market impact in models of the order...

Price manipulation Exponential decay Power-law decay Order book models Linear transient impact

Market impact in models of the order book

Jim Gatheral(joint work with Alexander Schied and Alla Slynko)

Market Microstructure: Confronting Many Viewpoints,December 8, 2010


Price manipulation in a simple model of market impact

We begin by showing that the modeling of market impact isconstrained by requiring no-price-manipulation.

In particular, we write down a simple model of market impactin which:

If the decay of market impact is exponential, market impactmust be linear in quantity.If decay of market impact is power-law and sensitivity toquantity is also power-law, no-price-manipulation imposesinequality constraints on the exponents.

This simple model is a natural generalization of other modelsthat have previously appeared in the literature.


Model specification

We suppose that the stock price St at time t is given by

St = S0 +

∫ t0

f (ẋs) G (t − s) ds +∫ t

0σ dZs (1)

where ẋs is our rate of trading in dollars at time s < t, f (ẋs)represents the impact of trading at time s and G (t − s) is adecay factor.

St follows an arithmetic random walk with a drift thatdepends on the accumulated impacts of previous trades.

The cumulative impact of (others’) trading is implicitly in S0and the noise term.

Drift is ignored.


We refer to f (·) as the instantaneous market impact functionand to G (·) as the decay kernel.(1) is a generalization of processes previously considered byAlmgren, Bouchaud and Obizhaeva and Wang.

Remark

The price process (1) is not the only possible generalization ofprice processes considered previously. On the one hand, it seemslike a natural generalization. On the other hand, it is notmotivated by any underlying model of the order book.


Model as limit of discrete time process

The continuous time process (1) can be viewed as a limit of adiscrete time process (see Bouchaud et al. [5] for example):

St =∑i 0 represents a purchase and δxi < 0 represents a sale.δt could be thought of as 1/ν where ν is the trade frequency.Increasing the rate of trading ẋi is equivalent to increasing thequantity traded each δt.


Cost of trading

Denote the number of shares outstanding at time t by xt .Then from (1), the cost C [Π] associated with a given tradingstrategy Π = {xt} is given by

C [Π] =

∫ T0

ẋt dt

∫ t0

f (ẋs) G (t − s) ds (2)

The dxt = ẋt dt shares liquidated at time t are traded onaverage at a price

St = S0 +

∫ t0

f (ẋs) G (t − s) ds

which reflects the residual cumulative impact of all priortrading.


The principle of No Price Manipulation

A trading strategy Π = {xt} is a round-trip trade if∫ T0

ẋt dt = 0

We define a price manipulation to be a round-trip trade Π whoseexpected cost C [Π] is negative.

The principle of no-price-manipulation

Price manipulation is not possible.

Remark

If price manipulation were possible, the optimal strategy would notexist.


A specific strategy

Consider a strategy where shares are accumulated at the (positive)constant rate v1 and then liquidated again at the (positive)constant rate v2. According to equation (2), the cost of thisstrategy is given by C11 + C22 − C12 with

C11 = v1 f (v1)

∫ θT0

dt

∫ t0

G (t − s) ds

C22 = v2 f (v2)

∫ TθT

dt

∫ tθT

G (t − s) ds

C12 = v2 f (v1)

∫ TθT

dt

∫ θT0

G (t − s) ds (3)

where θ is such that v1 θT − v2 (T − θT ) = 0 so

θ =v2

v1 + v2


Special case: Trade in and out at the same rate

One might ask what happens if we trade into, then out of aposition at the same rate v . If G (·) is strictly decreasing,

C [Π] = v f (v)

{∫ T/20

dt

∫ t0

G (t − s) ds +∫ T

T/2

dt

∫ tT/2

G (t − s) ds

−∫ T

T/2

dt

∫ T/20

G (t − s) ds

}

= v f (v)

{∫ T/20

dt

∫ t0

[G (t − s)− G (t + T/2− s)] ds

+

∫ TT/2

dt

∫ tT/2

[G (t − s)− G (T − s)] ds

}> 0

We conclude that if price manipulation is possible, it mustinvolve trading in and out at different rates.


