Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
On Lead-Lag Estimation
Mathieu Rosenbaum
CMAP-École Polytechnique
Joint works with
Marc Hoffmann, Christian Y. Robert and Nakahiro Yoshida
12 January 2011
Mathieu Rosenbaum
On Lead-Lag Estimation
1
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Outline
1
Introduction
2
Lead-Lag estimation from non synchronous data
3
Simulations
4
Real Data
5
A random matrices based approach
Mathieu Rosenbaum
On Lead-Lag Estimation
2
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Outline
1
Introduction
2
Lead-Lag estimation from non synchronous data
3
Simulations
4
Real Data
5
A random matrices based approach
Mathieu Rosenbaum
On Lead-Lag Estimation
3
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Motivation
Observation from practitioners in finance
Some assets are leading some other assets.
This means that a “lagger” asset may partially reproduce the
behavior of a “leader” asset.
This common behavior is unlikely to be instantaneous. It is
subject to some time delay called “lead-lag”.
Mathieu Rosenbaum
On Lead-Lag Estimation
4
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
A toy model for Lead-Lag
Bachelier model
For t ∈ [0, 1], and (B (1) , B (2) ) such that hB (1) , B (2) it = ρt,
set
(1) e
(2)
e0 + σ2 Bt ,
Xt := x0 + σ1 Bt , Y
t := y
et−θ , t ∈ [θ, 1]. Our lead-lag model is given by
Define Yt := Y
the bidimensional process (Xt , Yt ).
We have
(
Xt
Yt
(1)
= x0 + σ1 Bt
.
(1)
= y0 + ρ σ2 Bt−θ + σ2 (1 − ρ2 )1/2 Wt−θ
Mathieu Rosenbaum
On Lead-Lag Estimation
5
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Intuitive estimator in the Bachelier model (1)
Estimation idea (1)
Assume the data arrive at regular and synchronous time
stamps in the Bachelier model, i.e. we have data
(X0 , Y0 ), (X∆n , Y∆n ), (X2∆n , Y2∆n ), . . . , (X1 , Y1 ),
and suppose θ = k0 ∆n , k0 ∈ Z.
Let
Cn (k) :=
X
Xi∆n − X(i−1)∆n
Y(i+k)∆n − Y(i+k−1)∆n .
i
Mathieu Rosenbaum
On Lead-Lag Estimation
6
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Intuitive estimator in the Bachelier model (2)
Estimation idea (2)
Heuristically, we have
1/2 n
Cn (k) ≈ ∆−1
n E (X· − X·−∆n )(Y·+k∆n − Y·+(k−1)∆n ) + ∆n ξ .
Moreover,
∆n−1 E (X· −X·−∆n )(Y·+k∆n −Y·+(k−1)∆n ) =
0
if k =
6 k0
ρ σ1 σ2 if k = k0 .
Thus we can (asymptotically) detect the value k0 that defines
θ in the very special case θ = k0 ∆n by maximizing in k the
contrast sequence
Cn (k).
k
Mathieu Rosenbaum
On Lead-Lag Estimation
7
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Outline
1
Introduction
2
Lead-Lag estimation from non synchronous data
3
Simulations
4
Real Data
5
A random matrices based approach
Mathieu Rosenbaum
On Lead-Lag Estimation
8
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Covariation estimation
Previous-Tick estimation
Estimating covariation is an intricate problem as soon as non
synchronous data are considered.
Assume now we observe X at times (T X ,i ), i = 1, . . . and Y
at times (T Y ,i ), i = 1, . . ., with T X ,i ≤ T , T Y ,i ≤ T .
We build
X t = XT X ,i for t ∈ [T X ,i , T X ,i+1 ),
Y t = YT Y ,i for t ∈ [T Y ,i , T Y ,i+1 ).
For given h, the previous tick covariation estimator is
m
X
Vh =
X ih − X (i−1)h Y ih − Y (i−1)h .
i=1
Mathieu Rosenbaum
On Lead-Lag Estimation
9
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Drawback of this estimator
Epps effect
Systematic bias for this estimator.
Example : Assume that X and Y are two Brownian motions
with correlation ρ and that the observation times are arrival
times of two independent Poisson processes, then one can
show that
E[Vh ] → 0, as h → 0.
