order book resilience, price manipulations, and the positive portfolio problem - slides

8/9/2019 Order Book Resilience, Price Manipulations, And the Positive Portfolio Problem - Slides

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Order book resilience,

price manipulations, and thepositive portfolio problem

Alexander SchiedMannheim University

PRisMa WorkshopVienna, September 28, 2009

Joint work with Aurelien Alfonsi and Alla Slynko

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Spreading the order can reduce the overall price impact

time t

intradaystock

price


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Limit order book before market order

buyers bid offers sellers ask offers

bid-ask spread

best bidprice

best askprice

prices


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Limit order book before market order


bid-ask spread

best bidprice

best askprice

volume of sell market order


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Limit order book after market order


new bid-ask spread

new bestbid price

best askprice

volume of sell market order


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Resilience of the limit order book after market order


new bestask price

new bestbid price


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Overview

1. Linear impact, general resilience

2. Nonlinear impact,exponential resilience

3. Relations with Gatherals model

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ReferencesA. Alfonsi, A. Fruth, and A.S.: Optimal execution strategies in limitorder books with general shape functions. To appear in QuantitativeFinance

A. Alfonsi, A. Fruth, and A.S.: Constrained portfolio liquidation in a limit order book model. Banach Center Publ. 83, 9-25 (2008).

A. Alfonsi and A.S.: Optimal execution and absence of price manipulations in limit order book models. Preprint 2009.

A. Alfonsi, A.S., and A. Slynko: Order book resilience, price manipulations, and the positive portfolio problem. Preprint 2009.

J. Gatheral: No-Dynamic-Arbitrage and Market Impact. Preprint

(2008).A. Obizhaeva and J. Wang: Optimal Trading Strategy andSupply/Demand Dynamics. Preprint (2005)

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Limit order book model without large trader

unaffected best ask priceunaffected best bid price,is martingale



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Limit order book model after large trades

actual best ask priceactual best bid price


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Limit order book model at large trade

q

t = q(B t + B t )


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q

similarly for buy orders

sell order executed at average price B t

B t +xq dx

t = q(B t + B t )


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Limit order book model immediately after large trade


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Resilience of the limit order book

B t + t

: [0, [ [0, 1], (0) = 1, decreasing

tq

( t ) + decay of previous trades


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1. Linear impact, general resilienceStrategy:

N + 1 market orders: n shares placed at time tn s.th.

a) 0 = t0 t1 tN = T (can also be stopping times)

b) n is F t n -measurable and bounded from below,

c) we haveN

n =0

n = X 0

Sell order: n < 0Buy order: n > 0

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Actual best bid and ask prices

B t = B0

t +

1

q t n


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A simplied modelNo bid-ask spread

S 0t = una ff ected price, is (continuous) martingale.

S t = S 0t +1q t n


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Lemma 1. Suppose that S 0 = A0 . Then, for any strategy ,

cn ( ) cn ( ) with equality if k 0 for all k.

Thus: Enough to study the simplied model (as long as all trades nare positive)

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Lemma 1. Suppose that S 0 = A0 . Then, for any strategy ,


Thus: Enough to study the simplied model (as long as all trades nare positive)

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Lemma 2. In the simplied model, the expected execution costs of a strategy are

C( ) = E N

n =0

cn ( ) =12q

E C t ( ) ] + X 0 S 00 ,

where C t is the quadratic form

C t (x ) =N

m,n =0xn xm (|tn tm |), x R N +1 , t = ( t0 , . . . , t N ).

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The usual concept of viability from Hubermann & Stanzl (2004):

DenitionA round trip is a strategy with

N

n =0

n = X 0 = 0 .

A market impact model admits

price manipulation strategies

if there is a round trip with negative expected execution costs.

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In the simplied model, the expected costs of a strategy are

C( ) =1

2qE C t ( ) ] + X 0 S 00 ,

where

C t (x ) =N

m,n =0

xn xm (|tn tm |), x R N +1 , t = ( t0 , . . . , t N ).

There are no price manipulation strategies when C t is nonnegativedenite for all t = ( t0 , . . . , t N );

when the minimizer x of C t (x ) with i x i = X 0 exists, it yieldsthe optimal strategy in the simplied model; in particular, the

optimal strategy is then deterministic; when the minimizer x has only nonnegative components, it yieldsthe optimal strategy in the order book model.