Exponential decay

Suppose that the decay kernel has the form

G (τ) = e−ρ τ

Then, explicit computation of all the integrals in (3) gives

C11 = v1 f (v1)1

ρ2

{e−ρ θT − 1 + ρ θT

}C12 = v2 f (v1)

1

ρ2

{1 + e−ρT − e−ρ θT − e−ρ (1−θ) T

}C22 = v2 f (v2)

1

ρ2

{e−ρ (1−θ) T − 1 + ρ (1− θ) T

}(4)

We see in particular that the no-price-manipulation principle forcesa relationship between the instantaneous impact function f (·) andthe decay kernel G (·).


Exponential decay

After making the substitution θ = v2/(v1 + v2) and imposing theprinciple of no-price-manipulation, we obtain

v1 f (v1)

[e− v2 ρ

v1+v2 − 1 + v2 ρv1 + v2

]+v2 f (v2)

[e− v1 ρ

v1+v2 − 1 + v1 ρv1 + v2

]−v2 f (v1)

[1 + e−ρ − e−

v1 ρv1+v2 − e−

v2 ρv1+v2

]≥ 0 (5)

where, without loss of generality, we have set T = 1. We note thatthe first two terms are always positive so price manipulation is onlypossible if the third term (C12) dominates the others.


Example: f (v) =√

v

Let v1 = 0.2, v2 = 1, ρ = 1. Then the cost of liquidation is givenby

C = C11 + C22 − C12 = −0.001705 < 0

Since ρ really represents the product ρT , we see that for anychoice of ρ, we can find a combination {v1, v2,T} such that around trip with no net purchase or sale of stock is profitable. Weconclude that if market impact decays exponentially, theno-price-manipulation principle excludes a square rootinstantaneous impact function.

Can we generalize this?


Expansion in ρ

Expanding expression (5) in powers of ρ, we obtain

v1 v2 [v1 f (v2)− v2 f (v1)] ρ2

2(v1 + v2)2+ O

(ρ3)≥ 0

We see that price manipulation is always possible for small ρ unlessf (v) is linear in v and we have

Lemma

Under exponential decay of market impact, no-price-manipulationimplies f (v) ∝ v.

Remark

Alla Slynko has shown that this extends to any model of the form(1) in which G (0)


Empirical viability of exponential decay of market impact

Empirically, market impact is concave in v for small v .

Also, market impact must be convex for very large v

Imagine submitting a sell order for 1 million shares when thereare bids for only 100,000.

We conclude that the principle of no-price-manipulationexcludes exponential decay of market impact for anyempirically reasonable instantaneous market impact functionf (·).


Power-law decay

Suppose now that the decay kernel has the form

G (t − s) = 1(t − s)γ

, 0 < γ < 1

Then, explicit computation of all the integrals in (3) gives

C11 = v1 f (v1)T 2−γ

(1− γ) (2− γ)θ2−γ

C22 = v2 f (v2)T 2−γ

(1− γ) (2− γ)(1− θ)2−γ

C12 = v2 f (v1)T 2−γ

(1− γ) (2− γ){

1− θ2−γ − (1− θ)2−γ}

(6)


Power-law decay

According to the principle of no-price-manipulation , substitutingθ = v2/(v1 + v2), we must have

f (v1){

v1 v21−γ − (v1 + v2)2−γ + v12−γ + v22−γ

}+f (v2) v1

2−γ ≥ 0(7)

If γ = 0, the no-price-manipulation condition (7) reduces to

f (v2) v1 − f (v1) v2 ≥ 0

so again, permanent impact must be linear.

If γ = 1, equation (7) reduces to

f (v1) + f (v2) ≥ 0

So long as f (·) ≥ 0, there is no constraint on f (·) when γ = 1.


