On Lead-Lag Estimation

Transcription

1 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

2 Outline 1 Introduction Mathieu Rosenbaum On Lead-Lag Estimation 2

3 Outline 1 Introduction Mathieu Rosenbaum On Lead-Lag Estimation 3

4 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

5 A toy model for Lead-Lag Bachelier model For t [0, 1], and (B (1), B (2) ) such that B (1), B (2) t = ρt, set X t := x 0 + σ 1 B (1) t, Ỹ t := ỹ 0 + σ 2 B (2) t, Define Y t := Ỹt θ, t [θ, 1]. Our lead-lag model is given by the bidimensional process (X t, Y t ). We have { X t = x 0 + σ 1 B (1) t Y t = y 0 + ρ σ 2 B (1) t θ + σ 2(1 ρ 2 ) 1/2 W t θ. Mathieu Rosenbaum On Lead-Lag Estimation 5

6 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 (X 0, Y 0 ), (X n, Y n ), (X 2 n, Y 2 n ),..., (X 1, Y 1 ), and suppose θ = k 0 n, k 0 Z. Let C n (k) := i ( Xi n X (i 1) n )( Y(i+k) n Y (i+k 1) n ). Mathieu Rosenbaum On Lead-Lag Estimation 6

7 Intuitive estimator in the Bachelier model (2) Estimation idea (2) Heuristically, we have C n (k) 1 n E [ (X X n )(Y +k n Y +(k 1) n ) ] + 1/2 n ξ n. Moreover, 1 n E [ (X X n )(Y +k n Y +(k 1) n ) ] = { 0 if k k0 ρ σ 1 σ 2 if k = k 0. Thus we can (asymptotically) detect the value k 0 that defines θ in the very special case θ = k 0 n by maximizing in k the contrast sequence k Cn (k). Mathieu Rosenbaum On Lead-Lag Estimation 7

8 Outline 1 Introduction Mathieu Rosenbaum On Lead-Lag Estimation 8

9 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 = X T X,i for t [T X,i, T X,i+1 ), Y t = Y T Y,i for t [T Y,i, T Y,i+1 ). For given h, the previous tick covariation estimator is m ( ) ( ) V h = X ih X (i 1)h Y ih Y (i 1)h. i=1 Mathieu Rosenbaum On Lead-Lag Estimation 9

10 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[V h ] 0, as h 0. Mathieu Rosenbaum On Lead-Lag Estimation 10

11 A convergent estimator under asynchronicity Hayashi-Yoshida estimator Let Ii X = (T X,i, T X,i+1 ] and Ii Y = (T Y,i, T Y,i+1 ] The Hayashi-Yoshida estimator is U n = i,j X (I X i ) Y (I Y j )1 {I X i Ij Y }. This estimator does not need any selection of h and is convergent. Mathieu Rosenbaum On Lead-Lag Estimation 11

12 The lead-lag model Let θ > 0 (for simplicity, extensions are quite straightforward) and set F θ = (F θ t ) t 0, with F θ t = F t θ. Assumptions We have X = X c + A, Y = Y c + B. (X c t ) t 0 is a continuous F-local martingale, and (Y c t ) t 0 is a continuous F θ -local martingale. v n 0, v 1 n max { sup{ Ii X }, sup{ Ii Y } } 0. The T X,i are F vn -stopping times and the T Y,i are F θ+vn -stopping times. Mathieu Rosenbaum On Lead-Lag Estimation 12

13 Estimation strategy Estimator We set U n (θ) = i,j X (I X i ) Y (I Y j )1 {I X i (Ij Y ) θ }, with (Ij Y ) θ = (T Y,j θ, T Y,j+1 θ]. Eventually, θ n is defined as a solution of U n ( θ n ) = max θ G n U n (θ), where G n is a sufficiently fine grid. Mathieu Rosenbaum On Lead-Lag Estimation 13

14 Result Theorem As n, vn 1 ( θn θ ) 0, in probability, on the event { X c, Ỹ c T 0 }. Mathieu Rosenbaum On Lead-Lag Estimation 14

15 Outline 1 Introduction Mathieu Rosenbaum On Lead-Lag Estimation 15

16 Synchronous case Setup We consider 300 simulations of the Bachelier model with synchronous equispaced data with period n. t [0, 1], θ = 0.1, x 0 = ỹ 0 = 0, σ 1 = σ 2 = 1. The mesh size of the grid h n satisfies h n = n. We consider the following variations : - Mesh size : h n {10 3 (FG), (MG), (CG)}. - Correlation value : ρ {0.25, 0.5, 0.75}. Mathieu Rosenbaum On Lead-Lag Estimation 16

17 Results in the synchronous case θ n Other FG, ρ = MG, ρ = CG, ρ = FG, ρ = MG, ρ = CG, ρ = FG, ρ = MG, ρ = CG, ρ = 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

