Optimal trade execution in order books with stochasticliquidity∗

Antje Fruth† Torsten Schöneborn‡ Mikhail Urusov§

December 8, 2017

Abstract

In financial markets, liquidity changes randomly over time. We consider such random variationsof the depth of the order book and evaluate their influence on optimal trade execution strategies.If the stochastic structure of liquidity changes satisfies certain conditions, then the unique optimaltrading strategy exhibits a conventional structure with a single wait region and a single buy regionand profitable round trip strategies do not exist. In other cases, optimal strategies can featuremultiple wait regions and optimal trade sizes that can be decreasing in the size of the position tobe liquidated. Furthermore, round trip strategies can be profitable depending on bid-ask spreadassumptions. We illustrate our findings with several examples including the CIR model for theevolution of liquidity.

KEYWORDS: Market impact model, optimal order execution, limit order book, resilience, time-varying liquidity, stochastic order book depth, profitable round trip trading strategies.

1 Introduction

Liquidity is not constant throughout the day, but instead varies over time. Traders active in amarket are typically expected to continuously observe these changes in liquidity and adjust theirtrades accordingly. Some part of the liquidity changes is driven by deterministic changes in expectedliquidity levels, e.g., daily and weekly patterns as well as expected changes around important pointsin time such as news releases. These expected changes however do not explain liquidity variationfully. An unpredicted component of liquidity changes remains which can dominate the deterministiccomponent.

We extend existing limit order book models and introduce a stochastic depth of the order book. Inthis market, we consider an investor who wants to purchase a large asset position. If the order bookdynamics are driven by a general diffusion satisfying certain conditions, then we prove existence anduniqueness of the optimal trade execution strategy. This trading strategy exhibits a wait region / buyregion structure with a single wait region and a single buy region. If the investor finds herself in thewait region at a given point in time, then she does not place any orders at this point; if she is in thebuy region, then the investor buys just enough to bring her position from within the buy region tothe boundary of the wait region.∗We thank the Associate Editor and two anonymous referees for the comments that helped improve the paper.†Munich RE, Munich, Germany, mail@antje-fruth.de‡Deutsche Bank AG, London, UK, schoeneborn@math.tu-berlin.de§University of Duisburg-Essen, Essen, Germany, mikhail.urusov@uni-due.de

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If limit order book depth is not driven by a diffusion satisfying said conditions, then the classical waitregion / buy region structure with one region each does not need to hold. While optimal strategiesmay still be described in terms of wait and buy regions, there can be more than one of these regions.We provide several examples with such non-standard optimal trading strategies. Intuitively expectedfeatures do not need to hold any more. For example, the trade size at a given point in time canvary non-monotonically with the size of the remaining position: if a large or small position remains,then no order is placed, however a purchase order is placed if the remaining position is of mediumsize. To the best of our knowledge, a nonintuitive structure of such type in solutions of Markoviancontrol problems was never observed in the literature. In particular, this concerns also our previouspaper Fruth, Schöneborn, and Urusov (2014) on the subject, where the results imply that such aphenomenon cannot appear whenever the order book depth is a deterministic function of time.

In our model, the condition ensuring wait region / buy region structure also guarantees that roundtrip trading strategies cannot be profitable. If the condition is violated, then round trip strategiescan generate profits if the bid-ask spread is assumed to be zero; if a dynamic spread is assumed, thenprofits from round trip strategies remain unavailable.

The majority of the optimal trade execution literature considers one of two different market models.First, several models assume an instantaneous temporary price impact, e.g., Almgren and Chriss (2001)and Almgren (2003). In these models, the temporary price impact at time t is independent of all ordersexecuted at time prior to t and does not influence any order at a time after t, which greatly simplifiesthe analysis. Cheridito and Sepin (2014) and Almgren (2012) have studied stochastic temporary priceimpact in this setting and provide numerical methods for calculation of the optimal strategy and valuefunction, while Ankirchner, Jeanblanc, and Kruse (2014) and Ankirchner and Kruse (2015) add anadditional quadratic risk term and provide the solution in terms of a BSDE with a singular terminalcondition. Also see Dolinsky and Soner (2013), Bank, Dolinsky, and Gökay (2016) for the discussionof super-replication as well as Bank, Soner, and Voß (2017) for the discussion of hedging in such asetting with stochastic temporary price impact.

In a second group of models, inspired by a limit order book interpretation, resilience is finite anddepth and resilience are separately modelled. Our model falls into this second group. Due to thefinite resilience of the order book, the execution price at time t is influenced by orders filled at timesprior to t, and the execution at time t in turn influences the execution price of subsequent orders. Inour present paper, due to the fact that the order book depth is stochastic, the optimal strategies arecarried on a set of Lebesgue measure zero (we obtain a stop-and-go pattern depending on whetherwe are in the interior or on the boundary of the wait region). This is in contrast to the optimalstrategies in the first group of papers considering (stochastic) temporary impact, which are absolutelycontinuous with respect to the Lebesgue measure. Most of the existing literature on order book drivenoptimal liquidation assumes the liquidity parameters to be constant over time, see, e.g., Bouchaud,Gefen, Potters, and Wyart (2004), Obizhaeva and Wang (2013), Alfonsi, Fruth, and Schied (2010) andPredoiu, Shaikhet, and Shreve (2011). Alfonsi and Acevedo (2014), Bank and Fruth (2014) and Fruth,Schöneborn, and Urusov (2014) allow for deterministic changes in liquidity and are therefore closelyrelated to our paper. This paper is qualitatively different from the aforementioned papers, and themain differences are as follows. Due to the stochasticity in the depth of the order book, the optimalexecution strategies in the framework of this paper are no longer deterministic (the latter was the casein the aforementioned group of papers, see also Section 2). More surprisingly, the counterexamples tothe wait region / buy region structure mentioned above appear in the framework of our present paperonly.

Another paper that considers stochastically varying order book depth is Chen, Kou, and Wang (2015).They provide a numerical method for calculation of the optimal strategy and value function in discretetime with the depth of the limit order book driven by a discrete Markov chain. In contrast, we focuson analytical results in a continuous time setting with limit order book depth following a diffusiveprocess.

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Becherer, Bilarev, and Frentrup (2017) analyse yet another version of stochastic liquidity in a frame-work that resembles the framework in the aforementioned second group of models. They have a certainimpact process Y , which is analogous to our deviation process D (see (2)) with a constant resilienceρs ≡ β ∈ (0,∞), a constant liquidity process (Ks) normalised to one and additional Brownian noise.Stochasticity of liquidity is introduced in their model due to the mentioned Brownian noise as wellas due to a multiplicative price impact built via a nonlinear transform applied to the process Y (incontrast, we have an additive price impact in our modelling approach). Becherer, Bilarev, and Fren-trup (2017) find an explicit solution of the singular control problem, which corresponds to optimalexecution with infinite time horizon. Even though both in our present paper and in Becherer, Bi-larev, and Frentrup (2017) the optimal strategies are described via singular control problems, they arequalitatively different in that they even require different state spaces, which is due to the modellingdifferences.

A modelling framework that comprises instantaneous temporary price impact as in Almgren (2012),transient (or persistent) price impact like the one generated by a limit order book, and a certainrisk term that penalises slow liquidation is considered in Graewe and Horst (2016). While the partof the dynamics in Graewe and Horst (2016) that is linked to a limit order book corresponds toa constant (deterministic) order book depth, they have stochastic resilience. In this non-Markoviansituation, they prove that the value function is characterised as a unique solution to a three-dimensionalBSDE system. The differences with our approach are as follows. Firstly, as Graewe and Horst (2016)require a nondegenerate instantaneous price impact, their optimal strategies are, as in Almgren (2012),absolutely continuous with respect to the Lebesgue measure. Secondly, when we let the instantaneousprice impact go to zero in their model, we can obtain the optimal strategy of Obizhaeva and Wang(2013) in the limit, which corresponds to a constant order book depth, but we cannot get in this waythe strategies corresponding to genuinely stochastic order book depth.

Starting with Huberman and Stanzl (2004), profitable round trip strategies haven been studied in avariety of market models by Gatheral (2010), Alfonsi and Schied (2010), Alfonsi, Schied, and Slynko(2012) and Klöck, Schied, and Sun (2014) among others. To the best of our knowledge, all existingliterature on this topic assumes deterministic liquidity, while in our paper we consider it in ourstochastic liquidity model.

The remainder of this paper is structured as follows. We first compare this paper with our previouspaper Fruth, Schöneborn, and Urusov (2014) in Section 2. In Section 3, we introduce a limit orderbook model with stochastic depth and derive basic structural features in Section 4. We prove existenceand uniqueness of optimal strategies as well as the wait region / buy region structure in Section 5as long as the stochastic dynamics of the limit order book depth obeys certain conditions. We applythese results to several examples of diffusive processes in Section 6. If the conditions of Section 5are violated, then the optimal strategy does not need to be of wait region / buy region structureany more as we demonstrate in several examples in Section 7. In Section 8, we extend our model totwo-sided limit order books and investigate the returns of round trip trading strategies. We concludein Section 9 and provide proofs in Appendices A, B, C, D and E.

2 Connection to our previous work

This paper considers a liquidity model that extends the model examined in our previous paper Fruth,Schöneborn, and Urusov (2014). In this section, we discuss the connection between these two papers.1

Model: In our previous paper Fruth, Schöneborn, and Urusov (2014) we have analyzed an order bookmodel where the order book depth is assumed to be a deterministic function of time. The currentpaper extends the previous one by allowing the order book depth to be stochastic. This corresponds

1Many (but not all) of the results of both this and our previous paper are also contained in the PhD thesis Fruth(2011).

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to an additional dimension in our optimization problem, which we introduce in all notations. Theremainder of the model is identical in both papers.

We now compare the main results in the present paper to the main results in Fruth, Schöneborn, andUrusov (2014) in the two main topics of this work:

Topic 1: Existence, uniqueness and structure of optimal strategies. In both papers we prove existenceand uniqueness of an optimal trading strategy that has an intuitive wait region / buy region structure.In our previous paper we could show this under relatively general assumptions on the deterministicliquidity process. In the stochastic liquidity framework of the current paper, we need more restrictiveassumptions to be able to derive the wait region / buy region structure. The techniques that we needare completely different to the ones in the deterministic case. Subsequently we construct examplesof stochastic liquidity processes for which the intuitive wait region / buy region structure does nothold. For examples where it does hold, we numerically derive optimal strategies which are of courseno longer deterministic (as in the case of deterministic liquidity).

Topic 2: Profitable round trip trading strategies. In our previous paper we introduced the dynamicand zero spread models for the two-sided order book and study existence of profitable round tripstrategies in these models. The existence of profitable round trip strategies is sometimes also referredto as price manipulation. For a deterministic liquidity process, there is no price manipulation in thedynamic spread model, while in the zero spread model profitable round trips can occur dependingon the model parameters. In the present paper we study the same questions in the framework ofstochastic liquidity. Here, again, there is no price manipulation in the dynamic spread model, andthe line of argument from Fruth, Schöneborn, and Urusov (2014) applies directly. For the zero spreadmodel however we now need a different approach and discover a new connection between the absenceof profitable round trip strategies and the existence of wait region / buy region structure.

While this paper is self-contained, we refer the interested reader to our previous paper where weprovide more detail on financial motivation.

3 Model description

A limit order book model with time dependent depth was introduced in Fruth, Schöneborn, and Urusov(2014). In this previous paper we explain the model in depth and provide an economic motivation.In the following, we recapitulate the central components and notation and extend the model fromdeterministic order book depth to stochastic order book depth.