The limit v1 � v2 and 0 < γ < 1

In this limit, we accumulate stock much more slowly than weliquidate it. Let v1 = � v and v2 = v with �� 1. Then, in thelimit �→ 0, with 0 < γ < 1, equation (7) becomes

f (� v){�− (1 + �)2−γ + �2−γ + 1

}+ f (v) �2−γ

∼ −f (� v) (1− γ) �+ f (v) �2−γ ≥ 0

so for � sufficiently small we have

f (� v)

f (v)≤ �

1−γ

1− γ(8)

If the condition (8) is not satisfied, price manipulation is possibleby accumulating stock slowly, maximally splitting the trade, thenliquidating it rapidly.


Power-law impact: f (v) ∝ v δ

If f (v) ∼ v δ (as per Almgren et al. [1]), the no-price-manipulationcondition (8) reduces to

�1−γ−δ ≥ 1− γ

and we obtain

Small v no-price-manipulation condition

γ + δ ≥ 1


Cost of VWAP with power-law market impact and decay

From equation (6), the cost of an interval VWAP execution withduration T is proportional to

C = v f (v) T 2−γ

Noting that v = n/T , and putting f (v) ∝ (v/V )δ, the cost pershare is proportional to ( n

V

)δT 1−γ−δ

If γ + δ = 1, the cost per share is independent of T and inparticular, if γ = δ = 1/2, the impact cost per share is proportionalto√

n/V , which is the well-known square-root formula for marketimpact as described by, for example, Grinold and Kahn [10].


The square-root formula, γ and δ

The square-root market impact formula has been widely usedin practice for many years.

If correct, this formula implies that the cost of liquidating astock is independent of the time taken.

Fixing market volume and volatility, impact depends only size.

We can check this prediction empirically.

See for example Engle, Ferstenberg and Russell [7].

Also, according to Almgren, δ ≈ 0.6 and according toBouchaud γ ≈ 0.4.

Empirical observation

δ + γ ≈ 1!


The shape of the order book

Bouchaud, Mézard and Potters [6] derive the followingapproximation to the average density ρ(∆̂) of orders as a functionof a rescaled distance ∆̂ from the price level at which the order isplaced to the current price:

ρ(∆̂) = e−∆̂∫ ∆̂

0du

sinh(u)

u1+µ+ sinh(∆̂)

∫ ∞∆̂

due−u

u1+µ(9)

where µ is the exponent in the empirical power-law distribution ofnew limit orders.


Approximate order density

The red line is a plot of the order density ρ(∆̂) with µ = 0.6 (asestimated by Bouchaud, Mézard and Potters).

0 2 4 6 8 10

0.0

0.5

1.0

1.5

!̂

"(!̂)


Virtual price impact

Switching x− and y−axes in a plot of the cumulative order density givesthe virtual impact function plotted below. The red line corresponds toµ = 0.6 as before.

02

46

810

Quantity: !"0

#̂$(u)du

Price

impa

ct: #̂


Impact for high trading rates

You can’t trade more than the total depth of the book soprice impact increases without limit as n→ nmax .For a sufficiently large trading rate v , it can be shown that

f (v) ∼ 1(1− v/vmax)1/µ

Setting v = vmax (1− �) and taking the limit �→ 0,

f (v) ∼ 1�1/µ

as �→ 0.

Imagine we accumulate stock at a rate close to vmax := 1 andliquidate at some (lower) rate v .

This is the pump and dump strategy!


Impact for high trading rates continued

Substituting into condition (7) gives

1

�1/µ

{(1− �) v 1−γ − (1− �+ v)2−γ + (1− �)2−γ + v 2−γ

}+f (v) (1− �)2−γ ≥ 0

No-price-manipulation imposes that

h(v , γ) := v 1−γ − (1 + v)2−γ + 1 + v 2−γ ≥ 0


Graphical illustration

0.0 0.2 0.4 0.6 0.8 1.0

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

v

h(v,

γ)

γ = 2 − log(3) log(2)

γ = 0.5

γ = 0.3


Large size no price manipulation condition

From the picture, we see that h(v , γ) ≥ 0 implies

Large size no price manipulation condition

γ > γ∗ = 2− log 3log 2


Long memory of order flow

It is empirically well-established that order-flow is a longmemory process.

