Finite Sample Analysis of Weighted Realized Covariance with Noisy Asynchronous Observations

Transcription

1 Finite Sample Analysis of Weighted Realized Covariance with Noisy Asynchronous Observations Taro Kanatani JSPS Fellow, Institute of Economic Research, Kyoto University October 18, 2007, Stanford 1

2 Introduction Estimation of covariance between financial assets (co-volatility, crossvolatility) Portfolio risk, etc. High-frequency data (Intraday data) e.g. Hourly data,, 1 minute data,, transaction data Realized Volatility, Realized Covariance Serious problems for using RC with too high frequency data (transaction data): 1. Nonsynchronous observation 2. Microstructure noise (observation error) Weighted Realized Covariance (WRC) 2

3 No noise and synchronous observation True logarithmic price process: dp (t) }{{} 2 1 =Σ(t) }{{} 2 2 where z(t) is standard Brownian motion. dz (t), 0 t T }{{} 2 1 (Instantaneous or spot) volatility matrix: Ω(t) Σ(t)Σ(t) element by element (12 or 21 element) ω 12 (t) = i σ 1i (t)σ 2i (t) Discrete observation time points: 0=t 0 <t 1 < <t n < <t N 1 <t N = T 3

4 Realized covariance Estimation of integrated covariance T 0 ω 12 (t)dt e.g. T =1day, Daily Covariance N plim N Δp 1 (t n )Δp 2 (t n ) = ω 12 (t)dt n=1 0 }{{} Realized covariance Intuitively, T T dp 1 (t)dp 2 (t) = σ 1i (t)σ 2i (t) dz i (t) 0 0 }{{ 2 } i dt If true prices are observed synchronously, there is no problem... T 4

5 Nonsynchronous and noisy observation Nonsynchronous observation: financial assets are traded (observed) at different time points. 1st asset: 0=t 01 <t 11 < <t n1 < <t N1 1 <t N1 = T 2nd asset: 0=t 02 <t 12 < <t n2 < <t N2 1 <t N2 = T Nonsynchronous bias Hayashi-Yoshida (2005) estimator :) Market micirostructure noise: the price is observed with noise at each point. observed price {}}{ p o i (t n i ) = true price {}}{ p i (t ni ) + noise(observation error) {}}{ e i (t ni ) where e i (t ni ) IID(0,σ 2 i ) 5

6 Effect of the noise in short interval Define r o i (t n i )=Δp o i (t n i ), r i (t ni )=Δp i (t ni ), u i (t ni )=Δe i (t ni ) ri o (t n i )= r i (t ni ) }{{} + u i (t ni ) }{{} V (r)=o p (Δt) V (u)=o(1) In short interval, the noise term swamps the true return! True return is hidden by the noise!! 6

7 How to mitigate the noise? For volatility 1. Discard data (Use low frequency data) Optimal frequency: Bandi and Russell (2005) etc 2. Realized kernel: Barndorff-Nielsen et al. (2007) Two Scale Estimator (Zhang et al., 2005) Hansen and Lunde (2005) s estimator 3. Fourier estimator: Mancino and Sanfelici (2006) For cross-volatility 1. Optimal frequency: Griffin and Oomen (2006), Bandi and Russell (2006) 2. Realized kernel:? 3. Fourier estimator:? 7

8 Weighted realized covariance where WRC = N 1 N 2 r1 o (t n 1 )r2 o (t n 2 )w n1 n 2 =(r }{{ 1 o ) } n 1 =1 n 2 =1 1 N 1 = r 1 Wr 2 + r 1 Wu 2 + u 1 Wr 2 + u 1 Wu 2, r o i =(ro i (t 1 i ),..., r o i (t n i ),..., r o i (T )), r i =(r i (t 1i ),..., r i (t ni ),..., r i (T )), u i =(u i (t 1i ),..., u i (t ni ),..., u i (T )). N 1 N 2 {}}{ W r o 2 }{{} N 2 1 WRC nests 1. Lower frequency methods 2. Realized kernel: r 2 = r 1, w 11 =... = w NN, w 12 =... = w N 1N, w 21 =... = w NN 1,... (Toeplitz matrix) 3. Fourier estimator 8

