [ S ( T , 0 ) I ( τ > T )] + P ( τ > T )( − d − β ( K − M + m ) − − f ) (cid:9) + E [ L ]( S (
0, 0 ) + y + r ) P ( τ < T ) − E [( M − m − L )( d + β ( K − M + m + L ) − + f )] P ( τ < T )+ E [( M − m − L )] E [( S ( T , 0 ) I ( τ < T )] . (3.6)From Lemma 3.2, note that E [ S ( T , 0 ) I ( τ > T )] = S (cid:16) e µ T N (cid:0) ˜ β T (cid:1) − ˜ N ( − ˜ α T ) (cid:17) , E [ S ( T , 0 ) I ( τ < T )] = S (cid:16) e µ T N (cid:0) − ˜ β T (cid:1) + ˜ N ( − ˜ α T ) (cid:17) , P ( τ > T ) = N (cid:0) ˜ β T (cid:1) − ˜ N ( − ˜ α T ) , (3.7)Then, by plugging expressions in (3.7) into (3.6), we have the final expression as in (3.5). ρ with a Brownian motion Model Let us recall that L = ( M − m ) ρ , ρ ∈ [
0, 1 ] , where ρ is the proportion of the execution such that L = ( M − m ) ρ . In this subsection, we work on the simple case when ρ is a constant. Remark 3.
In practical situation, a constant ρ is an unrealistic assumption. A more realistic case for ρ is arandom model, such as ρ = + π arctan ( X ) where X follows a normal distribution, which could dependon y and m. Or, ρ could be a function of flows and time where we understand arrivals of orders as somecounting process. However, for simplicity, we consider ρ as a constant in this paper. In the following Lemma, we introduce the ECF (Expected Cash Flow) function when the priceprocess follows a Brownian motion.
Lemma 3.5.
Let us assume that the price process follows Brownian motion, S ( t , 0 ) = S (
0, 0 ) + µ t + σ B t ,where B t is a standard Brownian motion. When L = ( M − m ) ρ , ECF ( y , m ) in (3.2) can be expressed asfollows:ECF ( y , m ) = M (cid:0) S − ρ ( y − d ) ˜ N ( − α T ) − ( d + f − µ T )( − ρ + ρ(cid:101) ) + ( − (cid:101) ) ρ ( y + r ) (cid:1) − m (cid:0) ρ ( − (cid:101) )( d + f + y + r ) − ρ ( y − d ) ˜ N ( − α T ) + µ T ( − ρ + ρ(cid:101) ) (cid:1) − β (cid:2) m ( K − m ) − + ( M − m ) (cid:0) (cid:101) ( K − M + m ) − + ( − (cid:101) )( − ρ )( K − ( M − m )( − ρ )) − (cid:1)(cid:3) , where the notations α T , β T , (cid:101) are from Remark 2.Proof. This result is from the equation (3.2) of Lemma 3.3 with L = ( M − m ) ρ .Using this Lemma 3.5, we now study the optimal placement strategy. Next two theorems, The-orem 3.6 and Theorem 3.7 give us optimal strategies when µ < µ >
0, respectively. We findthe optimal ( y ∗ , m ∗ ) , which means that the optimal strategy at time 0 is to place a market order ofquantity m ∗ , and place remaining M − m ∗ orders by a limit order at the price of S (
0, 0 ) + y ∗ .8 heorem 3.6. Let ( y ∗ , m ∗ ) be the optimal strategy satisfyingECF ( y ∗ , m ∗ ) ≥ ECF ( y , m ) ∀ y >
0, 0 ≤ m ≤ M . Under the cash flow model from Lemma 3.5, let µ < . Then y ∗ = d. The behavior of m ∗ is as follows:1. If ρ = , m ∗ = .2. If ρ < and K ≥ M, • m ∗ = when ρ ( d + r + f ) + µ t ( − ρ ) > • m ∗ = M when ρ ( d + r + f ) + µ t ( − ρ ) <
3. If ρ < and K < M,m ∗ is one of { K , M − K / ( − ρ ) , M , m ∗ , m ∗ , m ∗ } wherem ∗ = K /2 − (cid:101) /2 β , m ∗ = M − K ( − ρ ) − (cid:101) ( − ρ ) β , m ∗ = M β ( − ρ ) + βρ K − (cid:101) β ( + ( − ρ ) ) , (cid:101) : = ρ ( d + r + f ) + µ t ( − ρ ) . Proof.
Part 1: Behavior of y ∗ : Note that ∂ ECF ∂ y ( M − m ) = ρ ( N ( − β t ) − ˜ N ( − α t )) (3.8) + e ( y − d ) µσ σ √ t ( φ ( α t ) − α t N ( − α t )) ( − ρ ( d + f + r − µ t ) + βγ ) (3.9) + ˜ N ( − α t ) ( y − d ) σ t ( − ρ ( d + f + r ) + βγ ) (3.10) γ : = min ( K − M + m , 0 ) − ( − ρ ) min ( K − ( M − m )( − ρ ) , 0 ) ( note : γ < ) When µ <
0, all three lines (3.8), (3.9), (3.10) are negative, so y ∗ = d . Part 2: Behavior of m ∗ : We’ve already shown that when µ < y ∗ = d . Therefore, ECF ( y = y ∗ , m ) can be simplified as following: ECF ( y = d , m ) = M ( S + ρ ( d + r ) − ( − ρ )( d + f − µ t )) − m (cid:101) + m β min ( K − m , 0 )+ β ( M − m )( − ρ ) min ( K − ( M − m )( − ρ ) , 0 ) , (3.11)where (cid:101) : = ρ ( d + r + f ) + µ t ( − ρ ) . We will divide the range of m to four cases to obtain theexpression of ECF ( y , m ) and to obtain m ∗ which maximizes ECF .1. K − ( M − m )( − ρ ) ≥ K − m ≥ m * against m , r =0.2 m K=10K=50K=70K=150 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 . . . . . . y * against m , r =0.2 m K=10K=50K=70K=150
Figure 1: m ∗ (Left) and y ∗ (Right) against µ for K =
10 (black solid line), K =
50 (red dashedline), K =
70 (green dotted line), K =
150 (blue dot-dashed line). ρ = β = M = r = f = (cid:101) = T = S = d = (cid:101) /2 = σ = m if (cid:101) > m if (cid:101) >
0. Note that if K ≥ M , for any m the conditions ( K − ( M − m )( − ρ ) ≥ K − m ≥ K ≥ M , m ∗ = (cid:101) > m ∗ = M if (cid:101) <
0. This includes thecase when ρ =
1: when ρ = (cid:101) = d + r + f >
0, so m ∗ = K − ( M − m )( − ρ ) ≥ m ≥ M − K / ( − ρ ) ), K − m < m ∗ = K /2 − (cid:101) /2 β .3. K − ( M − m )( − ρ ) < m < M − K / ( − ρ ) ), K − m ≥ m ∗ = M − K ( − ρ ) − (cid:101) ( − ρ ) β .4. K − ( M − m )( − ρ ) < m < M − K / ( − ρ ) ), K − m < m ∗ = M β ( − ρ ) + βρ K − (cid:101) β ( +( − ρ ) ) .To summarize, when K ≥ M , m ∗ = M depending on the sign of (cid:101) . If not, the maximumwill be obtained at one of { m ∗ , m ∗ , m ∗ , M , K , M − K / ( − ρ ) } .Theorem 3.6 shows that when µ < y ∗ = d , which means that the investor may put a limitorder at the best bid price. This is shown in behavior of y ∗ in the right panel of Figure 1, y ∗ = d when µ <