More precisely, the autocorrelation function of order signsdecays as a power-law.There is evidence that this autocorrelation results from ordersplitting.

Imposing linear growth of return variance in trading time(Bouchaud et al. [5]) in an effective model such as (1) forcespower-law decay of market impact with exponent γ ≤ 1/2.


Summary of prior work

By imposing the principle of no-price-manipulation, we showedthat if market impact decays as a power-law 1/(t − s)γ andthe instantaneous market impact function is of the formf (v) ∝ v δ, we must have

γ + δ ≥ 1

We excluded the combination of exponential decay withnonlinear market impact.


We observed that if the average cost of a (not-too-large)VWAP execution is roughly independent of duration, theexponent δ of the power law of market impact should satisfy:

δ + γ ≈ 1

By considering the tails of the limit-order book, we deducethat

γ ≥ γ∗ := 2− log 3log 2

≈ 0.415

Long memory of order flow imposes γ ≤ 1/2.


Schematic presentation of constraints

0.0 0.5 1.0 1.5

0.0

0.5

1.0

1.5

!

"

0.0 0.5 1.0 1.5

0.0

0.5

1.0

1.5

! + " # 1

! # !*

! $ 1 2

! = 0.4, " = 0.6! = 0.5, " = 0.5


The model of Alfonsi, Fruth and Schied

Alfonsi, Fruth and Schied [2] consider the following (AS) model ofthe order book:

There is a continuous (in general nonlinear) density of ordersf (x) above some martingale ask price At . The cumulativedensity of orders up to price level x is given by

F (x) :=

∫ x0

f (y) dy

Executions eat into the order book (i.e. executions are withmarket orders).

A purchase of ξ shares at time t causes the ask price toincrease from At + Dt to At + Dt+ with

ξ =

∫ Dt+Dt

f (x) dx = F (Dt+)− F (Dt)


We can define a volume impact process

Et := F (Dt)

which represents how much of the book has been eaten up byexecutions up to time t.

Depending on the model, either the spread Dt or the volumeimpact process Et revert exponentially at some rate ρ.

This captures the resiliency of the order book: Limit ordersarrive to replenish order density lost through executions.


Schematic of the model

Order density f(x)

f(Dt)

f(Dt+)

Et+ −Et

0 Dt Dt+

Et

Price level

When a trade of size ξ is placed at time t,

Et 7→ Et+ = Et + ξDt = F

−1(Et) 7→ Dt+ = F−1(Et+) = F−1(Et + ξ)


Example: The model of Obizhaeva and Wang

Obizhaeva and Wang [11] consider a block-shaped order book withconstant order density.

Order density f(x)f(Dt) f(Dt+)

Et+ −Et

0 Dt Dt+

Et

Price level


In the Obizhaeva Wang (OW) model, we thus have

f (x) = 1

F (x) = x

Et = Dt

∆Dt := Dt+ − Dt = ξ

Thus market impact ∆Dt is linear in the quantity ξ. Betweenexecutions, the spread Dt and the volume impact process Et bothdecay exponentially at rate ρ so that at time t after an executionat the earlier time s, we have

Dt = Ds+ e−ρ (t−s)


Cost of execution and optimal trading strategy

Given a trading strategy Π with trading at the rate ẋt , the cost ofexecution in the OW model is given by

C(Π) =∫ T

0ẋt dt

∫ t0

ẋs e−ρ (t−s) ds

When the trading policy ẋt = ut is statically optimal, theEuler-Lagrange equation applies:

∂

∂t

δCδut

= 0

Then, for some constant λ, we have the generalized Fredholmintegral equation

δCδut

=

∫ t0

us e−ρ (t−s) ds +

∫ Tt

us e−ρ (s−t) ds = λ (10)


Substitutingut = δ(t) + ρ+ δ(T − t)

where δ(·) is the Dirac delta function into (10) gives

δCδut

=

∫ t0

us e−ρ (t−s) ds +

∫ Tt

us e−ρ (s−t) ds

= e−ρ t +

∫ t0ρ e−ρ (t−s) ds +

∫ Tt

ρ e−ρ (s−t) ds + e−ρ (T−t)

= 2

so ut is the optimal strategy.