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Bochners theorem (1932):C t is always nonnegative denite ( is positive denite) if and

only if (| | ) is the Fourier transform of a positive Borel measure on R .

C t is even strictly positive denite ( is strictly positive denite)when the support of is not discrete.

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Examples

Example 1: Exponential resilience[Obizhaeva & Wang (2005), Alfonsi, Fruth, S. (2008)]

For the exponential resilience function

(t) = e t ,

(| | ) is the Fourier transform of the positive measure

(dt) =1

2 + t2

dt

Hence, is strictly positive denite.

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Optimal strategies for exponential resilience (t) = e t

! ! " # $!"

"%#

$

$%#

&'($"

! ! " # $!"

"%#

$

$%#

&'($#

! ! " # $!"

"%#

$

$%#

&'(!" ! ! " # $!"

"%#

$

$%#

&'(!#

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The optimal strategy can in fact be computed explicitly for any timegrid [Alfonsi, Fruth, A.S. (2008)]:

Letting 0 =

X 01 M 1 1

=X 0

21+ a 1 +

N n =2

1 a n1+ a n

,

the initial market order of the optimal strategy is

x0 = 0

1 + a1,

the intermediate market orders are given by

xn = 01

1 + an

an +11 + an +1

, n = 1 , . . . , N 1,

and the nal market order is

xN = 0

1 + aN .

all components of x are strictly positive

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For the equidistant time grid tn = nT/N the solution simplies:

x

0 = x

N =X 0

(N 1)(1 a) + 2and

x1 = = xN 1 =

0 (1 a).

! ! " ! #" # !"

"$!

#

#$!

%&'#"

! ! " ! #" # !"

"$!

#

#$!

%&'#!

! ! " ! #" # !"

"$!

#

#$!

%&'("

! ! " ! #" # !"

"$!

#

#$!

%&'(!

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The symmetry of the optimal strategy is a general fact:

Proposition 3. Suppose that is strictly positive denite and that the time grid is symmetric, i.e.,

t i = tN tN i for all i,

then the optimal strategy is reversible, i.e.,

xt i = x

t N i for all i.

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Example 2: Linear resilience (t) = 1 t for some 1/T

The optimal strategy is always of this form:

! ! " ! # $ % & '! !

"

!

#

$

%

&

()*+,-./+*(01

( ) * + , - .

/ 1 , 2 0 1

()*+,-./1()*(0.3

It is independent of the underlying time grid and consists of twosymmetric trades of size X 0 / 2 at t = 0 and t = T , all other trades arezero.

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Example 3: Power law resilience (t) = (1 + t)

! ! " ! #" #!"

"$!

#

#$!

%

&'(#"! ! " ! #" #!"

"$!

#

#$!

%

&'(#!

! ! " ! #" #!"

"$!

#

#$!

%

&'(%"

! ! "( !( #" # !"

"$!

#

#$!

%

&'(%!

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Example 4: Trigonometric resilienceThe function

cos xis the Fourier transform of the positive nite measure

=12

( + )

Since it is not strictly positive denite, we take

(t) = (1 )cos t + e t for some

2T .

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Trigonometric resilience (t) = 0 .999 cos(t / 2T ) + 0 .001e t

! " #!

! "

!

"

#!

#"

$!

%&'$!! " #!

! #!

! "

!

"

#!

#"

$!

$"

(!

%&'$!

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Example 5: Gaussian resilience

The Gaussian resilience function

(t) = e t2

is its own Fourier transform (modulo constants). The correspondingquadratic form is hence positive denite.

Nevertheless.....

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Gaussian resilience (t) = e t2

, N = 15

! ! " ! # $ % &" &!! "'!

"

"'!

"'#

"'$

"'%

&

&'!

&'#

&'$

()*&+

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, N = 20

! ! " ! # $ % &" &!! '

! !

! &

"

&

!

'

#

()*!"

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, N = 25

! ! " ! # $ % &" &!! !"

! &'

! &"

! '

"

'

&"

&'

!"

()*!'

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, N = 25

! ! " ! # $ % &" &!! !"

! &'

! &"

! '

"

'

&"

&'

!"

()*!'

absence of price manipulation strategies is not enough

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Denition [Hubermann & Stanzl (2004)]A market impact model admits

price manipulation strategies in the strong sense

if there is a round trip with negative expected liquidation costs.

Denition:A market impact model admits

price manipulation strategies in the weak sense

if the expected liquidation costs of a sell (buy) program can bedecreased by intermediate buy (sell) trades.

