Let’s face complexity September 4-8, 2017

1. Fakult¨at f¨ur Physik
Econophysics VI:
Price Cross–Responses in Correlated
Financial Markets
Thomas Guhr
Let’s Face Complexity, Como, 2017

2. Outline
Introduction
price formation
self–correlation of trade signs
price self–response
Empirical Analysis
cross-responses
cross-correlation of trade signs
Model
setup and construction
comparision of simulated and empirical results
impact functions

3. Introduction —price formation

4. Introduction —price formation
log mi(t+
)
log mi(t)
log mi(∞)
Time
Price
in a liquid market, shares can be
rapidly bought or sold with little
impact on stock price.
market liquidity measured by
spread between best ask and best
bid.

5. Introduction —price formation
market
orders take
liquidity
limit orders
provide
liquidity
log mi(t+
)
log mi(t)
log mi(∞)
Time
Price
lack of
short-run
liquidity
liquidity
restoration
in a liquid market, shares can be
rapidly bought or sold with little
impact on stock price.
market liquidity measured by
spread between best ask and best
bid.

6. Introduction —correlations of trade signs
How does liquidity influence the trades?
liquidity volume price cost total liquidity
($) ($) cost ($) cost ($)
high 10,000 2 20,000 20,000 0
5,000 2 10,000
low 2,000 2.2 4,400 21,500 1,500
3,000 2.5 7,500
order splitting
correlations of trade signs in single stocks
C0(l) = ⟨εn+l εn⟩ − ⟨εn⟩
2
C1(l) = ⟨εn+l εn ln Vn⟩
C2(l) = ⟨εn+l ln Vn+l εn ln Vn⟩
fitted by
C0(l) ≃
C0
lγ
, (l > 1) ,
where γ = 1/5 for France-Telecom.
Bouchaud, Gefen, Potters, Wyart,
Quantitative Finance 4, 176 (2004)
10
0
10
1
10
2
10
3
10
4
Time (trades)
10
−2
10
−1
10
0
10
1
C(l)
C2(l)
C1(l)
C0(l)

7. Introduction —price responses
Price response
measures how much price changes after
time l, on average, conditioned on an
initial buy or sell market order.
Rii (l) =
⟨(
Si (t + l) − Si (t)
)
εi (t)
⟩
t
Price reversion might seem to be at
odds with long-memory sign
correlation.
Decaying quantity, i.e. an impact
function, is required to reverse the
price.
1 10 100 1000
Time (Trades)
0
0.05
0.1
R(l)(Arbitraryunits)
Total
TF1
Barclays
Pechiney
Bouchaud, Gefen, Potters, Wyart, Quantitative Finance 4, 176 (2004)

8. Introduction —questions
What is the price response of one stock to the trades of the others?
Is there a sign cross–correlation? —What kind of?
so far: single stocks — but now: across stocks
our papers:
Shanshan Wang, Rudi Sch¨afer, Thomas Guhr,
The European Physical Journal B89, 105 (2016)
Shanshan Wang, Rudi Sch¨afer, Thomas Guhr,
The European Physical Journal B89, 207 (2016)
see also:
M. Benzaquen, I. Mastromatteo, Z. Eisler, J.P. Bouchaud, arXiv:1609.02395

9. Empirical Analysis —data sets we used
Trades and Quotes (TAQ) data set from NASDAQ stock market
stocks from S&P 500 index in the year 2008
intraday data with trading time from 9:40 to 15:50 (New York time)
for a given stock pair, we consider the common trading days
resolution of 1 second

10. Empirical Analysis —accumulated trade signs in 1 second interval
a time series of
trades
trade sign of n-th trade in time interval t is defined as
ε(t; n) =
{
sgn
(
S(t; n) − S(t; n − 1)
)
, if S(t; n) ̸= S(t; n − 1) ,
ε(t; n − 1) , otherwise .
accumulated trade sign in time interval t is
ε(t) =



sgn
(
N(t)∑
n=1
ε(t; n)
)
, if N(t) > 0 ,
0 , if N(t) = 0 .
ε(t) =



+1, more buy market orders,
0, lack of trading OR a balance
of buy and sell market orders
−1, more sell market orders.