Mathieu Rosenbaum
On Lead-Lag Estimation
10
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
A convergent estimator under asynchronicity
Hayashi-Yoshida estimator
Let IiX = (T X ,i , T X ,i+1 ] and IiY = (T Y ,i , T Y ,i+1 ]
The Hayashi-Yoshida estimator is
X
Un =
∆X (IiX )∆Y (IjY )1{I X ∩I Y 6=∅} .
i
j
i,j
This estimator does not need any selection of h and is
convergent.
Mathieu Rosenbaum
On Lead-Lag Estimation
11
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
The lead-lag model
Let θ > 0 (for simplicity, extensions are quite straightforward) and
set Fθ = (Ftθ )t≥0 , with Ftθ = Ft−θ .
Assumptions
We have
X = X c + A, Y = Y c + B.
(Xtc )t≥0 is a continuous F-local martingale, and (Ytc )t≥0 is a
continuous Fθ -local martingale.
∃vn → 0, vn−1 max sup{|IiX |}, sup{|IiY |} → 0.
The T X ,i are Fvn -stopping times and the T Y ,i are
Fθ+vn -stopping times.
Mathieu Rosenbaum
On Lead-Lag Estimation
12
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Estimation strategy
Estimator
We set
Un (θ) =
X
∆X (IiX )∆Y (IjY )1{I X ∩(I Y )−θ 6=∅} ,
i
j
i,j
with (IjY )−θ = (T Y ,j − θ, T Y ,j+1 − θ].
Eventually, θbn is defined as a solution of
n
U (θbn ) = max U n (θ),
n
θ∈G
where G n is a sufficiently fine grid.
Mathieu Rosenbaum
On Lead-Lag Estimation
13
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Result
Theorem
As n → ∞,
vn−1 θbn − θ → 0,
e c iT 6= 0 .
in probability, on the event hX c , Y
Mathieu Rosenbaum
On Lead-Lag Estimation
14
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Outline
1
Introduction
2
Lead-Lag estimation from non synchronous data
3
Simulations
4
Real Data
5
A random matrices based approach
Mathieu Rosenbaum
On Lead-Lag Estimation
15
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Synchronous case
Setup
We consider 300 simulations of the Bachelier model with
synchronous equispaced data with period ∆n .
t ∈ [0, 1], θ = 0.1, x0 = ye0 = 0, σ1 = σ2 = 1.
The mesh size of the grid hn satisfies hn = ∆n .
We consider the following variations :
- Mesh size : hn ∈ {10−3 (FG ), 3. 10−3 (MG ), 6. 10−3 (CG )}.
- Correlation value : ρ ∈ {0.25, 0.5, 0.75}.
Mathieu Rosenbaum
On Lead-Lag Estimation
16
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Results in the synchronous case
θbn
FG, ρ = 0.75
MG, ρ = 0.75
CG, ρ = 0.75
FG, ρ = 0.50
MG, ρ = 0.50
CG, ρ = 0.50
FG, ρ = 0.25
MG, ρ = 0.25
CG, ρ = 0.25
0.096
0
0
1
0
0
13
0
0
10
0.099
0
300
0
0
299
0
0
152
0
0.1
300
0
0
300
0
0
300
0
0
0.102
0
0
299
0
1
280
0
11
66
Other
0
0
0
0
0
7
0
137
124
Table 1 : Estimation of θ = 0.1 on 300 simulated samples for
ρ ∈ {0.25, 0.5, 0.75}.
Mathieu Rosenbaum
On Lead-Lag Estimation
17
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
One sample path, FG, ρ = 0.75
0.4
0.2
0.0
Contrast
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1.0
lag
Mathieu Rosenbaum
On Lead-Lag Estimation
18
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
One sample path, FG, ρ = 0.25
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0.00
Contrast
0.10
0.15
0.20
●
0.0
0.2
0.4
0.6
0.8
1.0
lag
Mathieu Rosenbaum
On Lead-Lag Estimation
19
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
One sample path, MG, ρ = 0.75
0.2
0.3
●
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Contrast
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Mathieu Rosenbaum
On Lead-Lag Estimation
20
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
One sample path, MG, ρ = 0.25
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Mathieu Rosenbaum
On Lead-Lag Estimation
21
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
One sample path, CG, ρ = 0.75
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lag
Mathieu Rosenbaum
On Lead-Lag Estimation
22
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
One sample path, CG, ρ = 0.25
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Mathieu Rosenbaum
On Lead-Lag Estimation
23
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Non synchronous case
Setup
We randomly pick 300 sampling times for X over [0, 1],
uniformly over a grid of mesh size 10−3 .