The model is built on a filtered probability space (Ω,F , (Fs)s∈[0,T ],P). As usual in dynamic program-ming we consider a general initial time t ∈ [0, T ] below. For the evolution of the trader’s asset positionover time interval [t, T ], we consider the set of admissible continuous-time increasing strategies

Actst :=

Θ: Ω× [t, T+]→ [0,∞) |(Fs)− adapted, increasing, bounded, càglàd with Θt = 0

and denote ξs := ∆Θs := Θs+ − Θs. In particular, absolutely continuous trading as well as impulsetrades are allowed. A strategy from Actst consists of a left-continuous process (Θs)s∈[t,T ] and anadditional random variable ΘT+ with ∆ΘT = ΘT+ − ΘT ≥ 0 being the last trade of the strategy.Let us emphasize that admissible strategies are bounded by definition, that is, for Θ ∈ Actst , we haveΘT+ ≤ const <∞ a.s. (the constant depends on a strategy). Denote by

Actst (x) :=

Θ ∈ Actst | ΘT+ = x a.s.

(1)

the admissible strategies that build up a position of x ∈ [0,∞) shares until time T almost surely. Forthe majority of this paper, we consider only one side of the limit order book (namely, the buy side)

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and hence only include increasing strategies in Actst . As we will see in Section 8, selling cannot reduceoverall purchase costs if the bid-ask spread is influenced by the trader.

In addition to continuous time, we will also consider trading in discrete time, i.e., at times

0 = t0 < t1 < ... < tN = T.

In this case, we constrain our admissible strategy set to

Adist :=

Θ ∈ Actst | Θs = 0 on [t, tn(t)] and

Θs = Θtn+ a.s. on (tn, tn+1) for n = n(t), ..., N − 1⊂ Actst

with n(t) := infn = 0, ..., N | tn ≥ t and define

Adist (x) :=

Θ ∈ Adist | ΘT+ = x a.s.

as the discrete analogue to Actst (x).

Let D be an “ask dislocation” process, i.e., the deviation of the current ask price from its steady statelevel, K the illiquidity process, and ρ the (time-varying) resilience speed.

Standing Assumption.(i) K is a (possibly time-inhomogeneous) (Fs)-Markov process with state space (0,∞) and finite firstmoments.(ii) ρ : [0, T ]→ (0,∞) is a strictly positive Lebesgue-integrable deterministic function.

We consider a block-shaped order book, where the height of the block (≡ order book depth) at eachtime s is qs := 1/Ks. If we, for instance, buy y shares at time s0, the deviation increases by y

qs0= Ks0y

and therefore Ds0+ = Ds0 +Ks0y. If we do not trade for some time afterwards, the deviation decays(moves towards zero), which is a feature of real-life limit order books called resilience. We assumeexponential resilience in our model, that is, if we do not trade on some time interval (s0, s1), wearrive at the deviation Ds1 = Ds0 +Ks0 exp

−∫ s1s0ρr dr

y. The following formulas (2)–(4) show the

interplay of the effects from buying shares (D increases) and from the resilience (D decreases) in ourmodel for a general strategy Θ. The deviation Ds results from past trades on [t, s) in the followingway

dDs = −ρsDsds+KsdΘs, Dt = δ. (2)

That is, for s ∈ [t, T ],

Ds =

∫[t,s)

Kue−∫ suρrdrdΘu + δe−

∫ stρudu (3)

and, taking into account the last trade ∆ΘT ,

DT+ =

∫[t,T ]

Kue−∫ TuρrdrdΘu + δe−

∫ Ttρudu. (4)

For any fixed t ∈ [0, T ], δ ≥ 0 and κ > 0, we define the cost function J(t, δ, ·, κ) : Actst → [0,∞] as

J(Θ) := J(t, δ,Θ, κ) := Et,δ,κ

[∫[t,T ]

(Ds +

Ks

2∆Θs

)dΘs

], (5)

i.e., the expected liquidity cost on the time interval [t, T ] when Dt = δ and Kt = κ. While we donot exclude the possibility of an infinite cost of a strategy Θ ∈ Actst , it is worth noting that, for anyΘ ∈ Adist , the cost is finite due to our standing assumption. Starting with (5) we meet the followingnotational convention, which will be used throughout the paper: Pt,κ is the probability measure under

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which the Markov process K starts at time t from κ, Et,κ is the expectation under Pt,κ, and we writeEt,δ,κ for the expectation when the expression contains the process D and the starting point at time tin (2) is δ.

Let us briefly recall how the right-hand side of (5) comes into play. Let the best ask price process (As)be modelled as As = Aus + Ds, where the unaffected best ask price (Aus ) is a càdlàg H1-martingale.Then, given that the limit order book has the block form, the total cost of a strategy Θ ∈ Actst (x) is∫

[t,T ]

(As + Ks

2 ∆Θs

)dΘs. A calculation involving integration by parts reveals that the expected total

cost equals

Et,δ,κ [AuTΘT+] + Et,δ,κ

[∫[t,T ]

(Ds +

Ks

2∆Θs

)dΘs

]= Aut x+ J(t, δ,Θ, κ)

with J(t, δ,Θ, κ) given by (5) (notice that Et,δ,κ∫

[t,T ]Θs dA

us = 0 because Au is an H1-martingale and

Θ is bounded). The first summand in the latter formula describes the expected cost that occurs dueto trading in the unaffected price. This cost depends on the strategy Θ ∈ Actst (x) only through thetotal number of shares x that the strategy acquires, and, due to the martingale property of Au, theexpression is trivial: the initial price times the number of shares. The second summand in the latterformula describes the expected liquidity cost, which occurs due to price impact. This cost significantlydepends on the strategy and is the object of our study.

Let us now define our value function for continuous trading time U cts : [0, T ]×[0,∞)2×(0,∞)→ [0,∞)as

U cts(t, δ, x, κ) := infΘ∈Actst (x)

J(t, δ,Θ, κ) (6)

and the value function for discrete trading time as

Udis(t, δ, x, κ) := infΘ∈Adist (x)

J(t, δ,Θ, κ) ≥ U cts(t, δ, x, κ). (7)

Denoting ξn := ξtn = ∆Θtn , we can also write the discrete time cost integral as a sum

Udis(t, δ, x, κ) = infΘ∈Adist (x)

Et,δ,κ

∑tn≥t

(Dtn +

Ktn

2ξn

)ξn

. (8)

Both value functions U = U cts and U = Udis fulfil the boundary conditions

U(T, δ, x, κ) =(δ +

κ

2x)x and U(t, δ, 0, κ) = 0. (9)

Going forward we will use U and At(x) as a notation to indicate that the corresponding statementholds for both the continuous and discrete time case. If a certain statement is referring to only onesetting, then we will explicitly use U cts and Actst (x) respectively Udis and Adist (x).

Finally, let us relate the setting in this paper with that in Fruth, Schöneborn, and Urusov (2014). Tothis end, let the illiquidity coefficient be described by a deterministic strictly positive Borel functionk : [0, T ]→ (0,∞). We introduce the cost and the value functions

Jk(·)(Θ) (≡ Jk(·)(t, δ,Θ)), U ctsk(·)(t, δ, x), Udisk(·)(t, δ, x)

similarly to (5)–(8) using the illiquidity k in place of K. These are the corresponding cost and valuefunctions in Fruth, Schöneborn, and Urusov (2014) (notice that in this case, as k is deterministic, theinfima over deterministic and adapted strategies coincide). Again, we will use just the notation Uk(·)to indicate that the corresponding statement holds for both the continuous and discrete time case.The following lemma is sometimes useful for performing comparisons with the case of deterministicallychanging illiquidity.

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Lemma 3.1 (Stochastic versus deterministic illiquidity).For all t ∈ [0, T ], δ ≥ 0, x ≥ 0, κ > 0, we have

U(t, δ, x, κ) ≤ UEt,κ[K(·)](t, δ, x).

Proof. U(t, δ, x, κ) is smaller than or equal to the infimum like the one in (6) respectively (7), but overdeterministic strategies. The latter infimum equals UEt,κ[K(·)](t, δ, x) due to (3) and (5).

4 Definition of WR-BR structure

In this section we define WR-BR structure (WR: wait region, BR: buy region) and derive fundamentalproperties related to it. A detailed introduction of WR-BR structure is provided in Fruth, Schöneborn,and Urusov (2014); we therefore keep the exposition brief in this section.

Before attacking the formal definition of WR-BR structure, we note that the four-dimensional valuefunction U can be reduced by one dimension due to the following scaling property (its proof is straight-forward).

Lemma 4.1 (Optimal strategies scale linearly).For all a ∈ [0,∞) we have

U(t, aδ, ax, κ) = a2U(t, δ, x, κ). (10)

Furthermore, if Θ∗ ∈ At(x) is optimal for U(t, δ, x, κ), then aΘ∗ ∈ At(ax) is optimal for U(t, aδ, ax, κ).

We now define the function V : [0, T ]× [0,∞)× (0,∞)→ [0,∞) by the formula

V (t, y, κ) := U(t, 1, y, κ). (11)

In what follows we will use the notation V cts and V dis where we need to differentiate explicitly betweenthe continuous and the discrete time settings. Notice that for δ > 0 we have by Lemma 4.1

U(t, δ, x, κ) = δ2V(t,x

δ, κ),

that is, the function V already determines the entire value function U . Technically, the formula abovedoes not yet allow us to draw conclusions about U(t, 0, x, κ) because, for δ = 0, the ratio y = x/δis undefined. The claim that V determines U is, however, correct without any restriction due tocontinuity of the value function U(t, ·, ·, κ) in the pair (x, δ) (see Proposition A.2).

We first define the buy and wait region and subsequently define the barrier function.

Definition 4.2 (Buy and wait region).For any t ∈ [0, T ] and κ > 0, we define the inner buy region as

Brt,κ :=y ∈ (0,∞) | ∃ξ ∈ (0, y) : U(t, 1, y, κ) = U (t, 1 + κξ, y − ξ, κ) +

(1 +

κ

2ξ)ξ,

and call the following sets the buy region and wait region at time t for the illiquidity coefficient κ:

BRt,κ := Brt,κ, WRt,κ := [0,∞) \Brt,κ

(the bar indicates closure in R).

The inner buy region at time t for illiquidity coefficient κ hence consists of all values y such thatimmediate buying at the state (1, y) is value preserving. Due to dynamic programming principle, anon-zero purchase can never create value (i.e., decrease cost), so we always have

U(t, 1, y, κ) ≤ U (t, 1 + κξ, y − ξ, κ) +(

1 +κ

2ξ)ξ.

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The wait region therefore contains all values y such that any non-zero purchase at (1, y) destroys value(i.e., increases cost). Let us note that BrT,κ = (0,∞), BRT,κ = [0,∞) and WRT,κ = 0. The waitregion / buy region conjecture can now be formalized as follows.

Definition 4.3 (WR-BR structure).The value function U has WR-BR structure if there exists a barrier function

c : [0, T ]× (0,∞)→ [0,∞]

such that for all t ∈ [0, T ] and κ > 0,

Brt,κ = (c(t, κ),∞)

with the convention (∞,∞) := ∅. For the value function Udis in discrete time to have WR-BRstructure, we only consider t ∈ t0, ..., tN and set cdis(t, κ) =∞ for t /∈ t0, ..., tN.

It is worth noting that the barrier can be infinite even in continuous time or in discrete time at timepoints t0, . . . , tN−1, that is, there can be certain t and κ, for which it is never optimal to perform ablock trade, regardless of how large the remaining position is. We refer to Propositions 5.8 and 5.9in Fruth, Schöneborn, and Urusov (2014) for sufficient conditions for infinite barrier in the case ofdeterministically varying K.