11. Empirical Analysis —time scale and sign accuracy
Trading of different stocks is not
synchronous ⇒ physical instead
of event time scale.
Wang, Sch¨afer, Guhr, Eur.
Phys. J. B89, 105 (2016)
-2
-1
0
1
2 (a)
ε(t;n)
empirical
theoretical
-2
-1
0
1
2 (b)
ε(t)
-2
-1
0
1
2
10:30:00 10:30:10 10:30:20 10:30:30 10:30:40 10:30:50 10:31:00
(c)
ε(t)
t/s
empirical, Eq. (2)
empirical, Eq. (3)
theoretical, Eq. (2)
for consecutive trades
for stamp of one second
for stamp of one second
ε(t; n) =



sgn
(
S(t; n) − S(t; n − 1)
)
,
if S(t; n) ̸= S(t; n − 1),
ε(t; n − 1), otherwise.
(1)
ε(t) =



sgn


N(t)∑
n=1
ε(t; n)

 ,
if N(t) > 0,
0, if N(t) = 0.
(2)
ε(t) =



sgn


N(t)∑
n=1
ε(t; n)v(t; n)

 ,
if N(t) > 0,
0, if N(t) = 0.
(3)
for consecutive
trades
accuracy for
emp. vs. theo.
empirical
signs
theoretical
signs
fig.(a) AAPL 20080107 86% from TotalView– derived from
average of 6 samples 85% ITCH data set Eq.(1)
for stamp of
one second
accuracy for
emp. Eq.(2) vs.
theo. Eq.(2)
accuracy for
emp. Eq.(3) vs.
theo. Eq.(2)
accuracy
difference
fig.(b) AAPL 20080107 82% 77% 5%
fig.(c) AAPL 20080602 87% 85% 2%
average of 6 samples 82% 80% 2%

12. Empirical Analysis — cross–responses and sign cross–correlators
midpoint price at time t is
mi (t) =
1
2
(ai (t) + bi (t)) .
logarithmic price change from t to t + τ is
ri (t, τ) = log mi (t + τ) − log mi (t) = log
mi (t + τ)
mi (t)
.
price cross–response function of stock i to stock j is
Rij (τ) =
⟨
ri (t, τ)εj (t)
⟩
t
.
cross–correlator of trade signs between stocks i and j is
Θij (τ) =
⟨
εi (t + τ)εj (t)
⟩
t
,
where
Θij (0) = Θji (0) and Θij (τ) = Θji (−τ) .

13. Empirical Analysis — cross–responses, sign cross–correlators
Rij(τ)
×10-5
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
τ/s
100
101
102
Θij(τ)
10-7
10-6
10
-5
10-4
10-3
10-2
10-1
i=AAPL, j=MSFT
i=MSFT, j=AAPL
i=XOM, j=CVX
i=GS, j=JPM
τ/s
100
101
102
103
i=AAPL, j=GS
i=GS, j=AAPL
i=GS, j=XOM
i=XOM, j=AAPL
×10-5
4
5
6
7
8
9
10
τ/s
100
101
102
103
10-7
10-6
10
-5
10-4
10-3
10-2
10-1
i=AAPL, j=AAPL
i=GS, j=GS
i=XOM, j=XOM
Including
εj (t) = 0
Θij (τ) =
θij
(
1 + (τ/τ
(0)
ij )2
)γij /2
{
0 < γij < 1, a long–memory process
γij > 1, a short–memory process
Wang, Sch¨afer, Guhr, Eur. Phys. J. B89, 105
(2016)
Sign Stock i Stock j γij
correlators inc. 0 exc. 0
AAPL MSFT 1.00 1.35
MSFT AAPL 1.15 1.15
XOM CVX 1.04 1.16
Cross GS JPM 1.00 1.00
AAPL GS 1.00 0.91
GS AAPL 1.00 1.00
GS XOM 1.04 1.10
XOM AAPL 1.09 1.42
AAPL AAPL 1.27 1.27
Self GS GS 1.17 1.18
XOM XOM 1.12 1.14