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Question: When does the minimizer x of

i,j

x i x j (|t i t j |) withi

x i = X 0

have only nonnegative components?

Related to the positive portfolio problem in nance:When are there no short sales in a Markowitz portfolio?

I.e. when is the solution of the following problem nonnegative

x M x m x min for x 1 = X 0 ,

where M is a covariance matrix of assets and m is the vector of returns?

Partial results, e.g., by Gale (1960), Green (1986), Nielsen (1987)

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Proposition 5. [Alfonsi, A.S., Slynko (2009)]When is strictly positive denite and trading times are equidistant,

then x0 > 0 and xN > 0.

Proof relies on Trench algorithm for inverting the Toeplitz matrix

M ij = (|i j | /N ), i, j = 0 , . . . , N

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Theorem 6. [Alfonsi, A.S., Slynko (2009)]

If is convex then all components of x are nonnegative.

If is strictly convex, then all components are strictly positive.

Conversely, x has negative components as soon as, e.g., isstrictly concave in a neighborhood of 0.

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Qualitative properties of optimal strategies?

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Qualitative properties of optimal strategies?

Proposition 8. [Alfonsi, A.S., Slynko (2009)]When is convex and nonconstant, the optimal x satises

x0 x1 and xN 1 x

N

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Proof: Equating the rst and second equations in M x = 0 1 givesN

j =0 x

j (t j ) =

N

j =0 x

j (|t j t1 |).

Thus,

x0 x1 =N

j =0 , j =1

xj (|t j t1 |) N

j =1

xj (t j )

= x0 (t1 ) x1 (t1 ) +N

j =2

xj (t j t1 ) (t j )

(x0 x1 ) (t1 ),

by convexity of . Therefore

(x0 x1 )(1 (t1 )) 0

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Proposition 8. [Alfonsi, A.S., Slynko (2009)]When is convex and nonconstant, the optimal x satises

x0 x1 and xN 1 xN

What about other trades? General pattern?

! !" " !" #" $" %" &"" &!""

&

!

'

#

(

$

)

%

*

+,-./012.-+34

+ , - . / 0 1

2 4 / 5 3 4

+,-./0124+,-+316

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No! Capped linear resilience (t) = (1 t)+ , = 2 /T

!!"# ! !"# !"$ !"% !"& ' '"#

! (

!

(

'!

'(

#!

#(

)!

)(

*+,-./01-,*23

* + , - . / 0

1 3 . 4 2 3

1/5672+1891*+,-./01-,*231:;'!!;'


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Proposition 9. [Alfonsi, A.S., Slynko (2009)]Suppose that (t) = (1 kt/T )+ and that k divides N . Then the

optimal strategy consists of k + 1 equal equidistant trades.

! ! " ! # $ % &" &!"

&

!

'

#

(

$

)

%

*

&"

+,-&"".-/,&"

Proof relies on Trench algorithm

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When k does not divide N , the situation becomes more complicated:

! ! " ! # $ % &" &!"

!

#

$

%

&"

&!

'()&""*)+($! ! " ! # $ % &" &!"

!

#

$

%

&"

&!

'()&""*)+(&,

! ! " ! # $ % &" &!"

!

#

$

%

&"

&!

'()#,*)+($! ! " ! # $ % &" &!"

!

#

$

%

&"

&!

'()#,*)+(&"

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1. Linear impact, general resilience

2. Nonlinear impact,exponential resilience

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Limit order book model without large trader


unaffected best ask priceunaffected best bid price,is martingale


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Limit order book model after large trades


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Limit order book model immediately after large trade


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Limit order book model with resilience


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f (x) = shape function = densities of bids for x < 0, asks for x > 0

B 0t = una ff ected bid price at time t, is martingale

B t = bid price after market orders before time t

D Bt = B t B0t

If sell order of t 0 shares is placed at time t:

D Bt changes to D Bt + , where

D Bt +

D Btf (x)dx = t

andB t + := B t + D Bt + D

Bt = B

0t + D

Bt + ,

= nonlinear price impact

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A0t = una ff ected ask price at time t, satises B0t A

0t

At = bid price after market orders before time t

D At = At A0tIf buy order of t 0 shares is placed at time t:

D At changes to D At + , where

D At +

D Atf (x)dx = t

andAt + := At + D At + D

At = A

0t + D

At + ,

For simplicity, we assume that the LOB has innite depth , i.e.,

|F (x)| as |x| , whereF (x) :=

x

0f (y) dy.