0. The left panel of Figure 1 describes behavior of m ∗ , the optimal market order size at t =
0. As shown in Theorem 3.6, when µ < m ∗ = m ∗ when K =
10, and m ∗ = K for K =
50, 70,and m ∗ = M when K = y ∗ and m ∗ when µ > T , the threshold for the time horizon T such thatif the investor’s time horizon T is bigger than T , then y ∗ > d . We also provide lower bound for T and stepwise-linear approximation of y ∗ as a function of T . Note that this work is about single-step. T is not the number of the time steps, but it is just the length of time forthe single step. heorem 3.7. Let ( y ∗ , m ∗ ) be defined as in Lemma 3.5. Then, when µ > ,y ∗ > d for T > T . The lower bound of T is thatT > d + f + r µ . Also, for T > T , as T (cid:38) T , the first-order approximation for y ∗ is given as:y ∗ ( T ) = d + κ ( T − T ) + o (( T − T ) ) , where κ : = − ∂ ECF ∂ T ∂ y ( d , T ) ∂ ECF ∂ y ( d , T ) . The details of κ are given in (3.17). In addition, there exists m ∗ as follows:1. If K ≥ M , m ∗ = .2. If K < M, m ∗ is one of { M , m ∗ , m ∗ , m ∗ , M , | K − M | , M − K / ( − ρ ) } where m ∗ = (cid:101) ( + (cid:101) ) M + ( − (cid:101) ) ( + (cid:101) ) K − (cid:101) ( + (cid:101) ) β , m ∗ = M − β ( (cid:101) + ( − (cid:101) )( − ρ )) K + (cid:101) β ( (cid:101) + ( − (cid:101) )( − ρ ) ) , m ∗ = β ( M ( (cid:101) + ( − ρ ) ( − (cid:101) )) + K ( − (cid:101) ) ρ ) − (cid:101) β ( + (cid:101) + ( − ρ ) ( − (cid:101) )) . (cid:101) : = (cid:0) ρ ( − (cid:101) )( d + f + y + r ) − ρ ( y − d ) ˜ N ( − α T ) + µ T ( − ρ + ρ(cid:101) ) (cid:1) , (cid:101) = (cid:0) N ( β T ) − ˜ N ( − α T ) (cid:1) . Proof.
Part 1: Behavior of y ∗ : First, note that as y → ∞ , from (3.8), (3.9), (3.10), ∂ ECF / ∂ y → − .Therefore, y ∗ < ∞ . Next, let’s find ∂ ECF / ∂ y | y = d . ∂ ECF ∂ y ( M − m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = d = ρ (cid:18) N (cid:18) µ T σ √ T (cid:19) − N (cid:18) − µ T σ √ T (cid:19)(cid:19) (3.12) + σ √ t (cid:18) φ (cid:18) − µ T σ √ T (cid:19) − µ T σ √ T N (cid:18) − µ T σ √ t (cid:19)(cid:19) ( − ρ ( d + f + r − µ T ) + βγ ) (3.13)Note that (3.12) >
0, and (3.13) > µ T > d + f + r + ( − βγ ) / ρ . (3.14)Note that ∂ ECF / ∂ y | y = d > y ∗ > d . Since from the proof of Theorem 3.6 we’veshown that γ < µ T > d + f + r + ( − βγ ) / ρ > d + f + r , so the lower bound for T to satisfythis condition is ( d + f + r ) / µ . (we ignore − βγ / ρ term since the definition of γ contains m , so itcan be correctly computed after we have the value of m ∗ . )11 ∂ t ∂ ECF ∂ y ( M − m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = d = − γβ (cid:18) σ T √ T ( φ ( h t ) − h t N ( − h T )) + µσ t N ( − h T ) (cid:19) (3.15) + ρσ T √ T (cid:18) ( d + f + r ) (cid:18) ( φ ( h T ) − h t N ( − h T )) + N ( − h T ) µ T σ √ T (cid:19) + µ T ( φ ( h T ) − h T N ( − h T )) (cid:19) (3.16)Note that both (3.15), (3.16) are positive. This implies that ∂ ECF ∂ y (cid:12)(cid:12)(cid:12) y = d is a increasing function of T ,which supports that for T such that ∂ ECF ∂ y (cid:12)(cid:12)(cid:12) y = d , T = T = ∂ ECF ∂ y (cid:12)(cid:12)(cid:12) y = d > T > T .Now, we will use the mean value theorem to show the behavior of y ∗ when T is close to T . Tothis end, the following conditions are necessary: (cid:12)(cid:12)(cid:12) ∂ y ECF (cid:12)(cid:12)(cid:12) needs to be strictly positive at y = d , T = T . ∂ y ECF ( y = d , T = T ) = − ρ ( M − m ) µ N ( − h t ) / σ , so | ∂ y ECF ( y = d , T = T ) | >
0. Next,since y ∗ satisfies ∂ y ECF ( y ∗ ) =
0, and thus, by the Implicit Function Theorem, there exists an openset U containing y = d , an open set V containing T = T , and a unique continuously differentiablefunction y ∗ ( T ) such that { ( y ∗ ( t ) , T | T ∈ V } = (cid:26) ( y , T ) ∈ U × V (cid:12)(cid:12)(cid:12)(cid:12) ∂ ECF ∂ y ( y , T ) = (cid:27) .In particular, y ∗ ( T ) → d as T → T . Furthermore, since T >
0, it is clear that ∂ y ECF is differen-tiable in a neighborhood of ( y = d , T = T ) , and, thus, we can apply the mean value theorem toshow that there exists δ ∈ (
0, 1 ) such that0 = ∂ ECF ∂ y ( y ∗ ( T ) , T ) = ∂ ECF ∂ y ( d + δ ( y ∗ ( T ) − d ) , T + δ ( T − T )) y ∗ ( T ) + ∂ ECF ∂ T ∂ y ( d + δ ( y ∗ ( T ) − d ) , T + δ ( T − T ))( T − T ) .Since ∂ ECF / ∂ y , ∂ ECF / ∂ y ∂ t are both continuous when T >
0, and there is an open set contain-ing ( d , T ) such that ∂ ECF / ∂ y is strictly negative and, furthermore, y ∗ ( T ) − dT − T = − ∂ ECF ∂ T ∂ y ( d + δ ( y ∗ ( T ) − d ) , T + δ ( T − T )) ∂ ECF ∂ y ( d + δ ( y ∗ ( T ) − d ) , T + δ ( T − T )) −−−−→ T → T − ∂ ECF ∂ T ∂ y ( d , T ) ∂ ECF ∂ y ( d , T ) : = κ ( T ) , κ ( T ) = φ ( − h T ) σ + σ ( N ( h T ) − N ( − h T )) µ √ T + ( − h T N ( − h T )) (cid:16) − d + f + r √ T + µ √ T + βγρ √ T + σ (cid:17) N ( h T ) √ T .(3.17) Remark 4.