14. Empirical Analysis — influence of zero trade signs
τ/s
100
101
102
103
Rij(τ)
×10
-5
-3
-2
-1
0
1
2
3
4
5
6
7
τ/s
100
101
102
103
Θij(τ)
-0.04
-0.02
0
0.02
0.04
0.06
0.08
for εj(t) = 0 included
for εj(t) = 0 excluded
for εj(t) = 0
stock i is MSFT, stock j is AAPL
price cross–responses for εj (t) = 0: R
(only 0)
ij (τ) = R
(inc. 0)
ij − R
(exc. 0)
ij
sign cross–correlators for εj (t) = 0: Θ
(only 0)
ij (τ) = Θ
(inc. 0)
ij − Θ
(exc. 0)
ij
inclusion of zero trade signs weakens rather than strengthens cross–responses and sign
cross–correlators.

15. Empirical Analysis — average cross–responses and average sign cross–correlators
passive cross–response measures how price of
stock i changes due to trades of all other
stocks j, on average
R
(p)
i (τ) =
⟨
Rij (τ)
⟩
j
active cross–response quantifies which effect
trades of stock j have on prices of all other
stocks i, on average
R
(a)
j (τ) =
⟨
Rij (τ)
⟩
i
passive and active cross–correlators of trade
signs
Θ
(p)
i (τ) =
⟨
Θij (τ)
⟩
j
Θ
(a)
i (τ) =
⟨
Θji (τ)
⟩
j in all averages i = j excluded

16. Empirical Analysis — average cross–responses and average sign cross–correlators
passive active
R
(p)
i(τ)
10-6
10
-5
10
-4
R
(a)
j(τ)
10-5
10-4
τ/s
100
101
102
103
104
Θ
(p)
i(τ)
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
i =AAPL, inc. 0
i =GS, inc. 0
i =XOM, inc. 0
i =AAPL, exc. 0
i =GS, exc. 0
i =XOM, exc. 0
τ/s
10
0
10
1
10
2
10
3
10
4
Θ
(a)
j(τ)
-0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
j =AAPL, inc. 0
j =GS, inc. 0
j =XOM, inc. 0
j =AAPL, exc. 0
j =GS, exc. 0
j =XOM, exc. 0
Passive cross–responses reverse
faster than active ones.
Sign cross–correlators are long
memory after averaging due to
noise reduction.
Sign cross– stock γi or γj
correlators i, j inc. 0 exc. 0
AAPL 0.68 0.73
Θ
(p)
i
(τ) GS 0.92 0.90
XOM 1.32 1.33
AAPL 0.90 0.91
Θ
(a)
j
(τ) GS 0.85 0.83
XOM 0.71 0.95