Let us remark that we always have c(T, κ) = 0. On the contrary, the barrier is always strictly positivefor t ∈ [0, T ) (whenever the value function U has WR-BR structure). The following is a directgeneralization of Proposition 5.7 in Fruth, Schöneborn, and Urusov (2014).

Proposition 4.4 (Wait region near zero).Assume that the value function U has WR-BR structure with the barrier c. Then, for any t ∈ [0, T )and κ > 0, we have c(t, κ) ∈ (0,∞].

Proof. Assume that for some t ∈ [0, T ) and κ > 0 we have c(t, κ) = 0. Let us fix some y > 0 anddefine

ξ := supξ ∈ (0, y) | U(t, 1, y, κ) = U(t, 1 + κξ, y − ξ, κ) +

(1 +

κ

2ξ)ξ≤ y.

As U(t, ·, ·, κ) is continuous (Proposition A.2), we get

U(t, 1, y, κ) = U(t, 1 + κξ, y − ξ, κ) +(

1 +κ

2ξ)ξ. (12)

If ξ < y, then, due to the scaling property of Lemma 4.1, the fact that (y − ξ)/(1 + κξ) ∈ Brt,κ, andthe splitting argument of Lemma A.1, we arrive at a contradiction with the definition of ξ. Thus,ξ = y, but then formula (12) contradicts Proposition A.3. This completes the proof.

5 The WR-BR theorem

In this section we show that the value function exhibits WR-BR structure if K is a diffusion satisfyingthe following assumption.

Assumption 5.1. (Special diffusion).K is a (possibly time-inhomogeneous) diffusion

dKs = µ(s,Ks) ds+ σ(s,Ks) dWKs , Kt = κ > 0, (13)

for an (Fs)-Brownian motionWK and µ, σ : [0, T ]×(0,∞)→ R such that, for any initial time t ∈ [0, T ]and starting point Kt = κ > 0, the stochastic differential equation has a weak solution which is uniquein law, is strictly positive and has finite first moments. Furthermore, for all t ∈ [0, T ] and κ > 0, wehave

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i) ηs := 2ρsKs

+ µ(s,Ks)K2s− σ2(s,Ks)

K3s

> 0 Pt,κ×µL-a.e. on Ω× [t, T ] (µL denotes the Lebesgue measure),

ii) Et,κ[

sups∈[t,T ] K2s

infs∈[t,T ] Ks

]<∞,

iii) Et,κ[(∫ T

0|ηs| ds

)(sups∈[t,T ]K

2s

)]<∞.

In Section 8 below we study profitable round trip strategies without assuming that the process η ispositive, but we will need part iii) of Assumption 5.1. That is why we write absolute value of η in iii).

Theorem 5.2. (WR-BR theorem).If Assumption 5.1 holds, then there is a unique optimal strategy, and we have WR-BR structure.

In fact, we will see in the proofs that existence and uniqueness of the optimal strategy both indiscrete and continuous time as well as WR-BR structure in discrete time hold under parts i)–ii) ofAssumption 5.1. We need part iii) only for WR-BR structure in continuous time. Overall, part i) ofAssumption 5.1 is the most crucial in the proofs because it is directly linked to the convexity of J .Parts ii) and iii) of Assumption 5.1 are required for more technical aspects of our proofs.

The proof of Theorem 5.2 is given in Appendix B. In Subsections 5.1 and 5.2 we only outline thestrategy of the proof. In Lemma 5.3 we show that Assumption 5.1 i)–ii) entails strict convexity of thecost functional J in the strategy. This is based on the representation

J(Θ) =1

2Et,δ,κ

[D2T+

KT− δ2

κ+

∫[t,T ]

ηsD2sds

](14)

(see the proof of Lemma 5.3 in Appendix B), which also proves to be useful in the study of profitableround trip strategies in Section 8. The strict convexity of J in turn guarantees existence and uniquenessof the optimal strategy. As we show in Subsection 5.2, the uniqueness excludes WR-BR-WR and othersituations: at any upper boundary of a buy region it must be equally optimal to wait and to executethe strictly positive trade to the lower boundary of the buy region. We first pursue this line ofargument for the discrete time case and then transfer it to continuous time and thus do not use theHamilton-Jacobi-Bellman equation.

Theorem 5.2 does not cover all models which result in WR-BR structure. E.g., restricting trading toonly two points in time always yields WR-BR structure irrespective of Assumption 5.1, see Proposi-tion 7.1. Furthermore, in the case of deterministically varying K we always have WR-BR structure indiscrete time and, for continuous K, in continuous time, see Fruth, Schöneborn, and Urusov (2014).In Section 6 we show that Assumption 5.1 and thus WR-BR structure hold in several standard models.In Section 7 we provide examples violating WR-BR structure, which highlights that some assumptionson K are necessary for it. As we will see in Section 8, part i) of Assumption 5.1 is also related to theabsence of profitable round trip trading strategies in a zero spread two-sided order book model.

5.1 Existence of a unique optimal strategy

Lemma 5.3. (Costs are convex in the strategy).Let Assumption 5.1 i)–ii) hold. Then, for all t ∈ [0, T ], δ ≥ 0 and κ > 0, the function J(·) ≡ J(t, δ, ·, κ)is finite and strictly convex on At.

The strict convexity of J guarantees the uniqueness of an optimal strategy provided it exists. Moreover,we can use the convexity together with a Komlós-type argument to get the existence of an optimalstrategy both in discrete and in continuous time:

9

Proposition 5.4. (Existence and uniqueness of an optimal strategy).Let Assumption 5.1 i)–ii) hold. Then, for all t ∈ [0, T ], δ ≥ 0, x ≥ 0 and κ > 0, there exists a uniqueoptimal strategy, i.e., there exists a unique Θ∗ = Θ∗(t, δ, x, κ) ∈ At(x) with

J (t, δ,Θ∗, κ) = infΘ∈At(x)

J (t, δ,Θ, κ) .

5.2 Wait and buy region structure

In order to prove WR-BR structure, we first exploit the existence and uniqueness of Proposition 5.4together with a technical result (Proposition B.1) in order to get WR-BR structure in discrete time:

Proposition 5.5. (Discrete time: WR-BR structure).Let Assumption 5.1 i)–ii) hold. Then the value function Udis has WR-BR structure.

The line of argument used in the proof of Proposition 5.5 does not extend directly to continuous time.We, therefore, transfer the discrete time result to continuous time using approximation techniques(Lemmas B.2 and B.3), and this is where we need part iii) of Assumption 5.1.

Proposition 5.6. (Continuous time: WR-BR structure).Let Assumption 5.1 i)–iii) hold. Then the value function U cts has WR-BR structure.

6 Example models with WR-BR structure

6.1 Analytical results

By Theorem 5.2, any model satisfying Assumption 5.1 has WR-BR structure. In this section, we showthat Assumption 5.1 is satisfied by several standard processes. We start with a deterministic K.

Proposition 6.1. (Deterministic case).Assume that K : [0, T ]→ (0,∞) is deterministic and two times continuously differentiable, ρ : [0, T ]→(0,∞) is continuously differentiable with K ′t+2ρtKt > 0 for all t ∈ [0, T ]. Then Assumption 5.1 holds,and the value function has WR-BR structure.

Proof. Condition i) is equivalent to K ′t + 2ρtKt > 0, and ii), iii) are clearly satisfied for deterministiccontinuous K.

We remark that Proposition 6.1 is not the best result one can get for a deterministic K. In Fruth,Schöneborn, and Urusov (2014) we prove that we have WR-BR structure whenever K is continuousand deterministic.

Let us now turn to a time-homogeneous geometric Brownian motion (GBM). Notice that, due to thehomogeneity in time, it is enough to verify the conditions in Assumption 5.1 only under measures P0,κ.

Proposition 6.2. (GBM case).Let K be a geometric Brownian motion

dKt = µKt dt+ σKt dWKt , K0 = κ > 0. (15)

Consider a constant resilience ρt ≡ ρ > 0 such that 2ρ+ µ− σ2 > 0. Then Assumption 5.1 holds, andthe value function has WR-BR structure.

10

The proof of this proposition is given in Appendix C. Also see Fruth (2011) for alternative conditionsensuring WR-BR structure in the GBM case.

At this point it is worth noting (see Fruth (2011)) that the barrier c(t, κ) in the GBM case has theform c(t)/κ. This is not desirable from the practical viewpoint because this prescribes to trade moreaggressively when κ is large. The intuition behind this effect is as follows. Even when the startingpoint κ of the process K is large, we do not have an incentive to wait because under the GBM modelK is going to remain large for some time. On the other hand, we profit from resilience while trading(in other words, we profit from the upcoming limit orders that arrive in place of the consumed oneswith the rate which is the resilience).

The natural generalisation of the GBM model is the CEV model (e.g., see Davydov and Linetsky(2001)), which is related to the squared Bessel process (e.g., see Carr and Linetsky (2006)). In thatmodel the hyperbolic form c(t)/κ for the barrier does not hold any longer. However, the previouslydeveloped intuition suggests to try a process with mean reversion. That is, a behaviour, where theprocess K is going to be quickly reduced whenever the starting point κ of K is big, looks particularlyappealing from the economic point of view. The natural model is then, speaking informally, a “squaredBessel process with mean reversion”, which is the Cox-Ingersoll-Ross (CIR) process.

Proposition 6.3. (CIR case).Let K be a Cox-Ingersoll-Ross process

dKt = µ(K −Kt) dt+ σ√Kt dW

Kt , K0 = κ > 0,

where K, µ, σ > 0. Consider a constant resilience ρt ≡ ρ > 0 such that

2ρ ≥ µ > 2σ2/K.

Then Assumption 5.1 holds, and the value function has WR-BR structure.

The proof of this proposition is presented in Appendix C.

6.2 Numerical results for CIR examples

To illustrate the WR-BR structure and the corresponding optimal strategy, we present the result of anumerical implementation of two specific parameter sets for the CIR process in this subsection.

One can use the Markov chain approximation method introduced by Kushner and his co-authors, e.g.,see Kushner and Dupuis (2001), to implement our optimization problem for the mean-reverting CIRprocess as introduced in Proposition 6.3. For a formal convergence proof and a detailed explanationof this numerical scheme see Fruth (2011), Chapter 3. For a numerical implementation of the CIRexample one needs to specify nine parameters:

ρ, µ, K, σ, T, ymax, κmax, ∆t, h.

The first four parameters specify the resilience speed and the parameters for the stochastic liquidityCIR process (mean-reversion speed, mean-reversion level and volatility). [0, T ]× [0, ymax]× [0, κmax]is the space, where we approximate the dimension-reduced value function V introduced in (11). ∆tand h correspond to the grid size in time and space respectively. It is crucial that ∆t is significantlysmaller than h for the numerical scheme to yield reasonable results.

In Figure 1, we state the result of our implementation of the CIR process for two different values ofmean-reversion speed µ. The size of the wait region is decreasing in time in both cases (see the plotson the top), i.e., buying becomes more aggressive as the investor runs out of time. This is also thecase for constant liquidity and this is what we intuitively expect. However, examples with the barrier

11

being increasing in time can also be constructed, see Figure 6.1 in Fruth, Schöneborn, and Urusov(2014).

The barrier profile c(t, κ) is increasing in κ for the high mean reversion case.2 This drives how theoptimal strategy Θt(ω) reacts to the illiquidity process Kt(ω): At times of growing Kt it falls behindthe constant liquidity strategy, while trading is accelerated when Kt decreases. In analogy to theterminology of aggressive in the money execution introduced in Schied and Schöneborn (2009), suchan execution strategy could be called aggressive in the liquidity. For the low mean reversion case, thebarrier is decreasing in κ, resulting in the opposite relationship (passive in the liquidity).