Note that κ contains γ , which depends on the value of m. For the fast approximation, we mayuse the lower bound of κ , κ ( T ) = φ ( − h T ) σ + σ ( N ( h T ) − N ( − h T )) µ √ T + ( − h T N ( − h T )) (cid:16) − d + f + r √ T + µ √ T + σ (cid:17) N ( h T ) √ T .12 art 2: Behavior of m ∗ Now, to find m ∗ , let us reorganize ECF ( y , m ) as following: ECF ( y , m ) = M (cid:101) − m (cid:101) − β (cid:20) ( M − m ) (cid:0) (cid:101) ( K − M + m ) − + ( − (cid:101) )( − ρ )( K − ( M − m )( − ρ )) − (cid:1) + m ( K − m ) − (cid:21) , (3.18)where (cid:101) : = (cid:0) ρ ( − (cid:101) )( d + f + y + r ) − ρ ( y − d ) ˜ N ( − α T ) + µ T ( − ρ + ρ(cid:101) ) (cid:1) , (cid:101) : = (cid:0) S − ρ ( y − d ) ˜ N ( − α T ) − ( d + f − µ T )( − ρ + ρ(cid:101) ) + ( − (cid:101) ) ρ ( y + r ) (cid:1) .Note that (cid:101) does not depend on m, and (cid:101) > (cid:0) ρ ( − (cid:101) )( d + f + r ) + ρ ( d ) ˜ N ( − α T ) + µ T ( − ρ + ρ(cid:101) ) (cid:1) + ρ y ( N ( − β T ) − ˜ N ( − α T )) , and N ( − β T ) − ˜ N ( − α T ) >
0. Now, investigate the behavior of (3.18) in different range of m to find m ∗ . Case 1-1: ( K − m ) ≥ ( K − M + m ) ≥ ( K − ( M − m )( − ρ )) ≥ . In this case, the (3.18)becomes a linear decreasing function of m . When K ≥ M , for any m this condition is satisfied so m ∗ = Case 1-2: ( K − m ) < ( K − M + m ) ≥ ( K − ( M − m )( − ρ )) ≥ K − (cid:101) β , which is < K , so in this range (3.18) is a decreasing function of m . Case 2-1: ( K − m ) ≥ ( K − M + m ) < ( K − ( M − m )( − ρ )) ≥ m ∗ = M − K /2 − (cid:101) /2 β(cid:101) , Case 2-2: ( K − m ) < ( K − M + m ) < ( K − ( M − m )( − ρ )) ≥ m ∗ = (cid:101) ( + (cid:101) ) M + ( − (cid:101) ) ( + (cid:101) ) K − (cid:101) ( + (cid:101) ) β Case 3: ( K − M + m ) ≥ ( K − ( M − m )( − ρ )) < Case 4-1: ( K − m ) ≥ ( K − M + m ) < ( K − ( M − m )( − ρ )) < m ∗ = M − β ( (cid:101) +( − (cid:101) )( − ρ )) K + (cid:101) β ( (cid:101) +( − (cid:101) )( − ρ ) ) Case 4-2: ( K − m ) < ( K − M + m ) < ( K − ( M − m )( − ρ )) < m ∗ = β ( M ( (cid:101) +( − ρ ) ( − (cid:101) ))+ K ( − (cid:101) ) ρ ) − (cid:101) β ( + (cid:101) +( − ρ ) ( − (cid:101) )) To summarize, when K ≥ M , m ∗ = { M , m ∗ , m ∗ , m ∗ , M , M − K , | M − K / ( − ρ ) |} .In Theorem 3.7, we have shown the behavior of y ∗ and m ∗ when µ is positive.First, let us focus on the behavior of y ∗ . As shown in the right panel of Figure 1, y ∗ = d until µ reaches a certain positive value (the lower bound is µ T > d + f + r + ( − βγ ) / ρ ) and then linearlyincreases with respect to µ .Also, in the right panel of Figure 2 it is observed that y ∗ = d until a certain time threshold, T ,and then increases linearly with respect to T . 13 .05 0.10 0.15 0.20 r =0.1 r =0.5 r =1 m* against T T 0.05 0.10 0.15 0.20 . . . . r =0.1 r =0.5 r =1 r =0.1, Approx. r =0.5, Approx. r =1, Approx.Approx.2 y* against T and approximations T Figure 2: Behavior of m ∗ (Left) and y ∗ (Right) against T. For the left graph, black solid line iswhen ρ = ρ = ρ =
1. Forthe graph in the right panel, y ∗ and it’s first-order approximation d + κ ( T − T ) are described insolid and dashed lines, respectively, using Black color( ρ = ρ = ) , and green color( ρ = M = r = f = (cid:101) = S = d = (cid:101) /2 = σ = µ = −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 b =0.001 b =0.005 b =0.01 m * against m for different b s m −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 . . . . b =0.001 b =0.005 b =0.01 y * against m for different b s m Figure 3: m ∗ (left) and y ∗ (right) against µ for β = β = β = K = ρ = M = r = f = (cid:101) = d = (cid:101) /2 = T = S = σ = s =0.1 s =0.15 s =0.2 m * against m , for multiple s s m −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 . . . . s =0.1 s =0.15 s =0.2 y * against m , for multiple s s m Figure 4: m ∗ (left), and y ∗ (right) against µ for σ = σ = σ = K = M = r = f = (cid:101) = d = (cid:101) /2 = T = S = ρ = y ∗ > d for a big enough µ T . In Theorem 3.7, wehave provided the existence of such a threshold, and named the threshold in the time horizon as T . We provided the stepwise-linear approximation of y ∗ as a function of T : y ∗ ( T ) ≈ d + κ ( T − T ) ,the first-order approximation given in Theorem 3.7. These approximations are also found in theright panel of Figure 2. Each y ∗ is described in solid lines, and the approximation is given in thedashed line using matching colors. Comparing with the result from [12], the blue dotted line inthe Right panel of Figure 2 is showing first order Taylor approximation of y ∗ using Theorem 3.6. of[12]. These approximations of y ∗ converges close to the actual y ∗ as T (cid:38) T . These approximationmethods provide the quick computation method that the investor can use to decide the limit orderplacement based on their time horizon, T .Now, let us focus on the behavior of y ∗ with various K s, the initial market depth. The rightpanel of Figure 1 shows that y ∗ increases as K increases, but the difference is relatively small for K =
10, 50, 70 and there is huge difference when K = K =
150 is special sincethis is only the case when K > M ( M =
100 in the simulation), which represents the case whenthe market depth K is bigger than the investor’s inventory M . In this case, the investor would notfear to put limit orders at the higher level, since even if limit orders do not get executed until time T , the market order at time T still will be beneficial since there is no impact on the supply curvebased on the quantity of the market order size M .Another interesting analysis of y ∗ is the change of behavior for different β s. Figure 3 shows be-havior of y ∗ int he right panel. y ∗ increases as β decreases. Also, recall that there is a lower boundof µ such that y ∗ > d after a certain value of µ , where the bound is µ T > d + r + f + ( − βγ ) / ρ (given in Equation (3.14)), where γ <