17. Empirical Analysis — market responses
including zero trade signs
τ = 1 s
stock j
I HC CD IT U F M E CS TS
stocki
IHCCDITUFMECSTS
τ = 2 s
stock j
I HC CD IT U F M E CS TS
stocki
IHCCDITUFMECSTS
τ = 60 s
stock j
I HC CD IT U F M E CS TS
stocki
IHCCDITUFMECSTS
τ = 300 s
stock j
I HC CD IT U F M E CS TS
stocki
IHCCDITUFMECSTS
τ = 1800 s
stock j
I HC CD IT U F M E CS TS
stocki
IHCCDITUFMECSTS
τ = 7200 s
stock j
I HC CD IT U F M E CS TS
stocki
IHCCDITUFMECSTS
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
excluding zero trade signs
τ = 1 s
stock j
I HC CD IT U F M E CS TS
stocki
IHCCDITUFMECSTS
τ = 2 s
stock j
I HC CD IT U F M E CS TS
stocki
IHCCDITUFMECSTS
τ = 60 s
stock j
I HC CD IT U F M E CS TS
stocki
IHCCDITUFMECSTS
τ = 300 s
stock j
I HC CD IT U F M E CS TS
stocki
IHCCDITUFMECSTS
τ = 1800 s
stock j
I HC CD IT U F M E CS TS
stocki
IHCCDITUFMECSTS
τ = 7200 s
stock j
I HC CD IT U F M E CS TS
stocki
IHCCDITUFMECSTS
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Responses are normalized as Rij (τ)/max(|Rij (τ)|)

18. Empirical Analysis — market responses
τ/s
10
0
10
1
10
2
10
3
10
4
R(τ)
×10-5
2
3
4
5
6
7
8
for εj(t) = 0 excluded
for εj(t) = 0 included
doubly averaged response for the market
R(τ) = ⟨⟨Rij (τ)⟩j ⟩i
excluding i = j.
99 stocks from 10 economic sectors in 2008
for each sector, first 9 or 10 stocks with
largest average market capitalization
Market efficiency is violated on short
time scales, but restored on longer
time scales.
Wang, Sch¨afer, Guhr, Eur. Phys. J. B89, 105 (2016)

19. Empirical Analysis —identifying influencing and influenced stocks
influencing stocks
300
COST
FCX
τ/s
VZ
60
COP
INTC
PG
AMZN
VZ
FCX
T
CSCO
COP
2
FCX
GS
GS
WMT
JNJ
AMZN
T
SLB
CVX
QCOM
1
INTC
PG
MSFT
WMT
JNJ
CSCO
INTC
T
FCX
XOM
AAPL
COP
CVX
SLB
QCOM
AMZN
PG
MSFT
INTC
WMT
CSCO
AMZN
XOM
16
AAPL
CVX
QCOM
T
COP
14
WMT
MSFT
CSCO
XOM
12
AAPL
CVX
QCOM
10
MSFT
stock j
XOM
8
AAPL
6
4
2
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.1
R
(a)
j(τ)
Industrials
Health Care
Consumer Discretionary
Information Technology
Utilities
Financials
Materials
Energy
Consumer Staples
Telecommunications Services
influenced stocks
300
MA
CRM
X
τ/s
60
FLR
ISRG
FCX
SHLD
WIN
WFR
FLR
SHLD
2
VNO
CF
WFR
CME
WYNN
DVN
MON
CME
Q
ICE
1
APA
ICE
FLS
OXY
WYNN
GS
EOG
X
EOG
S
DVN
Q
FTR
FLS
ICE
COP
ISRG
BEN
APA
WFR
SLB
16
S
HES
CF
MON
FTR
FLR
HES
14
FCX
WFR
NUE
OXY
12
CF
FLR
X
10
CF
FCX
stock i
8
NUE
X
6
4
2
0.05
0.1
0.15
0.2
0.25
0.35
0.3
R
(p)
i(τ)
Industrials
Health Care
Consumer Discretionary
Information Technology
Utilities
Financials
Materials
Energy
Consumer Staples
Telecommunications Services
99 stocks from 10 economic sectors in 2008
for each sector, first 9 or 10 stocks with largest average market capitalization
responses are normalized, Rij (τ)/max(|Rij (τ)|)
zero trade signs included

20. Empirical Analysis —questions
How to understand cross–responses between stocks?
What is the relation between the cross–responses and sign cross–correlators?
Why do active and passive average cross–responses differ?
we need a model!
our paper:
Shanshan Wang, Thomas Guhr, arXiv:1609.04890
see also:
M. Benzaquen, I. Mastromatteo, Z. Eisler, J.P. Bouchaud, arXiv:1609.02395