The optimal strategy

Trade blocks of stock at times t = 0 and t = T and tradecontinuously at rate ρ between these two times.


No price manipulation

The optimal strategy involves only purchases of stock, nosales.

Thus there cannot be price manipulation in the OW model.

The OW price process is a special case of (1) with linear priceimpact and exponential decay of market impact.

Consistent with our lemma, there is no price manipulation.

The OW model is also a special case of the AS model.


Optimal strategy in the AS model

The optimal strategy

Trade blocks of stock at times t = 0 and t = T and tradecontinuously between these two times.

The optimal size ξ0 of the initial block purchase satisfies

F−1(X − ξ0 ρT ) = F−1(ξ0) + F−1′(ξ0) ξ0

The optimal continuous trading rate is ρ ξ0 and the optimal size ofthe final block is just

ξT = X −

(ξ0 +

∫ T−0+

ρ ξ0 dt

)= X − ξ0 (1 + ρT )


No price manipulation

Once again, the optimal strategy involves only purchases ofstock, no sales.

Thus there cannot be price manipulation in the AS model.

What’s going on?

“Under exponential decay of market impact, no-price-manipulationimplies f (v) ∝ v”


A potential conundrum

From Alfonsi and Schied [3]

“Our result on the non-existence of profitable price manipulationstrategies strongly contrasts Gatheral’s conclusion that thewidely-assumed exponential decay of market impact is compatibleonly with linear market impact.”

“The preceding corollary shows that, in our Model 1, exponentialresilience of price impact is well compatible with nonlinear impact...This fact is in stark contrast to Gatheral’s observation that, in arelated but different continuous-time model, exponential decay ofprice impact gives rise to price manipulation as soon as priceimpact is nonlinear. ”


Expected price in the two models

Recall that under the price process (1),

E[St ] =∫ t

0f (ẋs) G (t − s) ds

In the AS model, the current spread Dt and the volume impactprocess Et are related as

Dt = F−1(Et)

so effectively, for continuous trading strategies,

E[St ] = F−1(∫ t

0ẋs e−ρ (t−s) ds

)


Cost of trading

The expected cost of trading in the two models is then given by

CJG =∫ T

0ẋt dt

∫ t0

f (ẋs) G (t − s) ds

and

CAS =∫ T

0ẋt dt F

−1(∫ t

0ẋs e−ρ (t−s) ds

)These two expression are identical in the OW case with

F (x) = x ; f (v) = v ; G (t − s) = e−ρ (t−s)

For more general F (·), the models are different.


VWAP equivalent models

Are there choices of F (·) for which we can match the expectedcost of a VWAP in the two models?

For a VWAP execution, we have ẋt = v , a constant. Then

CJG = v f (v)∫ T

0dt

∫ t0

G (t − s) ds

and

CAS = v∫ T

0dt F−1

(v

∫ t0

e−ρ (t−s) ds

)In [10], we show that CJG = CAS for all v if and only ifF−1(x) = xδ, f (v) = v δ and

G (τ) =δ ρ e−ρ τ

(1− e−ρ τ )1−δ


Thus for small τ ,

G (τ) ∼ δ ρ(ρ τ)γ

with γ = 1− δ and for large τ ,

G (τ) ∼ δ ρ e−ρ τ

Resolution

With power-law market impact ∝ v δ, and exponential resilience,decay of market impact is power-law with exponent γ = 1− δ.