We randomly pick 300 sampling times for Y likewise, and
independently of the sampling for X .
Fine grid case, with θ = 0.1 and ρ = 0.75.
Mathieu Rosenbaum
On Lead-Lag Estimation
24
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Results for the non synchronous case
θb
FG, ρ = 0.75
0.099
16
0.1
106
0.101
107
0.102
46
0.103
19
0.104
4
0.105
2
Table 2 : Estimation of θ = 0.1 on 300 simulated samples for
ρ = 0.75 and non-synchronous data.
Mathieu Rosenbaum
On Lead-Lag Estimation
25
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Outline
1
Introduction
2
Lead-Lag estimation from non synchronous data
3
Simulations
4
Real Data
5
A random matrices based approach
Mathieu Rosenbaum
On Lead-Lag Estimation
26
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
The data
Dataset
We study here the lead-lag relationship between the two following
assets :
The future contract on the DAX index, with maturity
December 2010,
The Euro-Bund future contract, with maturity December
2010.
Mathieu Rosenbaum
On Lead-Lag Estimation
27
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Dealing with microstructure noise
Methodology
We want to use high frequency data.
First approach : use of the Uncertainty Zones Model.
Here we just use signature plots in trading times. This enables
to take advantage of non synchronous data.
We keep one trade out of twenty.
We then compute the function U n over these trades.
Mathieu Rosenbaum
On Lead-Lag Estimation
28
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
7000
6000
Realized volatility
4000
5000
0.6
0.5
0.4
3000
0.3
0.2
Realized volatility
0.7
8000
0.8
Signature plots, October 13, for Bund (left) and FDAX
(right).
0
10
20
30
40
50
Period (in number of trades)
Mathieu Rosenbaum
0
10
20
30
40
50
Period (in number of trades)
On Lead-Lag Estimation
29
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
−2
−4
−6
−8
Contrast function
0
2
Function U n , October 13, between -10 and 10 minutes,
mesh=30 seconds
−600
−400
−200
0
200
400
600
Time shift value (seconds)
Mathieu Rosenbaum
On Lead-Lag Estimation
30
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
−8.0
−8.5
Contrast function
−7.5
−7.0
Function U n , October 13, between -5 and 5 seconds,
mesh=0.1 second
−4
−2
0
2
4
Time shift value (seconds)
Mathieu Rosenbaum
On Lead-Lag Estimation
31
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Bund and DAX, lead-lag estimation
Jour
1 Oct.
5 Oct.
6 Oct.
7 Oct.
8 Oct.
11 Oct.
12 Oct.
13 Oct.
14 Oct.
15 Oct.
Vol.(Bund)
2847
2213
2244
1897
2545
1050
2265
2018
2057
2571
Vol.(FDAX)
4215
3302
2678
3121
2852
1497
3018
3037
2625
3269
Mathieu Rosenbaum
LL.
-0.2
-1.1
-0.1
-0.5
-0.6
-1.4
-0.8
-0.8
0.0
-0.7
18
19
20
21
22
25
26
27
28
29
J.
Oct.
Oct.
Oct.
Oct.
Oct.
Oct.
Oct.
Oct.
Oct.
Oct.
On Lead-Lag Estimation
Vol.B.
1727
2527
2328
2263
1894
1501
2049
2606
1980
2262
Vol.F.