21. Price Impact Model —single stocks
log mi (1) = log mi (0) + Gii (1)f
(
vi (0)
)
εi (0) + ηii (0)
log mi (2) = Gii (1)f
(
vi (1)
)
εi (1) + ηii (1)
+Gii (2)f
(
vi (0)
)
εi (0) + ηii (0)
+ log mi (0)
log mi (t) =
∑
t′<t
Gii (t − t
′
)f
(
vi (t
′
)
)
εi (t
′
) +
∑
t′<t
ηii (t
′
)
+ log mi (−∞)
Bouchaud, Gefen, Potters, Wyart, Quantitative Finance 4, 176
(2004)
log mi(t+
)
log mi(t)
log mi(∞)
Time
Price
f (vi (t))
Gii (τ)
vi (t): traded volume
ηii (t): random variable
f (vi (t)): impact function
of traded volumes
Gii (τ): ‘bare’ impact
function of time
lags for a single
trade

22. Price Impact Model —across stocks
log mi (t) =
∑
t′<t
[
Gii (t − t
′
)f
(
vi (t
′
)
)
εi (t
′
) + ηii (t
′
)
]
+
∑
t′<t
[
Gij (t − t
′
)g
(
vj (t
′
)
)
εj (t
′
) + ηij (t
′
)
]
+ log mi (−∞)
log mi(t+
)
log mi(t)
log mi(∞)
Time
Price
liquidity→ f (vi (t))
information→ g(vj (t))
Gii (τ) → self–impact
Gij (τ) → cross–impact

23. Price Impact Model —across stocks
log mi (t) =
∑
t′<t
[
Gii (t − t
′
)f
(
vi (t
′
)
)
εi (t
′
) + ηii (t
′
)
]
+
∑
t′<t
[
Gij (t − t
′
)g
(
vj (t
′
)
)
εj (t
′
) + ηij (t
′
)
]
+ log mi (−∞)
for a buy market order
for a sell market order
Assume impact function
G(τ) =
Γ0
[
1 +
(
τ
τ0
)2
]β/2
+ Γ
Properties of impact function, e.g.
positive or negative impact,
temporary or permanent impact,
are determined by data fits.

24. Price Impact Model —across stocks
log mi (t) =
∑
t′<t
[
Gii (t − t
′
)f
(
vi (t
′
)
)
εi (t
′
) + ηii (t
′
)
]
+
∑
t′<t
[
Gij (t − t
′
)g
(
vj (t
′
)
)
εj (t
′
) + ηij (t
′
)
]
+ log mi (−∞)
log mi(t+
)
log mi(t)
log mi(∞)
Time
Price
liquidity→ f (vi (t))
information→ g(vj (t))
Gii (τ) → self–impact
Gij (τ) → cross–impact

25. Price Impact Model —across stocks
log mi (t) =
∑
t′<t
[
Gii (t − t
′
)f
(
vi (t
′
)
)
εi (t
′
) + ηii (t
′
)
]
+
∑
t′<t
[
Gij (t − t
′
)g
(
vj (t
′
)
)
εj (t
′
) + ηij (t
′
)
]
+ log mi (−∞)
R
(C)
ij (τ) =
⟨
r
(L)
ii (t, τ)εj (t)
⟩
t
=
∑
t≤t′<t+τ
Gii (t + τ − t
′
)
⟨
f
(
vi (t
′
)
)⟩
t
Θij (t
′
− t)
+
∑
t′<t
[
Gii (t + τ − t
′
) − Gii (t − t
′
)
] ⟨
f
(
vi (t
′
)
)⟩
t
Θji (t − t
′
)
log mi(t+
)
log mi(t)
log mi(∞)
Time
Price
liquidity→ f (vi (t))
information→ g(vj (t))
Gii (τ) → self–impact
Gij (τ) → cross–impact