In order to understand the difference between the barriers for low and high mean reversion in moredetail, let us first consider the case of constant K = κ (i.e., the model of Obizhaeva and Wang (2013)with constant, deterministic liquidity). If κ is large, then the market is easily dislocated, and we haveto accept large dislocations δ and hence have a small boundary c(t, κ). If on the other hand κ is smallthen we can maintain a smaller dislocation and hence larger c. The barrier c(t, κ) is hence decreasingin κ. This effect intuitively carries over to CIR processes with small mean reversion because K onaverage will not change significantly during the execution time horizon. However if the mean reversionof the CIR process is large, then we do expect a change of K over time depending on the current levelof Kt = κ: If κ is large and the mean reversion is high, then it is better to wait because the processK is going to quickly shrink to the smaller level K which will allow cheaper execution. Respectivelyif κ is small then we expect K to grow, resulting in more expensive execution in the future and hencean incentive to speed up execution now. The net result is a barrier c(t, κ) that is increasing in κ.

In the extreme case, we obtain an infinite barrier for large κ. A straightforward generalisation ofProposition 5.9 in Fruth, Schöneborn, and Urusov (2014) gives us the following sufficient conditionfor infinite barrier. If, for some 0 ≤ t1 < t2 ≤ T and κ > 0, the function t 7→ Et1,κKt is continuous on[t1, t2] and

κe−∫ t2t1ρu du > Et1,κKt2

(intuition: the increase in liquidity outweighs the resilience), then Brt1,κ = ∅, i.e., c(t1, κ) = ∞.Applying this statement in the CIR case and using the well-known formula

Et,κKs = K + e−µ(s−t)(κ− K) for s ≥ t,

we obtain that, for µ > ρ, we have infinite barrier for sufficiently large values of κ, which is what wesee on the top right plot in Figure 1.

There are flat stretches in the optimal strategy (see both plots in the bottom of Figure 1; the effect ismore pronounced in the plot on the right). During these time spans, the optimal ratio of outstandingshares over order book deviation is in the interior of the wait region. As opposed to the constantliquidity case (the dotted line in the bottom plots), the optimal strategy displays a stop-and-gopattern.

In Section 7 we will see counterintuitive examples of WR-BR-WR structure. In such WR-BR-WRsituations and δ = 0 we would not get the typical initial discrete trade in the optimal strategy, butthe optimal strategy would start with a flat stretch.

7 Example models without WR-BR structure

In this section, we provide a couple of examples that do not follow the WR-BR structure. We find thatcases of WR-BR-WR structure can occur: when a large number of shares remains to be purchased,it is optimal to wait, while buying is optimal if a smaller number of shares is remaining. We first

2The apparent lack of smoothness in the corresponding plot in Figure 1 is an artifact of the discretisation schemethat we used rather than a property of the actual barrier.

12

0 0.5 1 1.5 20

5

10

k

c(t

,k)

Low mean−reversion

0 0.05 0.1 0.15 0.2 0.250

0.5

1

1.5

2

t

Kt(ω

)

0 0.05 0.1 0.15 0.2 0.250

50

100

t

Θt(ω

)

0 0.5 1 1.5 20

5

10

kc(t

,k)

High mean−reversion

0 0.05 0.1 0.15 0.2 0.250

0.5

1

1.5

2

t

Kt(ω

)

0 0.05 0.1 0.15 0.2 0.250

50

100

t

Θt(ω

)

Figure 1: Implementation for the CIR process with ρ = 5, K = 1, σ2 = 2, T = 0.25, ymax = 30, κmax = 2,∆t = 0.00005, h = 0.05. Left: µ = 3. Right: µ = 10. On the top, the barriers κ 7→ c(t, κ) are plotted fort ∈

0, 1

4T, 1

2T, 3

4T(top-down). The Euler scheme is used to simulate a path Kt(ω) of the CIR process. In

the middle plots, Kt(ω) is compared with the mean-reverting level K = 1 (dotted). In the bottom plots,the optimal strategy for δ = 0, x = 100 and the simulated path Kt(ω) is compared to the constant liquiditystrategy (dotted).

13

present a two-scenario model with WR-BR-WR structure and discuss it in Subsection 7.1 in detail.This discussion sharpens intuition about the reasons of such a behaviour and provides an intuitiveexplanation why this phenomenon does not appear with deterministic liquidity dynamics. Using thisintuition Fruth (2011) provides a WR-BR-WR example in a certain two scenario continuous-timemodel.

Furthermore, the discussion in Subsection 7.1 inspires certain parameter choices in the CIR modelthat might also lead to WR-BR counterexamples. In Subsection 7.2 we perform a numerical checkfor one such parameter choice in the CIR model in discrete time and indeed recover another WR-BRcounterexample, now in the CIR framework. Our examples are in discrete time with trading occurringat three points in time. The following proposition shows that WR-BR structure always holds if tradingoccurs at two time points only.

Proposition 7.1. (WR-BR structure for two trading instances).Let N = 1, i.e., 0 = t0 < t1 = T , and denote a0 := e−

∫ t1t0ρsds. Then the value function has WR-BR

structure with

V dis (t0, y, κ) =1

2Et0,κ[KT ]y2 + a0y −

[(Et0,κ[KT ]−κa0)y−(1−a0)]

2

2κ+2Et0,κ[KT ]−4κa0if y > c (t0, κ)

0 otherwise

,

c (t0, κ) =

1−a0

Et0,κ[KT ]−κa0if Et0,κ[KT ] > κa0

∞ otherwise

.

Proof. We know that Udis(t1, δ, x, κ) = (δ + κ2x)x. The assertion follows from

Udis(t0, δ, x, κ) = minξ∈[0,x]

(δ +

κ

2ξ)ξ + Et0,κ

[Udis (t1, (δ + κξ)a0, x− ξ,KT )

].

Note that we have not made any specific assumptions on the distribution of KT in Proposition 7.1.

7.1 A two scenario WR-BR counterexample

Let us assume that the process K is not driven by a diffusion, but instead is given by a finite number ofscenarios. The case of a single scenario implies a deterministic evolution of K which always results ina WR-BR structure. We therefore focus on the second simplest case of two equally likely scenarios Aand B, i.e., Ω = ωA, ωB, P(ωA) = P(ωB) = 1/2, and consider three trading instances t0, t1, t2,i.e., we consider a discrete time example with N = 2. To fully specify this two scenario model, weneed to choose seven constants

a0 := e−∫ t1t0ρsds, a1 := e−

∫ t2t1ρsds, κ0 := Kt0 ,

κA1 := Kt1(ωA), κA2 := Kt2(ωA), κB1 := Kt1(ωB), κB2 := Kt2(ωB).

Proposition 7.2. With the parameter values given in Figure 2, we obtain WR-BR-WR structure,i.e., there are two threshold values 0 < cl < cu < ∞ such that the buy region at time t0 is given byBrt0 = (cl, cu].

A formal proof of Proposition 7.2 is given in Appendix D. We now present an intuitive explanation ofthe phenomenon described here, which will inspire the construction of a WR-BR counterexample inthe CIR model presented in Subsection 7.2.

14

æ

æ

ææ

æ æ

Κ1

A=1 Κ

2

A=1

Κ1

B=3

Κ2

B=2

Κ0=1.95

a0=0.9999 a1=0.5

Figure 2: Seven constants that specify the two scenario model with three trading instances.

How to compute cl and cu is explained in the proof. For the discussion below we notice that cl = 0.95and cu = 5.75. For a trade ξ, we define

Udis(t0, δ, x, κ0; ξ) :=(δ +

κ0

2ξ)ξ + E

[Udis(t1, (δ + κ0ξ)a0, x− ξ, κ1)

],

which is a cost of buying ξ shares at time t0 and optimal trading thereafter. It is easy to see thatUdis is piecewise quadratic in ξ.

To illustrate the dynamics of the optimal strategy, we take different x and plot

ξ 7→ Udis (t0, 1, x, 1.95; ξ)

in Figure 3. When the total order is as small as x = 0.9, it is optimal not to do an initial trade. Thetransition from wait to buy region is approximately at x = 0.95. For x = 1, we are in the buy regionand one optimally trades about two percent of the total order at time t0. But at x = 5.75, we switchfrom buy to wait region and stay in the wait region for all larger values of x. The graph for x = 5.75illustrates the non-uniqueness of the optimal strategy at the transition from buy to wait region.

Intuition might suggest that the larger the remaining position x at time t0, the larger the initialtrade ξ0. The downside of trading at time t0 is that the full initial impact δ is influencing the costfunctional (at later points in time this initial impact is partially decayed already). The upside is amore balanced distribution of new impact across an additional time point (any impact generated attime t0 will already be partially decayed at time t1). These two effects are the only drivers in thecase of deterministic K, and the second effect grows faster in the remaining position x than the firsteffect. If K evolves stochastically, then a third effect comes into play: trading at times after t0 canrespond to new information gained about K (such as whether scenario A or B occurred). This effectcan dominate the second effect for large remaining positions x.

Let us now analyze the situation for different values of κ0 while keeping the other model parametersincluding κA1 , κB1 , κA2 and κB2 fixed. Figure 4 indicates for each point (κ0, x) if it belongs to the buyor wait region. It is created by computing the optimal initial trade ξ(κ0, x) of Udis (t0, 1, x, κ0; ξ)analytically. WR-BR-WR structure occurs for κ0 ∈ (1.94, 2). The upper barrier from buy to waitregion has an asymptote at κ0 = 1.94. For the case κ0 = 1.95 that we discussed in Figure 3, the smalldots on the right-hand side of Figure 4 point out the transitions from wait to buy region and buy towait region respectively. For expensive κ0 ≥ 2, we are not trading irrespectively of the size of the totalorder. For inexpensive κ0 ≤ 1.94, we have the usual WR-BR situation. On the interval in between,the large investor has an incentive not to trade for large positions x. The resilience between t0 and t1is extremely low and waiting until t1 has the advantage of gaining information whether scenario Aor B has occurred. That is there is a tradeoff between gaining information by waiting until the nexttime instance and attracting resilience by trading right now.

15

0.005 0.010 0.015 0.020Ξ

1.0576

1.0578

1.0580

1.0582

Costs

x 0.9®Wait

0.01 0.02 0.03 0.04 Ξ

1.2498

1.2499

1.2500

1.2501

Costs

x 1® Buy

0.5 1.0 1.5 Ξ25.56

25.57

25.58

25.59

25.60

Costs

x 5.75® Buy or Wait

2 4 6 8 10Ξ

276.5

277.0

277.5

278.0

278.5

279.0

279.5

Costs

x 20®Wait

Figure 3: For the parameters from Figure 2 and total order size x = 0.9, 1, 5.75, 20, the graphs plot thedependence of the costs Udis (t0, 1, x, 1.95; ξ) on the initial trade ξ.

BR

WR

0.0 0.5 1.0 1.5 2.0 2.5 3.0Κ0

0.5

1.0

1.5

2.0

x

BR

1.94 1.95 1.96 1.97 1.98 1.99 2.00Κ0

1

2

3

4

5

6

7

x

Figure 4: For the parameters from Figure 2, but different values of κ0, we illustrate the wait and buy region.Looking more closely at the large dot (κ0, x) = (2, 1) yields the picture on the right-hand side. The buy regionhas the shape of a wedge.

16

Informally, one can view the two scenario model of this section as an approximation of a diffusivemodel that has almost half of its scenarios close to scenario A and another half close to scenario B.One can therefore expect that WR-BR structure is lost in such diffusive models.