0. Therefore, as β increases, the bound also increases, whichis observed in Figure 3.In addition, the behavior of y ∗ for various values of σ , the volatility of price process is shown inFigure 4. As σ increases, after µ gets bigger than a certain (positive) threshold value, y ∗ decreases.While for low (positive) µ , bigger σ implies a higher chance of the limit order execution. Howeverit is not necessarily true for high µ , and this explains why y ∗ decreases as σ increases for bigenough µ .Now, let us show the behavior of m ∗ . Figure 2 shows the behavior of m ∗ against T when µ > m ∗ follows the result of Theorem 3.7, and it is decreasing as T increases. This is15 r =0.2 r =0.4 r =0.6 r =0.8 r =1.0 m * against m , K=150 m −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 . . . . . . r =0.2 r =0.4 r =0.6 r =0.8 r =1.0 y * against m , K=150 m −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 m * against m , K=10 m −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 . . . . r =0.2 r =0.4 r =0.6 r =0.8 r =1.0 y * against m , K=10 m −0.10 −0.05 0.00 0.05 0.10 0.15 0.20 r =0.2 r =0.4 r =0.6 r =0.8 r =1.0 m * against m , K=10, GBM m −0.10 −0.05 0.00 0.05 0.10 0.15 0.20 . . . . r =0.2 r =0.4 r =0.6 r =0.8 r =1.0 y * against m , K=10, GBM m Figure 5: m ∗ (left), y ∗ (right) against for K = K
0, BM (middle) and K = ρ = ρ = ρ = ρ = ρ = M = r = f = (cid:101) = d = (cid:101) /2 = T = S = σ = σ = µ > T increases, the benefit from the limit order increases. When µ > T increases, and even if a limit order is notexecuted, the expected value of S T is higher than S , so the market order at t = m ∗ against µ when K =
150 (top) and K = ρ ’s. K =
10 represents the case when the market depth is low (the quantityof limit order on best bid queue when t = m ∗ shows a relatively smoothdecreasing trend as µ increases. For K = m ∗ is still decreasing but there isa sudden drop from m ∗ = M to m ∗ =
0. When the market depth is high, then all the market orderis covered by the current existing limit order. In this case, our inventory’s size does not affect thesupply curve. In other words, when K > M , the conclusion for m ∗ and y ∗ should be the samewhether M=1 or M=K-1. So in this case, the investor’s choice comes down to two choices: m = m = M (just use market order).The behavior of m ∗ for different K s with a fixed ρ is shown in the right panel of Figure 1. m ∗ , theoptimal quantity for the market order, decreases as µ increases. It is because as µ increases, thereis higher chance of LO execution and the market order at T is more preferred than MO at 0. Theslope for the decrease gets steeper when K , the market depth, increases. This is because the affectof market order size gets smaller as K increases.The change of m ∗ slope as β can be found in Figure 3. When β is smaller, the slope is sharper.This is because when β gets bigger, the market order size is more affective to the expected cashflow, so the difference in the market order size should be smaller compared to the case of a smaller β .The behavior of m ∗ for various values of σ is shown in Figure 4. As σ increases, it is observed that m ∗ increases. As described earlier, bigger σ implies a higher chance of the limit order execution.However it is not necessarily true for a high µ , and this explains why y ∗ decreases and m ∗ increasesas σ increases for a big enough µ .While in this paper we didn’t include theorems for the behavior of the optimal placement strat-egy using a geometric Brownian motion, we have checked that they’re showing a similar behavioras in a Brownian motion model. Bottom two panels of Figure 5 show m ∗ , y ∗ which maximize theECF given in Lemma 3.4. Since for the same price process, µ and σ for the geometric Brownianmotion model should be smaller than the those for BM, we are using 0.01 for σ and range of µ from − µ and σ ), y ∗ and m ∗ show similar behavior but it is smoother forthe geometric Brownian motion model.To summarize, we have investigated the behavior of optimal ( y ∗ , m ∗ ) against important LOBfeatures ( ρ , β , K ) and other parameters for price movement such as µ in a single period model. Inthe following section, we will extend this investigation into a multiple period model.17 A Multi-Period Model
In this section, we extend the single-period case to a multi-period case. In the single period case,the investor made a decision only at time t = T .As a natural extension, now we introduce n time steps between time 0 and time T . We find theoptimal placement strategy at the first step. The difference is that all un-executed orders are nottransferred to a market order at the end of the first period. Instead, now we solve a new optimalreplacement problem using the remaining order. In other word, the remaining shares after thefirst period becomes a new M for the second period, and so on.Let us denote the time steps as { t = t = T / n , t = T / n , . . . , t n = T } . At each t i , theinvestor is allowed to • cancel any remaining limit order, (this becomes new remaining inventory, M i ) • place a market order of size m i ( ≤ M i ), or • place the rest ( M i − m i ) using limit order at the price of S i + y i . ( S i : mid-price at time t i )Then we define the Multi-period Expected cash flow (MECF) as follows:For n = MECF ( n = S , y , m , M , T = t ) = ECF ( y , m ) , (4.1)where the ECF (expected case flow) from the single-period case is in (3.1).For n ≥