26. Price Impact Model —across stocks
log mi (t) =
∑
t′<t
[
Gii (t − t
′
)f
(
vi (t
′
)
)
εi (t
′
) + ηii (t
′
)
]
+
∑
t′<t
[
Gij (t − t
′
)g
(
vj (t
′
)
)
εj (t
′
) + ηij (t
′
)
]
+ log mi (−∞)
R
(C)
ij (τ) =
⟨
r
(L)
ii (t, τ)εj (t)
⟩
t
=
∑
t≤t′<t+τ
Gii (t + τ − t
′
)
⟨
f
(
vi (t
′
)
)⟩
t
Θij (t
′
− t)
+
∑
t′<t
[
Gii (t + τ − t
′
) − Gii (t − t
′
)
] ⟨
f
(
vi (t
′
)
)⟩
t
Θji (t − t
′
)
R
(S)
ij (τ) =
⟨
r
(I)
ij (t, τ)εj (t)
⟩
t
=
∑
t≤t′<t+τ
Gij (t + τ − t
′
)
⟨
g
(
vj (t
′
)
)⟩
t
Θjj (t
′
− t)
+
∑
t′<t
[
Gij (t + τ − t
′
) − Gij (t − t
′
)
] ⟨
g
(
vj (t
′
)
)⟩
t
Θjj (t − t
′
)
log mi(t+
)
log mi(t)
log mi(∞)
Time
Price
liquidity→ f (vi (t))
information→ g(vj (t))
Gii (τ) → self–impact
Gij (τ) → cross–impact

27. Price Impact Model —across stocks
cross–responses
Rij (τ) = R
(C)
ij (τ) + R
(S)
ij (τ) .
passive and active average cross–response functions
R
(p)
i (τ) =
⟨
R
(C)
ij (τ)
⟩
j
⟨
f
(p)
i (vi )
⟩
t
⟨
f
(p)
i (vi )
⟩
t
+
⟨
R
(S)
ij (τ)
⟩
j
⟨
g
(p)
i (vj )
⟩
t,j
⟨
g
(p)
i (vj )
⟩
t,j
,
R
(a)
i (τ) =
⟨
R
(C)
ji (τ)
⟩
j
⟨
f
(a)
i (vj )
⟩
t,j
⟨
f
(a)
i (vj )
⟩
t,j
+
⟨
R
(S)
ji (τ)
⟩
j
⟨
g
(a)
i (vi )
⟩
t
⟨
g
(a)
i (vi )
⟩
t
.
average cross–responses:
R
(p)
i (τ) = ⟨Rij (τ)⟩j
R
(a)
i (τ) = ⟨Rji (τ)⟩j
sign cross–correlators:
Θ
(p)
i (τ) = ⟨Θij (τ)⟩j
Θ
(a)
i (τ) = ⟨Θji (τ)⟩j
according to empirical analysis
⟨
f
(p)
i (vi )
⟩
t
∼ v
δip
i ,
⟨
g
(p)
i (vj )
⟩
t,j
∼ v
δjp
j
⟨
f
(a)
i (vj )
⟩
t,j
∼ v
δja
j ,
⟨
g
(a)
i (vi )
⟩
t
∼ v
δia
i
δip, δjp, δja, δia ∼ 0.5 ± 0.2 for small volumes of most stocks
⟨f
(p)
i (vi )⟩t , ⟨g
(p)
i (vj )⟩t,j , ⟨f
(a)
i (vj )⟩t,j , ⟨g
(p)
i (vi )⟩t → independent of time lag