Remark

The relationship δ + γ = 1 between the exponents emergesnaturally from a simple model of the order book!


Resolution

There is no contradiction.

Exponential resilience of an order book with power-law shapeinduces power-law decay of market impact (at least for shorttimes).

Both AS-style models and processes like (1) aregeneralizations of Obizhaeva and Wang’s model.

The AS models are motivated by considerations of dynamics ofthe order book.The class of AS models with exponential resilience has beenshown to be free of price manipulation.

Our results suggest that the empirically observed relationshipδ + γ ≈ 1 may reflect properties of the order book rather thansome self-organizing principle.


We now proceed to characterize decay kernels compatible withlinear instantaneous market impact.

The general case is left for future research.


Linear transient market impact

The price process assumed in [8] is

St = S0 +

∫ t0

h(vs) G (t − s) ds + noise

In [9], this model is on the one hand extended to explicitly includediscrete optimal strategies and on the other hand restricted to thecase of linear market impact. When the admissible strategy X isused, the price St is given by

St = S0t +

∫{s


Transaction-triggered price manipulation

Definition (Alfonsi, Schied, Slynko (2009))

A market impact model admits transaction-triggered pricemanipulation if the expected costs of a sell (buy) program can bedecreased by intermediate buy (sell) trades.

As discussed in [4], transaction-triggered price manipulation can beregarded as an additional model irregularity that should beexcluded. Transaction-triggered price manipulation can exist inmodels that do not admit standard price manipulation in the senseof Huberman and Stanzl definition.


First order condition

Theorem

Suppose that G is positive definite. Then X ∗ minimizes C(·) if andonly if there is a constant λ such that X ∗ solves the generalizedFredholm integral equation∫

G (|t − s|) dX ∗s = λ for all t ∈ T. (13)

In this case, C(X ∗) = 12 λ x. In particular, λ must be nonzero assoon as G is strictly positive definite and x 6= 0.


Condition for no transaction-triggered price manipulation

Theorem

Suppose that the decay kernel G (·) is convex, satisfies∫ 10 G (t) dt


An instructive example

We solve a discretized version of the Fredholm equation (with 512time points) for two similar decay kernels:

G1(τ) =1

(1 + t)2; G2(τ) =

1

1 + t2

0.2

0.4

0.6

0.8

1.0

G1(τ)

0.2

0.4

0.6

0.8

1.0

G2(τ)


G1(·) is convex, but G2(·) is concave near τ = 0 so there should bea unique optimal strategy with G1(·) as a choice of kernel butthere should be transaction-triggered price manipulation with G2(·)as the choice of decay kernel.


Schematic of numerical solutions of Fredholm equation

G1(τ) =1

(1+t)2G2(τ) =

11+t2

0.0 0.2 0.4 0.6 0.8 1.0

2

4

6

8

10

0.2 0.4 0.6 0.8 1.0

-6 ´ 107

-4 ´ 107

-2 ´ 107

2 ´ 107

In the left hand figure, we observe block trades at t = 0 and t = 1with continuous (nonconstant) trading in (0, 1). In the right handfigure, we see numerical evidence that the optimal strategy doesnot exist.


Now we give some examples of the optimal strategy with choicesof kernel that preclude transaction-triggered price manipulation.


Example I: Linear market impact with exponential decay

G (τ) = e−ρ τ and the optimal strategy u(s) solves∫ T0

u(s)e−ρ |t−s| ds = const.

We already derived the solution which is

u(s) = A {δ(t) + ρ+ δ(T − t)}

The normalizing factor A is given by∫ T0

u(t) dt = X = A (2 + ρT )

The optimal strategy consists of block trades at t = 0 and t = Tand continuous trading at the constant rate ρ between these twotimes.


Schematic of optimal strategy

The optimal strategy with ρ = 0.1 and T = 1

0.0 0.2 0.4 0.6 0.8 1.0

Time s

u(s)


Example II: Linear market impact with power-law decay

G (τ) = τ−γ and the optimal strategy u(s) solves∫ T0

u(s)

|t − s|γds = const.