2, we define the expected cash flow as a function of n , the number of time steps, S ,the stock mid-price at time t , y , the initial limit order placement, m , the initial market orderquantity, M , the inventory, and T , the time horizon.Note that the cash flow at a time step t i depends only on M i , S i , and the action at t i . Giventhis Markov structure, the optimal placement problem is a Markov decision problem where theexpected cost for taking each action at each step can be solved recursively. MECF ( n , S , y , m , M , T ) = m ( S ( − m ) − f ) + E (cid:20) L ( S (
0, 0 ) + y + r ) (cid:12)(cid:12)(cid:12)(cid:12) τ < Tn (cid:21) P (cid:18) τ < Tn (cid:19) (4.2) + MECF (cid:18) n − E [ S Tn (cid:12)(cid:12) τ < Tn ] , y , m , M − m − L , T ( n − ) n (cid:19) P (cid:18) τ < Tn (cid:19) (4.3) + MECF (cid:18) n − E [ S Tn (cid:12)(cid:12) τ > Tn ] , y , m , M − m , T ( n − ) n (cid:19) P (cid:18) τ > Tn (cid:19) (4.4)The first term of (4.2) is the cash flow from the market order, and the second term of (4.2) is thecash flow from executed limit order (recall that L is the amount of executed shares at the pricelevel S + y when τ , the first time the trader’s limit order becomes the best ask, is less than T / n ).Then, if τ < T / n , the M − m − L is the remaining inventory for the next time step, as describedin (4.3). Otherwise, if τ > T / n , then none of the limit order has been executed until T / n , and theremaining inventory for the next time step is M − m , as in (4.4).For the dynamic placement problem, at each time step, the investor may update parameters andsolve for the optimal placement. In this paper, we focus on solving for y and m assuming thatthe parameter remains the same during [ T ] . At each time step t i , after updating the parameter,the investor can use the same technique to find the optimal y i and m i .18 roposition 4.1. Let MECF be defined as in (4.1), (4.2) -(4.4). Let us assume that the price process, S t ,follows Brownian motion as in Lemma 3.5 and L = ( M − m ) ρ . Then, MECF(n, S, y, m, M, T) can besummarized as follows:For n = ,MECF ( n = S , y , m , M , T = t )= M (cid:0) S − ρ ( y − d ) ˜ N ( − α t ) − ( d + f − µ t )( − ρ + ρ(cid:101) ) + ( − (cid:101) ) ρ ( y + r ) (cid:1) − m (cid:0) ρ ( − (cid:101) )( d + f + y + r ) − ρ ( y − d ) ˜ N ( − α t ) + µ t ( − ρ + ρ(cid:101) ) (cid:1) − β (cid:2) m ( K − m ) − + ( M − m ) (cid:0) (cid:101) ( K − M + m ) − + ( − (cid:101) )( − ρ )( K − ( M − m )( − ρ )) − (cid:1)(cid:3) , α t = y − d + µ t σ √ t , β t = y − d − µ t σ √ t . ˜ N ( − α t ) = e ( y − d ) µ / σ N ( − α t ) , (cid:101) = (cid:0) N ( β t ) − ˜ N ( − α t ) (cid:1) For n ≥ ,MECF ( n , S , y n , m n , M , T )= m n ( S − d + β min ( K − m n , 0 ) − f )+( M − m n ) ρ (cid:110) ( S + y n + r ) (cid:16) N (cid:16) − β Tn (cid:17) + ˜ N (cid:16) − α Tn (cid:17)(cid:17)(cid:111) + MECF ( n − E [ S Tn | τ < T / n ] , y n , m n , ( M − m n )( − ρ ) , T ( n − ) n ) (cid:16) N (cid:16) − β Tn (cid:17) + ˜ N (cid:16) − α Tn (cid:17)(cid:17) + MECF ( n − E [ S Tn | τ > T / n ] , y n , m n , M − m n , T ( n − ) n ) (cid:16) N (cid:16) β Tn (cid:17) − ˜ N (cid:16) − α Tn (cid:17)(cid:17) (4.5) α t = y − d + µ t σ √ t , β t = y − d − µ t σ √ t . ˜ N ( − α t ) = e ( y − d ) µ / σ N ( − α t ) . Proof.
The result of this theorem is directly using the definition in (4.1), (4.2) -(4.4) and Lemma 3.1.In this theorem we introduced the expression of the Expected Cash flow for the multiple period.Now, let us define the optimal placement for the first time step, y ∗ n , m ∗ n , y ∗ n , m ∗ n : = arg max y ≥ d ,0 ≤ m ≤ M MECF ( n , S , y , m , T ) , (4.6)the optimal price level for the limit order placement (the limit order will be placed at S + y ∗ n )and the market order quantity m ∗ n ( ≤ M ) , when there are n time steps and T is the total investmenthorizon.Note that the action at each time step t i , y ni and m ni depends on the state of t i (remaining inven-tory and the mid-price at t i ), so the optimal policy at time t i , ( y ∗ ni , m ∗ ni ) , could be degenerated into (cid:0) ( y ∗ ni , m ∗ ni ) , ( y ∗ ni + , m ∗ ni + ) , . . . ( y ∗ nn − , m ∗ nn − ) (cid:1) . Then, the Dynamic Programming Principle leads to thefollowing backward recursion for the MECF: 19 y ∗ n , m ∗ n )= arg max y ≥ d ,0 ≤ m ≤ M (cid:20) m ( S − d + β min ( K − m , 0 ) − f ) + ( M − m ) ρ ( S + y + r ) (cid:16) N (cid:16) − β Tn (cid:17) + ˜ N (cid:16) − α Tn (cid:17)(cid:17) + max y , m MECF ( n − E [ S Tn | τ < T / n ] , y , m , ( M − m )( − ρ ) , T ( n − ) n ) (cid:16) N (cid:16) − β Tn (cid:17) + ˜ N (cid:16) − α Tn (cid:17)(cid:17) + max y , m MECF ( n − E [ S Tn | τ > T / n ] , y , m , M − m , T ( n − ) n ) (cid:16) N (cid:16) β Tn (cid:17) − ˜ N (cid:16) − α Tn (cid:17)(cid:17) (cid:21) .Using this Proposition 4.1, we are now going to solve the optimal placement decision at time 0.Because of the recursive nature of the multiple period, we are interested in analyzing the solutionof the first period. Optimal behaviors for the remaining period are updated at next time stepswith updated information (parameters), using the same method, sequentially. We now provideanalytical results for the optimal behavior. The following theorem gives analytical result for thecase when ρ = Theorem 4.2.