28. Price Impact Model —simulations and data fits
passive cross–responses
R
(p)
i(τ)
×10
-5
3
4
5
6
7
8
9
10
11
R
(p)
i(τ)
×10
-5
3
4
5
6
7
8
9
10
11
R
(p)
i(τ)
×10
-5
3
4
5
6
7
8
9
10
11
τ/s
10
0
10
1
10
2
R
(p)
i(τ)
×10
-5
3
4
5
6
7
8
9
10
11
τ/s
100
101
102
R
(p)
i(τ)
×10
-5
3
4
5
6
7
8
9
10
11
empirical
theoretical, Case (1)
theoretical, Case (2)
theoretical, Case (3)
w = 0.10 w = 0.30
w = 0.50 w = 0.70
w = 0.90
active cross–responses
R
(a)
i(τ)
×10
-5
4
6
8
10
12
14
R
(a)
i(τ)
×10-5
4
6
8
10
12
14
R
(a)
i(τ)
×10
-5
4
6
8
10
12
14
τ/s
10
0
10
1
10
2
R
(a)
i(τ)
×10
-5
4
6
8
10
12
14
τ/s
100
101
102
R
(a)
i(τ)
×10
-5
4
6
8
10
12
14
empirical
theoretical, Case (1)
theoretical, Case (2)
theoretical, Case (3)
w = 0.10 w = 0.30
w = 0.50 w = 0.70
w = 0.90
stock i is MSFT in 2008, and the pairwise stocks j are other 30 stocks with the largest average
number of daily trades in S&P 500 index of 2008.
Wang, Guhr, arXiv:1609.04890

29. Price Impact Model —impact functions
sketch of price impacts
after averaging,
Gij (τ) → G
(p)
i (τ), G
(a)
i (τ)
simulated impact function
G(τ) =
Γ0
[
1 +
(
τ
τ0
)2
]β/2
+ Γ
Wang, Guhr, arXiv:1609.04890
simulations of impact functions
τ/s
10
0
10
1
10
2
10
3
10
4
G(τ)
×10-4
0
0.5
1
1.5
2
2.5
3
3.5
Gii(τ)
G
(p)
i (τ)
G
(a)
i (τ)
100
101
102
103
104
10-6
10-5
10
-4
10-3
MSFT in 2008
impact
functions
Γ
(×10−10
)
Γ0
(×10−4
)
τ0
[s]
β
Gii (τ) 0.5 5.12 0.025 0.13
G
(p)
i
(τ) 0 0.25 70.873 0.49
G
(a)
i
(τ) 0 2.57 0.004 0.19

30. Summary
price formation:
interaction of market orders and limit orders
role of liquidity
empirical results:
cross–responses of stock pairs
trade sign cross–correlators
average cross–responses
average trade sign cross–correlators
market responses
influencing and influenced stocks
price impact model:
a self– and a cross–impact function
two response components related to the cross– and the
self–correlators, respectively
comparison of empirical and simulated results
self–, active and passive impact functions
Rij(τ)
×10
-5
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
τ/s
100
101
102
Θij(τ)
10
-7
10
-6
10
-5
10
-4
10-3
10
-2
10
-1
i=AAPL, j=MSFT
i=MSFT, j=AAPL
i=XOM, j=CVX
i=GS, j=JPM
τ/s
100
101
102
103
i=AAPL, j=GS
i=GS, j=AAPL
i=GS, j=XOM
i=XOM, j=AAPL
×10
-5
4
5
6
7
8
9
10
τ/s
100
101
102
103
10
-7
10
-6
10
-5
10
-4
10-3
10
-2
10
-1
i=AAPL, j=AAPL
i=GS, j=GS
i=XOM, j=XOM

31. Our papers
[1] Shanshan Wang, Rudi Sch¨afer, Thomas Guhr, Cross-response in correlated
financial markets: individual stocks, The European Physical Journal B89, 105
(2016)
[2] Shanshan Wang, Rudi Sch¨afer, Thomas Guhr, Average cross-responses in
correlated financial market, The European Physical Journal B89, 207 (2016)
[3] Shanshan Wang, Thomas Guhr, Microscopic understanding of cross-responses
between stocks: a two-component price impact model, arXiv:1609.04890