The solution is

u(s) =A

[s (T − s)](1−γ)/2

The normalizing factor A is given by

∫ T0

u(t) dt = X = A√π

(T

2

)γ Γ(1+γ2 )Γ(1 + γ2

)The optimal strategy is absolutely continuous with no block trades.However, it is singular at t = 0 and t = T .



The red line is a plot of the optimal strategy with T = 1 andγ = 1/2.

0.0 0.2 0.4 0.6 0.8 1.0

s

u(s)


Example III: Linear market impact with linear decay

G (τ) = (1− ρ τ)+ and the optimal strategy u(s) solves∫ T0

u(s) (1− ρ |t − s|)+ ds = const.

Let N := bρT c, the largest integer less than or equal to ρT . Then

u(s) = AN∑

i=0

(1− i

N + 1

) {δ

(s − i

ρ

)+ δ

(T − s − i

ρ

)}The normalizing factor A is given by∫ T

0u(t) dt = X = A

N∑i=0

2

(1− i

N + 1

)= A (2 + N)

The optimal strategy consists only of block trades with no tradingbetween blocks.



Positions and relative sizes of the block trades in the optimalstrategy with ρ = 1 and T = 5.2 (so N = bρT c = 5).

0 1 2 3 4 5

Time s

u(s)


Summary I

Optimal trade execution is of great importance in practice.

The optimal trading strategy depends on the model.

In Alfonsi-Schied style models with resiliency that depends onlyon the current spread, the minimal cost strategy is to trade ablock at inception, a block at completion and at a constantrate in between.We exhibited other models for which the optimal strategy ismore interesting.


Summary II

In some models, price manipulation is possible and there is nooptimal strategy.

It turns out that we also need to exclude transaction-triggeredprice manipulation.

We presented example of models for which price manipulationis possible.In the case of linear transient impact, we provided conditionsunder which transaction-triggered price manipulation isprecluded.

It remains for us to characterize optimal strategies in theempirically interesting case of nonlinear market impact.


References

[1] Robert Almgren, Chee Thum, Emmanuel Hauptmann and Hong Li

Equity market impactRisk Magazine July 57–62 (2005).

[2] Aurélien Alfonsi, Antje Fruth and Alexander Schied

Optimal execution strategies in limit order books with general shape functionsQuantitative Finance 10(2) 143–157 (2010).

[3] Aurélien Alfonsi and Alexander Schied.

Optimal execution and absence of price manipulations in limit order book models.SIAM Journal on Financial Mathematics 1 490–522 (2010).

[4] Aurélien Alfonsi, Alexander Schied and Alla Slynko

Order book resilience, price manipulation, and the positive portfolio problemhttp://papers.ssrn.com/sol3/papers.cfm?abstract id=1498514 (2009)

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[6] Jean-Philippe Bouchaud, Marc Mézard, and Marc Potters.

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[7] Robert F. Engle, Robert Ferstenberg, and Jeffrey Russell.

Measuring and modeling execution cost and risk.Technical report, University of Chicago Graduate School of Business, 2008.

[8] Jim Gatheral

No-dynamic-arbitrage and market impactQuantitative Finance 10(7) 749–759 (2010).

[9] Jim Gatheral, Alexander Schied and Alla Slynko

Transient linear price impact and Fredholm integral equationsTo appear in Mathematical Finance

[10] Jim Gatheral, Alexander Schied and Alla Slynko

Exponential resilience and decay of market impactTo appear in Econophysics of Order-driven Markets, Springer (2011).

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Active Portfolio Management.New York: The McGraw-Hill Companies, Inc. (1995).

[11] Anna Obizhaeva and Jiang Wang

Optimal trading strategy and supply/demand dynamics.MIT working paper (2005).

Price manipulation

Exponential decay

Power-law decay

Order book models

Linear transient impact

market impact in models of the order...

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