7.2 Cox-Ingersoll-Ross process in discrete time

In Section 6, we have considered examples of diffusive models and shown that they have WR-BRstructure if certain conditions are met. For the case of the CIR process3

dKs = µ(K −Ks

)ds+ σ

√KsdW

Ks ,

let us now consider three trading times t0, t1, t2 with

t0 = 0, t1 = 0.0072, t2 = 1.0072, ρ ≡ 1.3863,

µ = 0.6931, K = 1, σ = 5.2523. (16)

This example violates the conditions of Proposition 6.3. It is inspired by the two scenario modelpresented in Subsection 7.1. E.g., t1 is close to t0, and the high volatility makes illiquid scenarios withKt being very different from K likely to occur.

Using Proposition 7.1 and the density function of the CIR process together with a numerical integrationscheme, we can compute Udis (t0, 1, x, κ0; ξ) from dynamic programming. For each point (κ0, x), wecan calculate the costs of different trades ξ from an equidistant grid 0, dξ, ..., x. We can then inferthat the point (κ0, x) belongs to the wait region if the costs for ξ = 0 are smaller than the costs onthe remaining grid.

Executing this scheme for several points (κ0, x) yields Figure 5. As for the two scenario model, thereexist choices of κ0 that lead to WR-BR-WR structure. But instead of a wedge-shaped buy region, weget a tongue-shaped upper wait region, which is located around the mean-reversion level K = 1.

0 0.5 1 1.5 20

1

2

3

4

5

6

7

8

x

κ0

BR

WR

Figure 5: This figure shows a WR-BR-WR example for the CIR process with parameters (16) and threetrading instances. Points (κ0, x) ∈ 0.1, 0.2, ..., 2.1×0.2, 0.4, ..., 8 are considered. The wait region is shadedblack.

3See Fruth (2011) for a WR-BR-WR example for the time-inhomogeneous GBM and three trading instances.

17

8 Profitable round trip strategies

So far we have only considered one side of the limit order book. In this section, we extend our modeland include the other side of the limit order book. In such two-sided limit order books, round tripstrategies are possible and we determine under which conditions they can be profitable.

Without loss of generality we now consider the starting time 0. We model strategies that both buyand sell the asset as a pair (Θ, Θ), where Θ ∈ A0 and Θ ∈ A0 describe the number of shares whichthe investor bought respectively sold starting from time 0. The position at time t is given by Θt− Θt,and a round trip strategy is characterised by ΘT+ = ΘT+. Recall that, by definition of A0, ΘT+ andΘT+ are bounded random variables. If At and Bt are the best ask and best bid prices respectively,then the total cost of a strategy (Θ, Θ) is given by

C(Θ, Θ) :=

∫[0,T ]

(At +

Kt

2∆Θt

)dΘt −

∫[0,T ]

(Bt −

Kt

2∆Θt

)dΘt. (17)

We now present two different models for two-sided limit order books. The corresponding models fordeterministic K are discussed in Fruth, Schöneborn, and Urusov (2014). First, we consider a two-sidedlimit order book with bid-ask spread that depends on trading activity.

Model 8.1. (Dynamic spread model).The best ask and best bid price processes A and B in (17) are modelled as At := Aut + Dt andBt := But − Et, where the unaffected best ask and best bid price processes Au and Bu are càdlàgH1-martingales with But ≤ Aut for all t ∈ [0, T ], and

Dt := D0e−∫ t0ρsds +

∫[0,t)

Kse−∫ tsρududΘs, t ∈ [0, T+], (18)

Et := E0e−∫ t0ρsds +

∫[0,t)

Kse−∫ tsρududΘs, t ∈ [0, T+], (19)

with some given non-negative initial values D0 ≥ 0 and E0 ≥ 0.

Proposition 8.2. (Profitable round trips in the dynamic spread model).In the dynamic spread model round trip trading strategies cannot be profitable. That is, for all κ > 0,D0 ≥ 0 and E0 ≥ 0, for all admissible (Θ, Θ) with ΘT+ = ΘT+, we have

E0,κ[C(Θ, Θ)] ≥ 0.

Furthermore, the expected execution costs of a buy (or sell) program that builds up a deterministicposition of say x ∈ R shares cannot be decreased by intermediate sell (resp. buy) trades. That is, forall κ > 0, D0 ≥ 0 and E0 ≥ 0, for any admissible (Θ, Θ) with ΘT+ − ΘT+ = x > 0, there is anadmissible Θ with ΘT+ = x such that E0,κ[C(Θ, Θ)] ≥ E0,κ[C(Θ, 0)]; also the symmetric statementwith x < 0 holds true.

We omit the proof because it is a direct extension of the corresponding Proposition 3.4 in Fruth,Schöneborn, and Urusov (2014). Let us now consider an alternative model for a two-sided limit orderbook in which the spread is constantly zero.

Model 8.3. (Zero spread model).The best ask and best bid price processes in (17) are modelled as Alt := B

lt := Sut + D

lt , where the

unaffected price Su is a càdlàg H1-martingale, and

Dlt := D

l0e−∫ t0ρs ds +

∫[0,t)

Kse−∫ tsρu du(dΘs − dΘs), t ∈ [0, T+], (20)

with some given initial value Dl0 ∈ R.

18

There is a subtle difference in understanding price manipulation between the dynamic and zero spreadmodels. In the discussion of profitable round trip strategies in the dynamic spread model (see Propo-sition 8.2) we considered arbitrary initial values D0 ≥ 0 and E0 ≥ 0 in (18) and (19). In contrastto this, in the discussion of profitable round trip strategies in the zero spread model (Theorem 8.4,which follows) we will consider Dl0 = 0 in (20). Whenever Dl0 6= 0 we usually have profitable roundtrip strategies in the zero spread model, and this is due not to properties of the model, but ratherto the fact that both buy and sell orders are executed at the same price,4 therefore, profitable roundtrips will make use of the initial deviation Dl0 from the unaffected price Su and of the fact that, dueto the resilience, the absolute value of this deviation decreases to zero in the absence of trading. 5

In order to study profitable round trip strategies in the zero spread model let us introduce the notations

Θl := Θ− Θ (21)

for the composite strategy, which includes both buy and sell orders, and, by analogy with (5),

Jl(Θl) := Jl(t, δ,Θl, κ) := JlT (t, δ,Θl, κ) := Et,δ,κ

[∫[t,T ]

(Dls +

Ks

2∆Θls

)dΘls

](22)

for the cost function. As in (5), the subscript in Et,δ,κ means that we start at time t with Dlt = δ andKt = κ. The precise explanation of how (22) comes into play is similar to the explanation of (5) givenin the paragraph following the one that contains (5). Namely, consider a strategy (Θ, Θ) ∈ At × Atthat acquires Θ

lT+ = x shares on the time interval [t, T ] (x ∈ R is deterministic). The total cost of

this strategy is (cf. (17))∫[t,T ]

(Sus +Dls +

Ks

2∆Θs

)dΘs −

∫[t,T ]

(Sus +Dls −

Ks

2∆Θs

)dΘs.

A calculation involving integration by parts and using that Su is an H1-martingale as well as that Θand Θ are bounded reveals that the expected total cost equals

Sut x+ Et,δ,κ

[∫[t,T ]

(Dls +

Ks

2∆Θs

)dΘs −

∫[t,T ]

(Dls −

Ks

2∆Θs

)dΘs

].

Again, the first summand, which is trivial and moreover vanishes for round trip strategies, describesthe expected cost that occurs due to trading in the unaffected price. The second summand in thelatter formula, which describes the expected liquidity cost, is in general larger than Jl(t, δ,Θl, κ), butit is equal to Jl(t, δ,Θl, κ) whenever ΘT+ + ΘT+ equals the variation of Θl over [t, T ]. It remains tonotice that the latter can always be assumed without loss of generality (and, moreover, it does notmake sense economically to consider strategies (Θ, Θ) with ΘT+ +ΘT+ being strictly greater than thevariation of Θl over [t, T ] because this means that buying and selling happen simultaneously).

4If the spread between the bid and ask is a positive constant with trades dislocating both bid and ask prices by thesame amount then profitable round trip strategies are possible for sufficiently large Dl0 .

5 To explain this point in more detail, we assume that Dl0 6= 0, consider some κ > 0 and arbitrary ε ∈ (0, T ] andchoose

z =1− e−

∫ ε0 ρu du

E0,κKε + κ(1− e−∫ ε0 ρu du)

.

Now consider the strategy (using notation (21))

Θls = −zDl0I(0,ε](s), s ∈ [0, T+].

A straightforward calculation reveals that the expected cost is

1

2z(D

l0)2−(2− κz) + (2− 2κz)e−

∫ ε0 ρu du + zE0,κKε

<

1

2z(D

l0)2−(2− κz)

(1− e−

∫ ε0 ρu du

)+ zE0,κKε

< 0,

that is, (Θ, Θ) is a profitable round trip strategy.

19

We write JlT in (22) with the subscript T to emphasize the time horizon explicitly. We considerdiffusion setting (13) for all finite time horizons T <∞ and introduce the function η : R+×(0,∞)→ Rby the formula

η(s, κ) :=2ρsκ

+µ(s, κ)

κ2− σ2(s, κ)

κ3,

that is, we have ηs = η(s,Ks) for ηs as in Assumption 5.1 i).

Theorem 8.4. (Profitable round trips in the zero spread model).In the zero spread model suppose that Assumption 5.1 ii) holds for all finite T <∞ and

(A) the resilience is bounded away from zero (ρs ≥ ρ > 0) as well as, for all t ≥ 0 and κ > 0, thefunction s 7→ Et,κ[Ks], s ∈ [t,∞), is bounded.

We then have the following classification:

1. If η ≥ 0 everywhere, then all round trip strategies starting at any time t ≥ 0 with Dlt = 0 havenonnegative costs.

2. Under Assumption 5.1 iii), if η < 0 in some [t, t+ ∆t]× [κ− ε, κ+ ε], then there are profitableround trip strategies starting at t with Dlt = 0.6

Assumption (A) is satisfied for a wide range of processes K including stationary processes such asthe CIR process as well as the GBM process with non-positive drift (µ ≤ 0 in (15)) whenever theresilience is bounded away from zero.

We will see in the proof of Theorem 8.4 in Appendix E that the role of Assumption (A) is to ensurethat liquidation of a random but bounded position that the investor has at some time t+ ∆t can beachieved for arbitrarily small cost if Dlt+∆t+ = 0 and the time horizon T is large. The latter propertyseems natural to expect in reasonable models when the resilience is bounded away from zero. In fact,one might replace Assumption (A) with any other assumption that ensures the property stated above.

Parts ii) and iii) of Assumption 5.1 are required for some technical aspects of our proof. What isinstrumental there is the generalization of formula (14) to the case of not necessarily positive η. Themain message of Theorem 8.4 can be somewhat loosely described as follows: if η ≥ 0 everywhere,then the cost functional J is convex in the strategy and round trips cannot be profitable; if η < 0somewhere,7 then the cost functional is not convex and profitable round trip strategies exist.

The results of this section reveal a link between the models for two-sided limit order books in ourpresent setting (stochastic diffusive K): If Assumption 5.1 i) holds, then optimal strategies in thedynamic spread model are of WR-BR structure and profitable round trip strategies do not exist inthe zero spread model. If Assumption 5.1 i) is violated, then optimal strategies in the dynamic spreadmodel do not need to be of WR-BR structure, and round trip strategies in the zero spread model donot need to result in costs.

If only deterministic trading strategies are considered, then only the expected evolution of K mattersand, in the case µ(s, κ) is affine in κ, ηs := 2ρs

Ks+ µ(s,Ks)

K2s≥ 0 is sufficient to prevent free (or even

profitable) deterministic round trip strategies. As ηs = ηs − σ2(s,Ks)K3s

< ηs, we can have ηs ≥ 0 whileηs < 0. For some stochastic models for K, we therefore have only stochastic profitable round tripstrategies but no deterministic profitable round trip strategies.

