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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
buyers bid offers sellers ask offers
bid-ask spread
best bidprice
best askprice
volume of sell market order
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Limit order book after market order
buyers bid offers sellers ask offers
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
buyers bid offers sellers ask offers
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
buyers bid offers sellers ask offers
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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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Limit order book model at large trade
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 ,
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 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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Gaussian resilience (t) = e t2
, N = 20
! ! " ! # $ % &" &!! '
! !
! &
"
&
!
'
#
()*!"
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Gaussian resilience (t) = e t2
, N = 25
! ! " ! # $ % &" &!! !"
! &'
! &"
! '
"
'
&"
&'
!"
()*!'
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Gaussian resilience (t) = e t2
, 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
buyers bid offers sellers ask offers
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 at large trade
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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 ,
cn ( ) cn ( ) with equality if k 0 for all k.
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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Figure 11: f piecewise constant
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Figure 12: f piecewise constant
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Figure 13: 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