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If the large investor is inactive during the time interval [ t, t + s[,there are two possibilities:

Exponential recovery of the extra spread

D Bt = e ts r dr D Bs for s < t .

Exponential recovery of the order book volume

E Bt = e ts r dr E Bs for s < t ,

where

E Bt =

0

DBt

f (x) dx =: F (D Bt ).

In both cases: analogous dynamics for D A or E A

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Strategy:

N + 1 market orders: n shares placed at time n s.th.

a) the ( n ) are stopping times s.th. 0 = 0 1 N = T

b) n is F n -measurable and bounded from below,

c) we haveN

n =0

n = X 0

Will write( , )

and optimize jointly over and .

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When selling n < 0 shares, we sell f (x) dx shares at price B 0 n + xwith x ranging from D B n to D

B n + < D

B n , i.e., the costs are negative:

cn ( , ) := DB n +

D B n

(B 0 n + x)f (x) dx = n B0 n + D

B n +

D B n

xf (x) dx

When buying shares ( n > 0), the costs are positive:

cn ( , ) := n A0 n + D A n +

D A nxf (x) dx

The expected costs for the strategy ( , ) are

C( , ) = EN

n =0

cn ( , )

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Instead of the k , we will use

(1) k :=

k

k 1s ds, k = 1 , . . . , N .

The condition 0 = 0 1 N = T is equivalent to := ( 1 , . . . , N ) belonging to

A := := ( 1 , . . . , N ) R N +N

k =1

k =

T

0

s ds .

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A simplied model without bid-ask spreadS 0t = una ff ected price, is (continuous) martingale.

S t n = S 0t n + Dn

where D and E are dened as follows:

E 0 = D 0 = 0, E n = F (D n ) and D n = F 1 (E n ).

For n = 0 , . . . , N , regardless of the sign of n ,

E n + = E n n and D n + = F 1 (E n + ) = F 1 (F (D n ) n ) .

For k = 0 , . . . , N 1,

E k +1 = e k +1 E k + = e k +1 (E k k )

The costs arecn ( , ) = n S 0 n +

D n +

D nxf (x) dx

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Lemma 10. Suppose that S 0 = B 0 . Then, for any strategy ,


Moreover,

C( , ) := EN

n =0

cn ( , ) = E C ( , ) X 0 S 00

where

C ( , ) :=N

n =0 D

n +

D nxf (x) dx

is a deterministic function of A and R N +1 .

Implies that is is enough to minimize C ( , ) over A and

x = ( x0 , . . . , x N ) R N +1N

n =0

xn = X 0 .

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Theorem 11. Suppose f is increasing on R and decreasing on R + .Then there is a unique optimal strategy (, ) consisting of homogeneously spaced trading times,

i +1

i

r dr =1N

T

0r dr =: log a,

and trades dened via

F 1 (X 0 N 0 (1 a)) = F 1

(

0 ) aF 1

(a

0 )1 a ,

and 1 = = N 1 =

0 (1 a) ,

as well asN = X 0 0 (N 1)0 (1 a) .

Moreover, i > 0 for all i.

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Robustness of the optimal strategy[Plots by C. Lorenz (2009)]First gure:

f (x) = 11 + |x|

Figure 1: f , F , F 1 , G and optimal strategy

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Figure 2: f (x) = |x|

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Figure 3: f (x) = 18 x2

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Figure 4: f (x) = exp( (|x| 1)2 ) + 0 .1

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Figure 5: f (x) = 12 sin( |x|) + 1

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Figure 8: f random

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Figure 9: f random

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Figure 10: f piecewise constant

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Continuous-time limit of the optimal strategy

Initial block trade of size 0 , where

F 1 X 0 0 T

0s ds = F 1 (0 ) +

0f (F 1 (0 ))

Continuous trading in ]0, T [ at rate

t = t

0

Terminal block trade of size

T = X 0

0

0

T

0

t dt

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Conclusion Market impact should decay as a convex function of time

Exponential or power law resilience leads to bathtub solutions

! !" " !" #" $" %" &"" &!""

&

!

'

#

(

$

)

%

*

+,-./012.-+34

+ , - . / 0 1

2 4 / 5 3 4

+,-./0124+,-+316

which are extremely robust

Many open problems

order book resilience, price manipulations, and the positive portfolio problem - slides

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