6As pointed out above profitable round trip strategies exist also for Dlt being different from zero, but the relevantquestion in the zero spread model is the one for Dlt = 0.

7Let us also notice that, if η(t, κ) < 0 at some point (t, κ) and the functions ρ, µ and σ are continuous, then η < 0in some [t, t+ ∆t]× [κ− ε, κ+ ε].

20

9 Concluding discussions

Summary

We propose a limit order book model with stochastic liquidity that captures random fluctuationsof the limit order book depth. If the stochastic liquidity in this model follows a diffusion processmeeting certain conditions, then optimal trade execution follows the classical wait region / buy regionstructure often observed in limit order book models with static or deterministically time dependentliquidity. For other stochastic liquidity processes, the optimal trade execution strategy can take moregeneral forms; for example, multiple wait regions can occur, and optimal trade sizes do not need todepend monotonically on the size of the position that remains to be liquidated. In our framework, theconditions for the wait region / buy region structure also result in all round trip strategies generatingpositive costs even if the zero spread model is assumed.

Follow-up questions and open problems

As discussed after Theorem 8.4, the absence of profitable round trips in the zero spread model islinked to the convexity of the cost functional J in the strategy. This link holds true also in the caseof deterministic K (at least for absolutely continuous K, in which case (14) applies).8 But in the caseof deterministic K we have WR-BR structure whenever K is continuous (see Theorem 7.7 in Fruth,Schöneborn, and Urusov (2014)), that is, in the case of deterministic K we can go beyond convexityof J and still have WR-BR structure. By analogy one might expect the same in our present settingof diffusive K and conjecture that the convexity assumption in Theorem 5.2 is superfluous. However,as the WR-BR counterexamples in Section 7 reveal, this assumption cannot be dropped entirely forstochastic liquidity. This suggests follow-up questions such as:

• Is it possible to relax Assumption 5.1 and still guarantee WR-BR structure? Putting it dif-ferently, how far can one go beyond convexity of J (if possible at all) and still have WR-BRstructure?

• Other than (intuitive) WR-BR and (counterintuitive, but possible) WR-BR-WR structures,what kind of structures can appear in general? E.g., is it possible to have multiple buy regions?

• How is the optimal strategy changed when a different cost function J is used, e.g., to incorporaterisk aversion of the trader?

These questions are not easy to treat in a mathematically rigorous way and are natural open problemsfor future research.

A Auxiliary results

The following simple result is recalled from Fruth, Schöneborn, and Urusov (2014). It shows that ourformulas for the price deviation and for the cost are economically sensible.

Lemma A.1 (Splitting argument).Doing two separate trades ξα, ξβ > 0 at the same time s has the same effect as trading at onceξ := ξα + ξβ, i.e., both alternatives incur the same cost and the same price deviation Ds+.

8It is worth noting at this point that the corresponding results for deterministic K, Proposition 8.3 and Corollary 8.5in Fruth, Schöneborn, and Urusov (2014), are proved differently from Theorem 8.4.

21

Proof. The cost is in both cases(Ds +

Ks

2ξ

)ξ = Ds(ξα + ξβ) +

Ks

2(ξ2α + 2ξαξβ + ξ2

β)

=

(Ds +

Ks

2ξα

)ξα +

(Ds +Ksξα +

Ks

2ξβ

)ξβ

and the price deviation Ds+ = Ds +Ks(ξα + ξβ) after the trade is the same in both cases as well.

Proposition A.2 (Continuity of the value function).For each t ∈ [0, T ] and κ > 0, the function

U(t, ·, ·, κ) : [0,∞)2 → [0,∞)

is continuous.

The proof is similar to the one of Proposition 5.5 in Fruth, Schöneborn, and Urusov (2014).

Proposition A.3 (Trading never completes early).For all t ∈ [0, T ), δ ≥ 0, x > 0 and κ > 0, the value function satisfies

U(t, δ, x, κ) <(δ +

κ

2x)x,

i.e., it is never optimal to buy the whole remaining position at any time t ∈ [0, T ).

Proof. The result immediately follows from Lemma 3.1 and the corresponding result for determinis-tically varying K, see Proposition 5.6 in Fruth, Schöneborn, and Urusov (2014).

B Proof of the WR-BR theorem

As discussed in Section 5, Theorem 5.2 follows from the results formulated in Subsections 5.1 and 5.2,which we now prove.

Proof of Lemma 5.3. Let t, δ and κ be fixed. Clearly, Assumption 5.1 ii) implies Et,κ sups∈[t,T ]Ks <∞, hence J(·) is finite on the whole At. We demonstrate below that

J(Θ) =1

2Et,δ,κ

[D2T+

KT− δ2

κ+

∫[t,T ]

ηsD2sds

](23)

with (ηs) as in Assumption 5.1 i). The right-hand side is strictly convex in the process (Ds)s∈[t,T ].Thus, for two different strategies Θ′,Θ′′ ∈ At with correspondingD′, D′′ both starting inD′t = D′′t = δ,we have D(νΘ′ + (1− ν)Θ′′) = νD′ + (1− ν)D′′, hence J(νΘ′ + (1− ν)Θ′′) < νJ(Θ′) + (1− ν)J(Θ′′)for all ν ∈ (0, 1) as desired. Hence, we only need to show (23).

Define the local martingale Ms :=∫

[t,s∧T ]D2uσ(u,Ku)

2K2u

dWKu for s ∈ [t,∞). That is, τn = s ≥ t |

〈M〉s ≥ n is an increasing sequence of stopping times such that τn ∞ a.s. and the stopped processMτn is a martingale for every n. In particular, Et,δ,κ[MT∧τn ] = 0. Due to the monotone convergencetheorem and τn ≥ T a.s. for large n,

J(Θ) = limn→∞

Et,δ,κ

[∫[t,T∧τn]

(Ds +

Ks

2∆Θs

)dΘs

]. (24)

22

Using dΘs = dDs+ρsDsdsKs

and ∆Θs = ∆DsKs

, we get

J(Θ) = limn→∞

Et,δ,κ

[∫[t,T∧τn]

Ds + 12∆Ds

KsdDs +

∫[t,T∧τn]

ρsD2s

Ksds+

∫[t,T∧τn]

12∆DsρsDs

Ksds

].

The last integral is zero because D has at most countably many jumps. With integration by parts forcàglàd processes,∫

[t,T∧τn]

Ds

KsdDs =

D2(T∧τn)+

K(T∧τn)− δ2

κ−∫

[t,T∧τn]

Dsd

(D

K

)s

−∑

s∈[t,T∧τn]

(∆Ds)2

Ks.

Use d(DK

)s

= 1KsdDs +Dsd

(1Ks

)and rearrange terms to get

∫[t,T∧τn]

Ds

KsdDs =

1

2

D2(T∧τn)+

K(T∧τn)− δ2

κ−∫

[t,T∧τn]

D2sd

(1

Ks

)−

∑s∈[t,T∧τn]

(∆Ds)2

Ks

.

Applying Itô’s formula

d

(1

Ks

)=

(σ2(s,Ks)

K3s

− µ(s,Ks)

K2s

)ds− σ(s,Ks)

K2s

dWKs

yields ∫[t,T∧τn]

(Ds +

Ks

2∆Θs

)dΘs =

1

2

[D2

(T∧τn)+

KT∧τn− δ2

κ+

∫[t,T∧τn]

ηsD2sds+MT∧τn

]. (25)

Lebesgue’s dominated convergence theorem together with Assumption 5.1 ii) guarantee

Et,δ,κ

[D2

(T∧τn)+

KT∧τn

]−−−−→n→∞

Et,δ,κ[D2T+

KT

], (26)

while, by the monotone convergence theorem, we have

Et,δ,κ

[∫[t,T∧τn]

ηsD2sds

]−−−−→n→∞

Et,δ,κ

[∫[t,T ]

ηsD2sds

]. (27)

Now (24) and (25) together with (26)–(27) complete the proof.

Proof of Proposition 5.4. Thanks to Lemma 5.3, we only need to prove existence. Let t, δ and κ befixed. We start by showing that there exists a sequence of strategies

(Θn)⊂ At(x) that converges in

some sense to a strategy Θ∗ ∈ At(x) and minimizes the cost J , i.e., limn→∞ J(

Θn)

= infΘ∈At(x) J(Θ).

We conclude by deducing that limn→∞ J(Θn) = J(Θ∗).

Let(Θj)⊂ At(x) be a minimizing sequence for J . Due to the Komlós theorem in the form of

Lemma 3.5 from Kabanov (1999), there exists a Cesaro convergent subsequence (Θjm). That is,

Θn

:=1

n

n∑m=1

Θjm

converges to some strategy Θ∗ ∈ At in the following sense. For Pt,κ-almost every ω, the measuresΘn(ω) on [t, T ] converge weakly to the measure Θ∗(ω). In what follows we call such a convergence

23

pathwise weak convergence in time. Equivalently, for almost every ω, we have limn→∞Θn

s = Θ∗swhenever s ∈ [t, T ] with ∆Θ∗s = 0. We set Θ∗T+ = x redefining Θ∗T+ if necessary. Notice that this does

not disturb the weak convergence. Thus, Θ∗ ∈ At(x). Moreover,(

Θn)⊂ At(x) is again a minimizing

sequence for J because J is convex.

It remains to show that Θ∗ attains the infimum. Applying (23) yields

J(

Θn)

=1

2Et,δ,κ

[(DnT+

)2KT

− δ2

κ+

∫[t,T ]

ηs (Dns )

2ds

], (28)

J (Θ∗) =1

2Et,δ,κ

[(D∗T+

)2KT

− δ2

κ+

∫[t,T ]

ηs (D∗s)2ds

], (29)

where Dn and D∗ are the price deviation processes that correspond to Θnand Θ∗. By the (pathwise

weak in time) convergence of Θnto Θ∗, for almost every ω, we get limn→∞Dn

s = D∗s for every points ∈ [t, T ], where Θ∗ is continuous, as well as for s = T+.9 Fatou’s lemma and (28)–(29) now implyJ (Θ∗) ≤ lim infn→∞ J

(Θn), which means that Θ∗ is an optimal strategy.

Proposition B.1. (WR-BR structure is equivalent to trading towards the barrier).Assume that for each (t, δ, x, κ) there exists a unique optimal strategy

(Θ∗s(t, δ, x, κ))s∈[t,T+] ∈ At(x).

Then the following statements are equivalent.

(a) The value function has WR-BR structure.

(b) There exists c : [0, T )× (0,∞)→ (0,∞] such that for all (t, δ, x, κ)

∆Θ∗t (t, δ, x, κ) = max

0,x− c(t, κ)δ

1 + κc(t, κ)

. (30)

In particular, ∆Θ∗t (t, δ, x, κ) is continuous in δ and x.

(c) For all (t, δ, κ), the function x 7→ ∆Θ∗t (t, δ, x, κ) is increasing.

Furthermore, if these equivalent statements hold, then the function c in (b) coincides with the barrierc of Definition 4.3.

Proof. First we prove the equivalence of (a) and (b). Statement (c) follows immediately from (b). Weconclude by showing that (c) implies (b). The scaling property (Lemma 4.1) yields

∆Θ∗t (t, δ, x, κ) = δ∆Θ∗t

(t, 1,

x

δ, κ).

Therefore, we only need to discuss the case δ = 1. Fix arbitrary t ∈ [0, T ], κ ∈ (0,∞).