For y ∗ n , m ∗ n defined as in Proposition 4.1 and (4.6) and when ρ = , y ∗ n = d , m ∗ n = forall n when µ < .Proof. For n =
1, proof is done in Proposition 3.6. Note that y ∗ = d , m ∗ =
0, max
MECF = MECF ( n = S , y = d , m = M , T ) = M ( S + d + r ) .Now, let us prove that for n ≥
2, if y ∗ ( n − ) = d , m ∗ ( n − ) = MECF ( n − S , y , m , M , T ) = M ( S + d + r ) , then y ∗ n = d , m ∗ n = MECF ( n , S , y , m , M , T ) = M ( S + d + r ) .To prove this, let us denote f ( y , m ) : = m ( S − d + β min ( K − m , 0 ) − f ) + ( M − m )( S + y + r ) P ( τ < T / n )+ ( M − m ) E [ S Tn I ( τ > T / n )] + ( M − m )( d + r ) P ( τ > T / n ) .Then, from Eq. (4.7), max y , m MECF ( n , S , y , m , M , T ) = max y , m f ( y , m ) . We now need to find y , m which maximize f ( y , m ) .Let us recall that t : = T / n . Using Lemma 3.1, f ( y , m ) can be written as f ( y , m ) = MS + m ( − d + β min ( K − m , 0 ) − f ) + ( M − m )( d + r )+ ( M − m )( y − d )( N ( − β t ) + ˜ N ( − α t )) + ( M − m ) (cid:0) µ t N ( β t ) + ( − ( y − d ) − µ t ) ˜ N ( − α t ) (cid:1) ,And when we take a derivative with respect to y , ∂ f ( y , m ) / ∂ y = ( M − m ) (cid:0) N ( − β t ) − ˜ N ( − α t ) − µ t σ N ( − α t ) (cid:1) <
0, so y ∗ = d .Now, f ( y = d , m ) = M ( S + d + r ) + m ( − d + β min ( K − m , 0 ) − f − r ) , which is linearly de-creasing when m ≤ K , and when m > K , f ( y = d , m ) becomes a second order concave polynomialof m which takes its maximum at K /2 − ( d + f + r ) / β < K , so for m > K , f ( y = d , m ) is also adecreasing function of m . Therefore, f ( y = d , m ) is a decreasing function of m for 0 ≤ m ≤ M , so m ∗ =
0. Also, max
MECF ( n , S , y , m , M , T ) = MECF ( n , S , y = d , m = M , T ) = M ( S + d + r ) .We have proved that that if y ∗ ( n − ) = d , m ∗ ( n − ) = MECF ( n − S , y , m , M , T ) = M ( S + d + r ) , then y ∗ n = d , m ∗ n = MECF ( n , S , y , m , M , T ) = M ( S + d + r ) . Byinduction, this proves the theorem. 20n Theorem 4.2, we’ve shown the analysis for the case when ρ = µ <
0, which means thatthe limit order execution is guaranteed when the price process reaches the price level of the limitorder, and the drift is negative. This could represent the liquid market condition. In this case, theoptimal strategy is to place all using the limit order at the best ask, and this does not change forthe number of time steps.Now, let us show that y ∗ = d for any ρ ∈ [
0, 1 ] for the negative drift. Due to the complexity,we’ve proved for the case when n = Theorem 4.3.
For y ∗ n defined as in Proposition 4.1 and (4.6), y ∗ = d when µ < .Proof. For n =
1, recall that
ECF ( y = d , m , M ) = M ( S + ρ ( d + r ) − ( − ρ )( d + f − µ t )) − m (cid:101) + m β min ( K − m , 0 )+ β ( M − m )( − ρ ) min ( K − ( M − m )( − ρ ) , 0 ) ,Let us denote F ( M , m ) : = ECF ( y = d , m , M ) − MS .where (cid:101) : = ρ ( d + r + f ) + µ t ( − ρ ) .For n =
2, let M A : = ( M − m )( − ρ ) and M B : = ( M − m ) . MECF can be written as MECF ( n = S , y , m , M , T )= MS + m ( − d + β min ( K − m , 0 ) − f )+ ( M − m ) ρ (cid:110) ( y + r ) (cid:16) N (cid:16) − β T (cid:17) + ˜ N (cid:16) − α T (cid:17)(cid:17)(cid:111) + E [ S T I ( τ < T /2 )]( M − m )( − ρ ) + max ≤ m ≤ M A F ( M A , m ) (cid:16) N (cid:16) − β T (cid:17) + ˜ N (cid:16) − α T (cid:17)(cid:17) + E [ S T I ( τ > T /2 )]( M − m ) + max ≤ m ≤ M B F ( M B , m ) (cid:16) N (cid:16) β T (cid:17) − ˜ N (cid:16) − α T (cid:17)(cid:17) . (4.7)By taking derivative with respect to y , ∂∂ y MECF ( n = S , y , m , M , T )= ρ ( M − m ) (cid:16) N (cid:16) − β T (cid:17) − ˜ N (cid:16) − α T (cid:17)(cid:17) + e ( y − d ) µσ σ (cid:113) T (cid:16) φ ( α T ) − α T N ( − α T ) (cid:17) (cid:18) − ρ ( r + d − µ T )( M − m ) + max ≤ m ≤ M B F ( M B , m ) − max ≤ m ≤ M A F ( M A , m ) (cid:19) + ( y − d ) ˜ N (cid:16) − α Tn (cid:17) σ T (cid:18) − ρ ( r + d )( M − m ) + max ≤ m ≤ M B F ( M B , m ) − max ≤ m ≤ M A F ( M A , m ) (cid:19) Since first line is negative, if we show that ( max ≤ m ≤ M B F ( M B , m ) − max ≤ m ≤ M A F ( M A , m )) < ρ ( r + d )( M − m ) , we can show that the entire expression is negative. The proof follows:21ax ≤ m ≤ M B F ( M B , m ) − max ≤ m ≤ M A F ( M A , m )= ρ M B ( ρ ( d + r ) − ( − ρ )( d + f − µ T ))+ max ≤ m ≤ M B [ − m (cid:101) + m β min ( K − m , 0 ) + β ( M B − m )( − ρ ) min ( K − ( M B − m )( − ρ ) , 0 )] − max ≤ m ≤ M A [ − m (cid:101) + m β min ( K − m , 0 ) + β ( M