2326
3162
2554
3128
1784
2065
2462
2864
2632
2346
LL
-2.1
-1.6
-0.5
-0.1
-1.2
-0.4
-0.1
-0.6
-1.3
-1.6
32
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Outline
1
Introduction
2
Lead-Lag estimation from non synchronous data
3
Simulations
4
Real Data
5
A random matrices based approach
Mathieu Rosenbaum
On Lead-Lag Estimation
33
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Model (1)
Dynamics
We consider two processes (Xt )t∈[0,1] (leader) and (Yt )t∈[0,1]
(lagger) such that
Z t
Xt − X0 =
Ks+θ dWs+θ ,
0
Z t∧θ
Z t∨θ
Z t
Yt − Y0 = ρ
Ks dW̃s + ρ
Ks dWs +
Ls dWs0 .
0
θ
0
The interval [θ, 1] is the set of time where the lead-lag
relation is in force.
For s ∈ [θ, 1],
dYs = ρdXs−θ + Ls dWs0 .
Mathieu Rosenbaum
On Lead-Lag Estimation
34
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Model (2)
Observations
We consider m + 1 equidistant data for each process :
Xi/m , Yi/m ), for i = 0, . . . , m.
m = pbp a c where p is a positive integer and a > 0.
Later, p will be the order of magnitude of the number of
“days” the processes will be observed and m + 1 the number
of data per day. This parameter will drive the asymptotic.
Mathieu Rosenbaum
On Lead-Lag Estimation
35
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Increments
We consider increments of the processes on grids with mesh 1/p.
Notation
For i = 1, . . . , p, and l = 0, . . . , bp a c
∆(l,p) Xi = Xi/p+l/m − X(i−1)/p+l/m .
∆(0,p) Xi and ∆(l,p) Yi are centered Gaussian with variance
X
vi,0
Z
i/p
2
Ks+θ
ds,
=
Y
vi,l
(i−1)/p
Z
i/p+l/m
=
(ρ2 Ks2 + L2s )ds.
(i−1)/p+l/m
Random vector of interest :
Z (l,p) = p 1/2 ∆(0,p) X1 , . . . , ∆(0,p) Xp , ∆(l,p) Y1 , . . . , ∆(l,p) Yp−1
Mathieu Rosenbaum
On Lead-Lag Estimation
>
.
36
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Theoretical covariance
Let bθcp = bpθc /p. Z (l,p) is a Gaussian vector of size 2p − 1 with
5-diagonal covariance matrix Σ(l,p) . For l = 0, . . . , bm(θ − bθcp )c :

1 ≤ i ≤ p, 1 ≤ j ≤ p, i = j




p + 1 ≤ i ≤ 2p − 1, p + 1 ≤ j ≤ 2p − 1, i = j








(Σ(l,p) )i,j
(Σ(l,p) )i,j
(Σ(l,p) )i,j
(Σ(l,p) )i,j
(Σ(l,p) )i,j
(Σ(l,p) )i,j
1 ≤ i ≤ p, p + 1 ≤ j ≤ 2p − 1, j − p = i + p bθcp
1 ≤ i ≤ p, p + 1 ≤ j ≤ 2p, j − p = i + p bθcp + 1
p + 1 ≤ i ≤ 2p − 1, 1 ≤ j ≤ p, i − p = j + p bθcp
p + 1 ≤ i ≤ 2p − 1, 1 ≤ j ≤ p, i − p = j + p bθcp + 1
X
= pvi,0
Y
= pvj−p,l
XY
= pvi,l,1
XY
= pvi,l,2
XY
= pvj,l,1
XY
= pvj,l,2
with
XY
vi,l,1
Z
i/p−(θ−bθcp )+l/m
=ρ
2
Ks+θ
ds and
(i−1)/p
XY
vi,l,2
Z
i/p
=ρ
2
Ks+θ
ds.
i/p−(θ−bθcp )+l/m
The parameter θ appears in the location of the diagonals.
Analogous result for l = bm(θ − bθcp )c + 1, . . . , bp a c.
Mathieu Rosenbaum
On Lead-Lag Estimation
37
Introduction
Lead-Lag estimation from non synchronous data
Simulations
Real Data
A random matrices based approach
Result
Theorem
Using random matrices theory results, we can build another
estimator of the lead-lag parameter and provide its asymptotic
theory.
Mathieu Rosenbaum
On Lead-Lag Estimation
38