(a) ⇒ (b) The assertion holds for x = 0. Assume x ∈ (0, c(t, κ)], where c is the barrier in Defini-tion 4.3. Then the WR-BR structure implies that for all ξ ∈ (0, x)

U(t, 1, x, κ) < U (t, 1 + κξ, x− ξ, κ) +(

1 +κ

2ξ)ξ.

Therefore it cannot be optimal to trade immediately at time t.9See also Lemma 7.3 of Fruth, Schöneborn, and Urusov (2014).

24

Assume c(t, κ) < ∞ and x ∈ (c(t, κ),∞). Then the WR-BR structure implies that thereexists ξ ∈ (0, x) such that

U(t, 1, x, κ) = U(t, 1 + κξ, x− ξ, κ

)+(

1 +κ

2ξ)ξ.

Due to the uniqueness of the optimal strategy, we get

∆Θ∗t (t, 1, x, κ) = ξ + ∆Θ∗t

(t, 1 + κξ, x− ξ, κ

)> 0.

For ξ < x−c(t,κ)1+κc(t,κ) , we have x−ξ

1+κξ> c(t, κ) and thus

∆Θ∗t

(t, 1 + κξ, x− ξ, κ

)> 0.

Consequently, ∆Θ∗t (t, 1, x, κ) ≥ x−c(t,κ)1+κc(t,κ) . Two trades executed immediately after each other

have the same effect as one trade of their combined size (see Lemma A.1). Due to this splittingargument, we have

∆Θ∗t (t, 1, x, κ) =x− c(t, κ)

1 + κc(t, κ)+ ∆Θ∗t

(t, 1 + κ

x− c(t, κ)

1 + κc(t, κ), x− x− c(t, κ)

1 + κc(t, κ), κ

).

Observe that the second summand equals zero because

x− x−c(t,κ)1+κc(t,κ)

1 + κ x−c(t,κ)1+κc(t,κ)

= c(t, κ).

Thus, we proved (b) with c(t, κ) = c(t, κ).

(b) ⇒ (a) Assume x ∈ (0, c(t, κ)]. Then (30) implies ∆Θ∗t (t, 1, x, κ) = 0. Together with the unique-ness of the optimal strategy we can therefore conclude that x /∈ Brt,κ, as, for all ξ ∈ (0, x),

U(t, 1, x, κ) < U (t, 1 + κξ, x− ξ, κ) +(

1 +κ

2ξ)ξ.

Assume c(t, κ) <∞ and x ∈ (c(t, κ),∞). Then (30) implies

∆Θ∗t (t, 1, x, κ) ∈ (0, x).

The optimality of Θ∗ leads to the conclusion x ∈ Brt,κ because

U(t, 1, x, κ) = U (t, 1 + κ∆Θ∗t (t, 1, x, κ), x−∆Θ∗t (t, 1, x, κ), κ)

+(

1 +κ

2∆Θ∗t (t, 1, x, κ)

)∆Θ∗t (t, 1, x, κ).

We thus established that Brt,κ = (c(t, κ),∞), i.e., we have WR-BR structure with the barrierc(t, κ) = c(t, κ).

(c) ⇒ (b) Definec(t, κ) := inf x ∈ (0,∞)|∆Θ∗t (t, 1, x, κ) > 0 ,

where inf ∅ := ∞. We are done for c(t, κ) = ∞. Let c(t, κ) < ∞. Then the definition of c(t, κ)guarantees ∆Θ∗t (t, 1, x, κ) = 0 for all x < c(t, κ), and Property (c) implies ∆Θ∗t (t, 1, x, κ) > 0for all x > c(t, κ). Suppose for a contradiction that

∆Θ∗t (t, 1, c(t, κ), κ) > 0.

Due to the uniqueness and the splitting argument, we then have, for ε ∈ (0,∆Θ∗t (t, 1, c(t, κ), κ)),

∆Θ∗t (t, 1, c(t, κ), κ) = ε+ ∆Θ∗t (t, 1 + κε, c(t, κ)− ε, κ) = ε < ∆Θ∗t (t, 1, c(t, κ), κ) .

25

Therefore, ∆Θ∗t (t, 1, x, κ) = 0 for all x ≤ c(t, κ).

We still need to prove ∆Θ∗t (t, 1, x, κ) = x−c(t,κ)1+κc(t,κ) for x > c(t, κ). Let us first assume that

∆Θ∗t (t, 1, x, κ) > x−c(t,κ)1+κc(t,κ) . Once more, we make use of the uniqueness and the splitting argument

in order to get a contradiction

∆Θ∗t (t, 1, x, κ) =x− c(t, κ)

1 + κc(t, κ)+ ∆Θ∗t

(t, 1 + κ

x− c(t, κ)

1 + κc(t, κ), x− x− c(t, κ)

1 + κc(t, κ), κ

)=

x− c(t, κ)

1 + κc(t, κ)< ∆Θ∗t (t, 1, x, κ).

Finally, assume ∆Θ∗t (t, 1, x, κ) < x−c(t,κ)1+κc(t,κ) . That is,

x−∆Θ∗t (t,1,x,κ)1+κ∆Θ∗t (t,1,x,κ) > c(t, κ) and we again arrive

at a contradiction:

∆Θ∗t (t, 1, x, κ) = ∆Θ∗t (t, 1, x, κ) + ∆Θ∗t

(t, 1 + κ∆Θ∗t (t, 1, x, κ), x−∆Θ∗t (t, 1, x, κ), κ

)> ∆Θ∗t (t, 1, x, κ).

This concludes the proof.

Proof of Proposition 5.5. According to Propositions 5.4 and B.1, we only need to show that the op-timal initial trade ∆Θ∗tn (tn, δ, x, κ) is increasing in x, where Θ∗ denotes the corresponding optimalstrategy. Due to the scaling property of the value function (Lemma 4.1),

∆Θ∗tn (tn, δ, x, κ) = δ∆Θ∗tn

(tn, 1,

x

δ, κ).

Due to the splitting argument (Lemma A.1) and the uniqueness of the optimal strategy, ∆Θ∗tn (tn, 1, ·, κ)must be increasing and continuous apart from a possible discontinuity in the form of a jump back tozero. That is there might exist y > 0 with ∆Θ∗tn (tn, 1, y−, κ) > 0 and ∆Θ∗tn (tn, 1, y+, κ) = 0. In thefollowing, we exclude such discontinuities using a Komlós argument as in the proof of Proposition 5.4.

Suppose for a contradiction that such a discontinuity exists in y > 0. Let us take some monotonesequences y1,j y and y2,j y and define Θi,j := Θ∗(tn, 1, y

i,j , κ) for i ∈ 1, 2. Let us choose ε > 0such that ∆Θ1,j

tn ≥ ε > 0 for all sufficiently large j. Without loss of generality we assume that thelatter inequality holds for all j. As V dis is continuous in y (see Proposition A.2),

J(tn, 1,Θ

1,j , κ)

= V dis(tn, y

1,j , κ)−−−→j→∞

V dis (tn, y, κ) .

Define bj := yy1,j 1. Then we have

0 ≤ J(tn, 1, bjΘ

1,j , κ)− J

(tn, 1,Θ

1,j , κ)

≤ J(tn, bj , bjΘ

1,j , κ)− J

(tn, 1,Θ

1,j , κ)

= (b2j − 1)J(tn, 1,Θ

1,j , κ)−−−→j→∞

0.

Therefore,(bjΘ

1,j)is a minimizing sequence of strategies that build up the position of y shares, i.e.,

bjΘ1,j ∈ Adistn (y) and

limj→∞

J(tn, 1, bjΘ

1,j , κ)

= V dis (tn, y, κ) .

As in the proof of Proposition 5.4, we can define Θ ∈ Adistn (y) as the pathwise weak in time limit ofthe averaged sum over a subsequence of

(bjΘ

1,j)such that J(tn, 1,Θ, κ) = V dis(tn, y, κ), i.e., Θ is an

optimal strategy. Due to the construction of Θ, with ε > 0 from above, we have

∆Θtn (tn, 1, y, κ) ≥ ε > 0.

26

Similarly, one constructs an optimal strategy Θ ∈ Adistn (y) using the sequence(

yy2,j Θ2,j

)of strategies

with zero initial trade: as we now treat the discrete time case, the initial trade remains zero also inthe weak limit

∆Θtn (tn, 1, y, κ) = 0.

Thus, Θ and Θ are different. This contradicts the uniqueness of the optimal strategy.

Lemma B.2. (Approximation via step functions).Let Assumption 5.1 ii)–iii) hold. For Θ ∈ Actst (x), let ΘN ∈ Adist (x) be its approximation from belowby an equidistant grid step function. More precisely, define T 0

t := t, T,

T N+1t := T Nt ∪

(s+

T − t2N+1

)∧ T | s ∈ T Nt

and

ΘNs :=

0 if s = tΘu+ if s ∈

(u, u+ T−t

2N

], u ∈ T Nt

x if s = T+

.

Then J (t, 1,Θ, κ) = limN→∞ J(t, 1,ΘN , κ

).

Proof. We proceed as in the end of the proof of Proposition 5.4. That is, we only need to showthat ΘN converges pathwise weakly in time to Θ. Due to T Nt ⊂ T N+1

t , ΘN is increasing in N . Forall s ∈ [t, T+], the sequence

(ΘNs

)N∈N is bounded above by Θs. Hence, it is convergent. Due to the

definition of ΘN , we must even have limN→∞ΘNs = Θs for all s ∈ [t, T ] with ∆Θs = 0. Now the result

follows from (23) and the dominated convergence theorem (apply Assumption 5.1 ii) and iii)).

Lemma B.3. (Cesaro weak convergence).Fix t ∈ [0, T ], κ ∈ (0,∞) and for various x ∈ [0,∞) consider(

ΘN (t, 1, x, κ))N∈N ⊂ A

ctst (x).

Then there exists a subsequence Nj(t, κ), which does not depend on x, and a set of strategies Θ(t, 1, ·, κ)such that for all x ∈ [0,∞) ∩Q

1

m

m∑j=1

ΘNj (t, 1, x, κ)w−−−−→

m→∞Θ (t, 1, x, κ) . (31)

In (31) the notation “ w−→” stands for the pathwise weak convergence in time (cf. the proof of Propo-sition 5.4).

Proof. As Q is countable, we can write [0,∞) ∩ Q = x1, x2, .... For each x ∈ [0,∞), the Komlóstheorem guarantees the existence of a subsequence Nj (t, x, κ) such that the desired pathwise weakconvergence in time holds. That is we get (N

(1)j )j∈N ⊂ N for x1 and extract the subsequenceN (2)

j for x2

from N(1)j , etc. We remark that the Komlós theorem gives not only Cesaro convergent subsequences,

but subsequences such that all their subsequences are Cesaro convergent to the same limit. TheCantor diagonal sequence Nj := N

(j)j then guarantees the Cesaro weak convergence of ΘNj (t, 1, x, κ)

for all x ∈ [0,∞) ∩Q.

Proof of Proposition 5.6. As in the proof of Proposition 5.5, we only need to exclude the jump backto zero of x 7→ ∆Θ∗t (t, 1, x, κ). Let ΘN ∈ Adist (x) be the approximation of Θ∗ ∈ Actst (x) by stepfunctions from below as in Lemma B.2. Then

J (t, 1,Θ∗, κ) = limN→∞

J(t, 1,ΘN , κ

).

27

Let Θ∗N be the unique optimal strategy within Adist (x) for the time grid T Nt , i.e.,

J(t, 1,ΘN , κ

)≥ J

(t, 1,Θ∗N , κ

)≥ J (t, 1,Θ∗, κ) .

Hence,J (t, 1,Θ∗, κ) = lim

N→∞J(t, 1,Θ∗N , κ

).