A − m )( − ρ ) min ( K − ( M A − m )( − ρ ) , 0 )] (4.8)Let’s denote m b : = arg max F ( M B , m ) . Let’s divide into two cases: m b ≤ M A = M B ( − ρ ) and M A = M B ( − ρ ) < m b < M B . Case 1, m b ≤ M A = M B ( − ρ ) : In this case, max ≤ m ≤ M A F ( M A , m ) ≥ F ( M A , m = m b ) , somax ≤ m ≤ M B F ( M B , m ) − max ≤ m ≤ M A F ( M A , m ) ≤ F ( M B , m = m b ) − F ( M A , m = m b ) , therefore ( ) ≤ ρ M B ( ρ ( d + r ) − ( − ρ )( d + f − µ T ))+ (cid:104) − m b (cid:101) + m b β min ( K − m b , 0 ) + β ( M B − m b )( − ρ ) min ( K − ( M B − m b )( − ρ ) , 0 ) (cid:105) + (cid:104) − m b (cid:101) + m b β min ( K − m b , 0 ) + β ( M A − m b )( − ρ ) min ( K − ( M A − m b )( − ρ ) , 0 ) (cid:105) = ρ M B ( d + r ) − ρ ( − ρ )( d + f − µ T ) M B + β ( M B − m b )( − ρ ) min ( K − ( M B − m b )( − ρ ) , 0 ) − β ( M A − m b )( − ρ ) min ( K − ( M A − m b )( − ρ ) , 0 ) < ρ M B ( d + r ) ≤ ( M − m ) ρ ( d + r ) . Case 2, M A = M B ( − ρ ) < m b < M B : In this case, max ≤ m ≤ M A F ( M A , m ) ≥ F ( M A , m = M A ) , so max ≤ m ≤ M B F ( M B , m ) − max ≤ m ≤ M A F ( M A , m ) ≤ F ( M B , m = m b ) − F ( M A , m = m A ) , therefore ( ) ≤ ρ M B ( ρ ( d + r ) − ( − ρ )( d + f − µ T ))+ (cid:104) − m b (cid:101) + m b β min ( K − m b , 0 ) + β ( M B − m b )( − ρ ) min ( K − ( M B − m b )( − ρ ) , 0 ) (cid:105) − [ − M A (cid:101) + M A β min ( K − M A , 0 )]= ρ M B ( d + r ) − ρ ( − ρ )( d + f − µ T ) M B + (cid:20) − m b (cid:18) ρ ( d + r + f ) + µ T ( − ρ ) − β min ( K − m b , 0 ) (cid:19) + β ( M B − m b )( − ρ ) min ( K − ( M B − m b )( − ρ ) , 0 ) (cid:21) − (cid:20) − M A ρ ( d + r + f ) − M A µ T ( − ρ ) + M A β min ( K − M A , 0 ) (cid:21) < ρ M B ( d + r ) + ρ ( − ρ ) µ T M B − m b µ t ( − ρ ) + M A µ T ( − ρ ) < ρ M B ( d + r ) ≤ ρ ( M − m )( d + r ) .So we proved that ( max ≤ m ≤ M B F ( M B , m ) − max ≤ m ≤ M A F ( M A , m )) < ρ ( r + d )( M − m ) , whichimplies ∂∂ y MECF ( n , S , y , m , M , T ) <
0. Therefore, y ∗ = d .22 emark 5. While we’ve only proved for the case for two-period case, y ∗ n = d appears to be true for all n. Now, let us study what happens to m ∗ n . Let us show the behavior of m ∗ . Theorem 4.4.
For m ∗ n defined as in Proposition 4.1, m ∗ ≤ m ∗ when µ < . For m ∗ ,1. If ρ = , m ∗ = .2. If ρ < and K ≥ M ‘ , • m ∗ = when ρ ( d + r + f ) + µ t ( − ρ ) > • m ∗ = M ‘ when ρ ( d + r + f ) + µ t ( − ρ ) <
3. If ρ < and K < M ‘ ,m ∗ is one of { K , M ‘ − K / ( − ρ ) , M ‘ , m ∗ , m ∗ , m ∗ } wherem ∗ = K /2 − (cid:101) ‘3 /2 β , m ∗ = M ‘ − K ( − ρ ) − (cid:101) ‘3 ( − ρ ) β , m ∗ = M ‘ β ( − ρ ) + βρ K − (cid:101) ‘3 β ( + ( − ρ ) ) , where (cid:101) ‘3 : = ρ ( d + r + f ) + ( − ρ ) µ T and M ‘ : = M − m ,m : = arg max ≤ m ≤ M ECF ( n = S , y = d , m , M , T /2 ) , which can be computed using Theorem 3.6.Proof. Note that the MECF for n = MECF ( n = S , y = d , m , M , T )= MS + m ( − d + β min ( K − m , 0 ) − f ) + ( M − m ) ρ { ( d + r ) } + µ T ( M − m )( − ρ ) + ( M − m )( ρ ( d + r ) − ( − ρ )( d + f − µ T /2 ))+ max ≤ m ≤ M A [ − m (cid:101) + m β min ( K − m , 0 ) + β ( M A − m )( − ρ ) min ( K − ( M A − m )( − ρ ) , 0 )] ,(4.9) M A = ( M − m )( − ρ ) . m which minimizes MECF ( n = S , y = d , m , M , T ) actually minimizes the following withrespect to m : − m β min ( K − m , 0 ) − m (cid:101) ‘3 + β ( M ‘ − m )( − ρ ) min ( K − ( M (cid:48) − m )( − ρ ) , 0 ) ,while the solution for n = − m β min ( K − m , 0 ) − m (cid:101) + β ( M − m )( − ρ ) min ( K − ( M − m )( − ρ ) , 0 ) ,with respect to m. Note that (cid:101) : = ρ ( d + r + f ) + µ t ( − ρ ) , (cid:101) ‘3 : = ρ ( d + r + f ) + ( − ρ ) µ T and M ‘ : = M − m .Using the proof of Theorem 3.6. we have 23 l l l l l l rho=0.4,n=1rho=0.8,n=1rho=1,n=1rho=0.4,n=2rho=0.8,n=2rho=1,n=2 m* for n=2 (points) and n=1 (line) m −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 . . . . . . . l l l l l y* for n=2 (points) and n=1 (line) m l rho=0.4,n=1rho=0.8,n=1rho=1,n=1rho=0.4,n=2rho=0.8,n=2rho=1,n=2 Figure 6: m ∗ (left) and y ∗ (right) for multi-period (dots) and single period (lines). M = r = f = (cid:101) = d = (cid:101) /2 = T = S = σ = ρ = m ∗ = ρ < K ≥ M ‘ , • m ∗ = ρ ( d + r + f ) + µ t ( − ρ ) > • m ∗ = M ‘ when ρ ( d + r + f ) + µ t ( − ρ ) <
03. If ρ < K < M ‘ , m ∗ is one of { K , M ‘ − K / ( − ρ ) , M , m ∗ , m ∗ , m ∗ } where m ∗ = K /2 − (cid:101) ‘3 /2 β , m ∗ = M ‘ − K ( − ρ ) − (cid:101) ‘3 ( − ρ ) β , m ∗ = M ‘ β ( − ρ ) + βρ K − (cid:101) ‘3 β ( + ( − ρ ) ) . Remark 6.