18 One sample path, FG, ρ = 0.75 Contrast lag Mathieu Rosenbaum On Lead-Lag Estimation 18

19 One sample path, FG, ρ = lag Contrast Mathieu Rosenbaum On Lead-Lag Estimation 19

20 One sample path, MG, ρ = 0.75 Contrast lag Mathieu Rosenbaum On Lead-Lag Estimation 20

21 One sample path, MG, ρ = lag Contrast Mathieu Rosenbaum On Lead-Lag Estimation 21

22 One sample path, CG, ρ = 0.75 Contrast lag Mathieu Rosenbaum On Lead-Lag Estimation 22

23 One sample path, CG, ρ = lag Contrast Mathieu Rosenbaum On Lead-Lag Estimation 23

24 Non synchronous case Setup We randomly pick 300 sampling times for X over [0, 1], uniformly over a grid of mesh size 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 = Mathieu Rosenbaum On Lead-Lag Estimation 24

25 Results for the non synchronous case θ FG, ρ = 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

26 Outline 1 Introduction Mathieu Rosenbaum On Lead-Lag Estimation 26

27 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 Mathieu Rosenbaum On Lead-Lag Estimation 27

28 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

29 Signature plots, October 13, for Bund (left) and FDAX (right). Realized volatility Realized volatility Period (in number of trades) Period (in number of trades) Mathieu Rosenbaum On Lead-Lag Estimation 29

30 Function U n, October 13, between -10 and 10 minutes, mesh=30 seconds Contrast function Time shift value (seconds) Mathieu Rosenbaum On Lead-Lag Estimation 30

31 Function U n, October 13, between -5 and 5 seconds, mesh=0.1 second Contrast function Time shift value (seconds) Mathieu Rosenbaum On Lead-Lag Estimation 31

32 Bund and DAX, lead-lag estimation Jour Vol.(Bund) Vol.(FDAX) LL. J. Vol.B. Vol.F. LL 1 Oct Oct Oct Oct Oct Oct Oct Oct Oct Oct Oct Oct Oct Oct Oct Oct Oct Oct Oct Oct Mathieu Rosenbaum On Lead-Lag Estimation 32

33 Outline 1 Introduction Mathieu Rosenbaum On Lead-Lag Estimation 33

34 Model (1) Dynamics We consider two processes (X t ) t [0,1] (leader) and (Y t ) t [0,1] (lagger) such that X t X 0 = Y t Y 0 = ρ t K s+θ dw s+θ, 0 t θ 0 t θ t K s d W s + ρ K s dw s + L s dw s. θ 0 The interval [θ, 1] is the set of time where the lead-lag relation is in force. For s [θ, 1], dy s = ρdx s θ + L s dw s. Mathieu Rosenbaum On Lead-Lag Estimation 34

35 Model (2) Observations We consider m + 1 equidistant data for each process : ( Xi/m, Y i/m ), for i = 0,..., m. m = p p a 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

36 Increments We consider increments of the processes on grids with mesh 1/p. Notation For i = 1,..., p, and l = 0,..., p a (l,p) X i = X i/p+l/m X (i 1)/p+l/m. (0,p) X i and (l,p) Y i are centered Gaussian with variance i/p vi,0 X = Ks+θ 2 ds, (i 1)/p i/p+l/m v i,l Y = (ρ 2 Ks 2 + L 2 s )ds. (i 1)/p+l/m Random vector of interest : Z (l,p) = p 1/2( (0,p) X 1,..., (0,p) X p, (l,p) Y 1,..., (l,p) Y p 1 ). Mathieu Rosenbaum On Lead-Lag Estimation 36

37 Theoretical covariance v XY Let θ p = pθ /p. Z (l,p) is a Gaussian vector of size 2p 1 with 5-diagonal covariance matrix Σ (l,p). For l = 0,..., m(θ θ p ) : 1 i p, 1 j p, i = j (Σ (l,p) ) i,j = pvi,0 X p + 1 i 2p 1, p + 1 j 2p 1, i = j (Σ (l,p) ) i,j = pvj p,l Y 1 i p, p + 1 j 2p 1, j p = i + p θ p (Σ (l,p) ) i,j = pv XY i,l,1 1 i p, p + 1 j 2p, j p = i + p θ p + 1 (Σ (l,p) ) i,j = pvi,l,2 XY p + 1 i 2p 1, 1 j p, i p = j + p θ p (Σ (l,p) ) i,j = pvj,l,1 XY p + 1 i 2p 1, 1 j p, i p = j + p θ p + 1 with i,l,1 = ρ i/p (θ θ p )+l/m (i 1)/p (Σ (l,p) ) i,j = pv XY j,l,2 i/p Ks+θ 2 XY ds and vi,l,2 = ρ Ks+θ 2 ds. i/p (θ θ p )+l/m The parameter θ appears in the location of the diagonals. Analogous result for l = m(θ θ p ) + 1,..., p a. Mathieu Rosenbaum On Lead-Lag Estimation 37

38 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