That is, for each x ∈ [0,∞), (Θ∗N (t, 1, x, κ))N∈N is a minimizing sequence, and for each N ∈ N, x 7→∆Θ∗Nt (t, 1, x, κ) is increasing thanks to Proposition 5.5.

Apply Lemma B.3 to Θ∗N (t, 1, x, κ) (for all rational x). The resulting strategy Θ(t, 1, x, κ) as in (31)is optimal (apply convexity of the cost function together with (23) and the dominated convergencetheorem). As the optimal strategy is unique, Θ(t, 1, x, κ) must coincide with Θ∗(t, 1, x, κ) for allx ∈ [0,∞) ∩Q. Furthermore, as we already proved WR-BR structure in discrete time, for all N ands ∈ [t, T ], the function x 7→ Θ∗Ns (t, 1, x, κ) is increasing. Due to the pathwise weak convergence asin (31), for all s ∈ [t, T ], the function x 7→ Θ∗s(t, 1, x, κ) is increasing over rational x. In particular,x 7→ ∆Θ∗t (t, 1, x, κ) ≡ Θ∗t+(t, 1, x, κ) is increasing over rational x. As we only need to exclude thedownward jump, it suffices to have this monotonicity over the rational numbers.

C Proofs for GBM and CIR cases with WR-BR structure

Proof of Proposition 6.2. i) We have ηt = 1Kt

(2ρ+ µ− σ2

)> 0.

ii) Set qt := 1Kt

. Thanks to Hölder’s inequality,

E0,κ

(

supt∈[0,T ]Kt

)2

inft∈[0,T ]Kt

≤ E0,κ

[supt∈[0,T ]

K4t

] 12

E0,κ

[supt∈[0,T ]

q2t

] 12

. (32)

The explicit formula for GBM, Kt = K0eσWK

t +(µ− σ2

2

)t, yields

E0,κ

[supt∈[0,T ]

K4t

]≤ κ4 max

1, e

4(µ− σ2

2

)TE0,κ

[exp

(4σ sup

t∈[0,T ]

WKt

)].

The latter expression is finite due to the fact that (supt∈[0,T ]WKt ) has the same distribution as |WK

T |,which is a consequence of the reflection principle for a Brownian motion. The second expectationin (32) is finite because qt = 1

Ktis also a GBM (with drift (σ2 − µ) and volatility σ).

iii) Due to the form of ηt, it is enough to consider

E0,κ

∫ T

0

(supt∈[0,T ]

Kt

)21

Ktdt

≤ T E0,κ

(

supt∈[0,T ]Kt

)2

inft∈[0,T ]Kt

,where the right-hand side is finite according to ii).

Proof of Proposition 6.3. Such a CIR process stays a.s. strictly positive, as the Feller condition µK ≥σ2/2 is met. Moreover, it turns out that ηt = 1

Kt(2ρ− µ) + 1

K2t

(µK− σ2) > 0 due to our assumptions.Conditions ii) and iii) both hold by showing

E0,κ

(

supt∈[0,T ]Kt

)2

(inft∈[0,T ]Kt

)2 <∞.

28

Thanks to Hölder’s inequality, with qt = 1Kt

, we have

E0,κ

(

supt∈[0,T ]Kt

)2

(inft∈[0,T ]Kt

)2 ≤ E0,κ

[supt∈[0,T ]

K8t

] 14

E0,κ

[supt∈[0,T ]

q83t

] 34

. (33)

As the drift of the CIR process is bounded above, we can isolate the local martingale part of K anduse the Burkholder-Davis-Gundy inequalities.10 With appropriate positive constants cn, we obtain

E0,κ

[supt∈[0,T ]

K8t

]≤ c1

κ8 +

(µKT

)8+ E0,κ

[supt∈[0,T ]

∣∣∣∣∫ t

0

σ√Ks dW

Ks

∣∣∣∣8]

(34)

≤ c2

κ8 +(µKT

)8+ E0,κ

(∫ T

0

σ2Ks ds

)4 .

The latter expectation is finite because all positive moments of the CIR process are finite (e.g., seeFilipovic and Mayerhofer (2009)).

It remains to show that the second term on the right-hand side of (33) is finite. By Itô’s formula, theprocess qt = 1

Kthas the dynamics

dqt =(µqt −

(µK − σ2

)q2t

)dt− σq

32t dW

Kt .

With these preparations, we proceed similarly to (34):

E0,κ

[supt∈[0,T ]

q83t

]≤ c3

κ− 83 +

(µ2T

4(µK − σ2

)) 83

+ E0,κ

[supt∈[0,T ]

∣∣∣∣∫ t

0

σq32s dW

Ks

∣∣∣∣83

]≤ c4

κ− 83 +

(µ2T

4(µK − σ2

)) 83

+ E0,κ

(∫ T

0

σ2q3s ds

) 43

.

We are done because E0,κ

[(∫ T

0q3s ds)

43

]≤ c5

∫ T0E0,κ[q4

s ] ds, and the fourth moment of the inverseCIR process is finite whenever µK > 2σ2 (e.g., see Ahn and Gao (1999) for an explicit calculation ofnegative moments of the CIR process).

D Proof of Proposition 7.2

The optimal strategy is determined by ξ0, ξA1 and ξB1 . As c(t1, κA1 ) = c(t1, κB1 ) = 1 =: c(t1) by

Proposition 7.1, we see that ξA1 > 0 if and only if ξB1 > 0.

Let us now consider a given trade ξ0 at time t0 and assume optimal trading thereafter. This resultsin a cost of

Udis(t0, δ, x, κ0; ξ0) :=(δ +

κ0

2ξ0

)ξ0 + E

[Udis(t1, (δ + κ0ξ0)a0, x− ξ0, κ1)

].

10For every m > 0, there exist universal positive constants km and Km such that

kmE[〈M〉mτ

]≤ E

[(maxt≤τ|Mt|

)2m]≤ KmE

[〈M〉mτ

]for every continuous local martingale M with M0 = 0 and every stopping time τ . E.g., see Karatzas and Shreve (2000),Chapter 3, Theorem 3.28.

29

It is easy to see that Udis is piecewise quadratic in ξ0. For the section of ξ0 where the optimal ξ1 ispositive (ξ1 > 0), a straightforward calculation shows that the quadratic coefficient is negative. Udistherefore cannot attain its minimum in the interior of this section; the optimal strategy thereforesatisfies ξ0 = 0 or ξ1 = 0.

Using Proposition 7.1, we easily calculate that for trading only at times t0 and t2, we have

cl := c0,2(t0, κ0) < a0 = c(t1)a0.

Hence (cl, a0] must be a subset of the buy region Brt0 . For y > a0, we need to compare the cost U0,2

of optimally trading only at times t0 and t2 with the cost U1,2 of optimally trading only at times t1and t2. Using the parameter values given in Figure 2, we find that the quadratic coefficient of U0,2

is larger than the quadratic coefficient of U1,2; therefore there must be an intersection point cu > clwhere U1,2 = U0,2. We then have for y ≤ cl that U1,2 = U0,2 and the optimal strategy trades neitherat t0 nor t1, for cl < y < cu that U0,2 < U1,2 and the unique optimal strategy trades at t0 but notat t1, for y = cu that U0,2 = U1,2 and there are two optimal strategies (one trading at t0 but not t1,and one trading at t1 but not t0), and for y > cu that U0,2 > U1,2 and the unique optimal strategytrades at t1 but not at t0.

E Proof of Theorem 8.4

We can extend the proof of (23) to the zero spread model and find that the cost function Jl satisfies

Jl(t, δ,Θl, κ) =1

2Et,δ,κ

[(DlT+)2

KT− δ2

κ+

∫[t,T ]

ηs(Dls)2 ds

](35)

with ηs ≡ η(s,Ks) as in Assumption 5.1 i). More precisely, instead of monotone convergencein (24) we need to use dominated convergence, which applies because Θ and Θ are bounded andEt,κ[sups∈[t,T ]Ks] < ∞ (the latter follows from Assumption 5.1 ii)), and again dominated conver-gence works in (26) (based on Assumption 5.1 ii)). As for (27), we use monotone convergence in thefirst case (η ≥ 0), while dominated convergence applies in the second case (due to Assumption 5.1 iii)).In particular, when we start at time t with Dlt = 0, we have

Jl(t, 0,Θl, κ) =1

2Et,0,κ

[(DlT+)2

KT+

∫[t,T ]

ηs(Dls)2 ds

],

which establishes the statement in the first case (η ≥ 0 everywhere).

Similarly to (35) we establish that, for any stopping time τ with t ≤ τ ≤ T , it holds that

JlT (t, δ,Θl, κ) =

1

2Et,δ,κ

[(Dlτ+)2

Kτ− δ2

κ+

∫[t,τ ]

ηs(Dls)2 ds

]+ Et,δ,κ

[∫(τ,T ]

(Dls +

Ks

2∆Θls

)dΘls

].

(36)We now make use of (36) to construct a profitable round trip strategy in the second case. Starting at(t, κ) with Dlt = 0, let us define the stopping time

τ := (t+ ∆t) ∧ inf s ≥ t | Ks /∈ (κ− ε, κ+ ε)

and consider the following trading strategy. First, buy x > 0 units of the asset at time t. This resultsin Dlt+ = κx. At time τ , we have Dlτ = κxe−

∫ τtρs ds and sell y =

DlτKτ

units of the asset, resulting in

Dlτ+ = 0. We do nothing in (τ, t + ∆t) and then liquidate the position x − y with a uniform speed

between t + ∆t and T . Notice that the position x − y is random (it depends on Kτ ), but bounded

30

(due to the construction of τ). Summarizing, we consider the following round trip strategy: Θlt = 0,

Θls = x for s ∈ (t, τ ], Θ

ls = x− y for s ∈ (τ, t+ ∆t],

Θls = x− y +s− t−∆t

T − t−∆t(y − x), s ∈ (t+ ∆t, T ],

and ΘlT+ = Θ

lT = 0. An application of (36) in this case yields

JlT (t, 0,Θl, κ) =

1

2Et,0,κ

[∫[t,τ ]

ηs(Dls)2 ds

]+ Et,0,κ

[∫(t+∆t,T ]

Dls dΘls

].

The first term on the right-hand side is strictly negative (we are considering the second case) anddoes not depend on T . Below we present a calculation showing that the second term goes to zero asT goes to infinity, which means that, for a sufficiently large T , we constructed a round trip strategywith strictly negative cost, i.e., with strictly positive profit.

Relying on Assumption (A) we finally show that

Et,0,κ

[∫(t+∆t,T ]

Dls dΘls

]−−−−→T→∞

0 (37)

for the strategy described above. Recall that Dlτ+ = 0, hence Dlt+∆t+ = 0. That is, for s ∈ (t+∆t, T ],we have

Dls =y − x

T − t−∆t

∫ s

t+∆t

Kue−∫ suρr dr du,

therefore,

Et,0,κ

[∫(t+∆t,T ]

Dls dΘls

]= Et,κ

[(y − x)2

(T − t−∆t)2

∫ T

t+∆t

∫ s

t+∆t

Kue−∫ suρr dr du ds

]

≤ const

(T − t−∆t)2Et,κ

[∫ T

t+∆t

∫ s

t+∆t

Kue−∫ suρr dr du ds

], (38)

where we used that the random variable (y − x)2 is bounded. Further,

Et,κ

[∫ T

t+∆t

∫ s

t+∆t

Kue−∫ suρr dr du ds

]=

∫ T

t+∆t

(∫ T

u

e−∫ suρr dr ds

)Et,κ[Ku] du

≤ 1

ρ

[∫ T

t+∆t

Et,κ[Ku] du

]≤ const (T − t−∆t).

Together with (38), we obtain (37). This completes the proof.

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