Note that (cid:101) < (cid:101) ‘3 , and M ‘ < M, which makes m ∗ ≤ m ∗ , m ∗ ≤ m ∗ , m ∗ ≤ m ∗ . From thisinequality, it is observed that m ∗ , the optimal Market Order quantity at time for the two-step case is lessthan (or equal to ) m ∗ , the optimal Market Order quantity at time for the one-step case. Behaviors of y ∗ (line) and y ∗ (points) for different ρ values (black: ρ = ρ = ρ = µ < y ∗ = d asdescribed in Theorem 4.3. For µ >
0, it appears that it still increases in a linear (step-wise linear)way, but its slope against ρ is more sensitive to the value of ρ for n =
2. More specifically, when ρ = y ∗ < y ∗ , but when ρ = y ∗ > y ∗ . This is explained by the chance of the max ECF for the single period.For the single period, when limit orders are not executed, the investor use a market order attime T . Then, the expected stock price does not change with respect to ρ , and the change of y ∗ w.r.t. ρ in the single period is minimal.However, the max ECF (for the single period) changes when ρ changes. When n =
2, if thelimit order placed at t = . Therefore, y ∗ changes more sensitively to the change of ρ than y ∗ . We observe that when thenumber of time step increases, the slope change will more various.The left panel of Figure 6 shows behaviors of m ∗ and m ∗ against µ for different ρ s. It appearsthat m ∗ ≤ m ∗ , which implies that the optimal quantity for the market order at time 0 decreaseswhen n , number of time step, increases. This is because as n increases, it is better to divide themarket order into multiple chunks to reduce the negative impact from the size of the market order.Existence of K suggests that placing a smaller amount of a market order twice is better than placinga big amount of at once even though the total MO amount is same. Throughout this paper, we have obtained explicit solutions and approximations for the optimalexecution problem under the liquidity cost. First, we have derived the explicit solution for theoptimal size of the market order, m ∗ , for the single order. In addition, the approximation of theoptimal price level for the limit order, y ∗ , has been calculated. The behavior of y ∗ and m ∗ havebeen investigated under various market conditions. Finally, we have extended this problem intomultiple-period problem where investors can change their decision at multiple time steps. Wehave shown the change of the optimal behavior for different number of time steps.To summarize, we have investigated the behavior of optimal ( y ∗ , m ∗ ) against important LOBfeatures and other parameters for price movement in single and multiple period models. Insteadof modeling the whole LOB movement, we have taken essential features of LOB to simplify theoptimization without losing the important traits of LOB.There are other important features in the Limit Order Book closely related to this paper whichhave not been studied yet. For instance, the consideration of movement of bid-ask spread, theestimation and the update of parameters, the effect from correlated assets, the realistic diffusivemodels with jumps, are all worth further development. The models in this study could enlightensuch developments. References [1] A. Alfonsi, A. Fruth, and A. Schied. Optimal execution strategies in limit order books withgeneral shape functions.
Quantitative Finance , 10(2):143–157, 2010.[2] A. Alfonsi, A. Schied, and A. Slynko. Order book resilience, price manipulation, and thepositive portfolio problem.
SIAM Journal on Financial Mathematics , 3(1):511–533, 2012.[3] R. Almgren and N. Chriss. Optimal execution of portfolio transactions.
Journal of Risk , 3:5–40,2001.[4] P. Bank and D. Baum. Hedging and portfolio optimization in financial markets with a largetrader.
Mathematical Finance: An International Journal of Mathematics, Statistics and FinancialEconomics , 14(1):1–18, 2004.[5] D. Bertsimas and A. W. Lo. Optimal control of execution costs.
Journal of Financial Markets , 1(1):1–50, 1998. 256] M. Blais and P. Protter. An analysis of the supply curve for liquidity risk through book data.
International Journal of Theoretical and Applied Finance , 13(06):821–838, 2010.[7] A. Cartea, S. Jaimungal, and J. Ricci. Buy low, sell high: a high frequency trading perspective.
SIAM J. Financ. Math. , 5(1):415–444, 2014.[8] U. C¸ etin, R. A. Jarrow, and P. Protter. Liquidity risk and arbitrage pricing theory. In
Handbookof Quantitative Finance and Risk Management , pages 1007–1024. Springer, 2010.[9] R. Cont and A. Kukanov. Optimal order placement in limit order markets.
Available at ssrn2155218 , 2013.[10] J. Donier. Market impact with autocorrelated order flow under perfect competition.
Availableat SSRN 2191660 , 2012.[11] Z. Eisler, J.-P. Bouchaud, and J. Kockelkoren. The price impact of order book events: marketorders, limit orders and cancellations.
Quantitative Finance , 12(9):1395–1419, 2012.[12] J. E. Figueroa-L ´opez, H. Lee, and R. Pasupathy. Optimal placement of a small order in adiffusive limit order book.
High Frequency , 1(2):87–116, 2018.[13] R. Frey and P. Patie. Risk management for derivatives in illiquid markets: A simulation study.In
Advances in finance and stochastics , pages 137–159. Springer, 2002.[14] J. Gatheral. No-dynamic-arbitrage and market impact.
Quantitative finance , 10(7):749–759,2010.[15] J. Gatheral, A. Schied, and A. Slynko. Transient linear price impact and fredholm integralequations.
Mathematical Finance: An International Journal of Mathematics, Statistics and FinancialEconomics , 22(3):445–474, 2012.[16] O. Gu´eant. Optimal execution and block trade pricing: a general framework.
Applied Mathe-matical Finance , 22(4):336–365, 2015.[17] F. Guilbaud and H. Pham. Optimal high-frequency trading with limit and market orders.
Quantitative Finance , 13(1):79–94, 2013.[18] X. Guo, A. De Larrard, and Z. Ruan. Optimal placement in a limit order book.
Mathematicsand Financial Economics , 2016.[19] A. Jacquier and H. Liu. Optimal liquidation in a level-i limit order book for large-tick stocks.
Preprint. Available at arXiv:1701.01327 [q-fin.TR] , 2017.[20] M. Jeanblanc, M. Yor, and M. Chesney.
Mathematical methods for financial markets . Springer,2009.[21] C. Maglaras, C. Moallemi, and H. Zheng. Optimal execution in a limit order book and anassociated microstructure market impact model.
Preprint available at ssrn 2610808 , 2015.[22] A. A. Obizhaeva and J. Wang. Optimal trading strategy and supply/demand dynamics.
Journal of Financial Markets , 16(1):1–32, 2013.2623] M. Potters and J.-P. Bouchaud. More statistical properties of order books and price impact.
Physica A: Statistical Mechanics and its Applications , 324(1-2):133–140, 2003.[24] S. Predoiu, G. Shaikhet, and S. Shreve. Optimal execution in a general one-sided limit-orderbook.