9 Bias of WRC Denote { wn diag 1 n 2 wn off 1 n 2 IC n1 n 2 = if (t n1 1,t n1 ] (t n2 1,t n2 ] otherwise. min{tn1,t n2 } max{t n1 1,t n2 1} ω 12 (t)dt, IC = 0 otherwise. Bias of WRC: E[WRC IC]= n 1,n 2 (w diag T 0 ω 12 dt, if (t n1 1,t n1 ] (t n2 1,t n2 ] n 1 n 2 1)IC n1 n 2 If w diag n 1 n 2 =1for all n 1,n 2, WRC is unbiased. (w diag n 1 n 2 =1and w off n 1 n 2 = 0 Hayashi and Yoshida (2005) estimator) 9

10 MSE of WRC where E[WRC IC] 2 = E[r 1 Wr 2 IC] 2 }{{} A + E[r 1 Wu 2] 2 }{{} B + E[u 1 Wr 2] 2 }{{} C A = ( w diag n 1 n 2 IC n1 n 2 ) 2 + w 2 n1 n 2 IV n1 IV n2 + Bias 2 + E[u 1 Wu 2] 2 }{{} D B = σ 2 2 wn1 n 2 (2w n1 n 2 w n1 n 2 1 w n1 n 2 +1)IV n1 (IV ni C = σ 2 1 wn1 n 2 (2w n1 n 2 w n1 1n 2 w n1 +1n 2 )IV n2 tni t ni 1 ω ii dt) D = σ 2 1 σ2 2 wn1 n 2 { 4wn1 n 2 + w n1 1n w n1 1n w n1 +1n 2 1 +w n1 +1n (w n1 1n 2 + w n1 +1n 2 + w n1 n w n1 n 2 +1) } 10

11 For feasible evaluation of MSE: weight function A specific form of weight function: Hayashi-Yoshida, Fourier estimator, Error function weight, Kernels We limit our discussion within unbiased estimators (wn diag 1 n 2 =1). Bias =0and ( wn diag ) 2 1 n 2 IC n1 n 2 is unknown constant therefore we never need to evaluate it when minimizing MSE! 11

12 For feasible evaluation of MSE: Approximation and estimation Assumption: Volatility does not change so much over [0,T] (Bandi and Russell, 2005) where IV i = T 0 ω ii (t)dt. IV ni IV iδt ni T Estimates of σ 2 i, IV i Bandi and Russell (2005), Barndorff-Nielsen et al. (2007) etc. 12

13 Minimization of feasible MSE where θ = arg min θ (A + B + C + D ) A = T 2 IV 1 IV 2 w 2 n1 n 2 Δt n1 Δt n2 B = T 1 IV 1 σ2 2 wn1 n 2 (2w n1 n 2 w n1 n 2 1 w n1 n 2 +1)Δt n1 C = T 1 IV 2 σ1 2 wn1 n 2 (2w n1 n 2 w n1 1n 2 w n1 +1n 2 )Δt n2 D = σ1 2 { σ2 2 wn1 n 2 4wn1 n 2 + w n1 1n w n1 1n w n1 +1n 2 1 +w n1 +1n (w n1 1n 2 + w n1 +1n 2 + w n1 n w n1 n 2 +1) } { 1 if (tn1 w n1 n 2 = 1,t n1 ] (t n2 1,t n2 ] f(t n1,t n2 ; θ) otherwise, 13

14 An example of WRC: Fourier estimator Fourier estimator (Malliavin and Mancino, 2002) 1 if t n1 = t n2 w n1 n 2 = otherwise, sin (n+1)(t n 1 tn 2 ) 2 cos n(t n 1 tn 2 ) 2 n sin (t n 1 tn 2 ) 2 where n is the number of Fourier coefficients. The (nonsynchronous) bias corrected version is w n1 n 2 = otherwise. 1 if (t n1 1,t n1 ] (t n2 1,t n2 ] sin (n+1)(t n 1 tn 2 ) 2 cos n(t n 1 tn 2 ) 2 n sin (t n 1 tn 2 ) 2 14

15 An example of WRC: Error function weight Error function weight 1 ( ) if (t n1 1,t n1 ] (t n2 1,t n2 ] w n1 n 2 = exp (t n 1 t n2 ) 2 h otherwise. where h>0. Extreme cases h 0, WRC = HY h, WRC =(r1 o) 1 N1 1 N r o 2 2 =(po 1 (T ) po 1 (0))(po 2 (T ) po 2 (0)) 15

16 An example of WRC: kernels in BHLS(2007) unbiased kernel w n1 n 2 = 1 ( ) if (t n1 1,t n1 ] (t n2 1,t n2 ], tn1 t n2 k if t n1 t n2 <H, H 0 otherwise. k(x) Bartlett 1 x Epanechnikov 1 x 2 Parzen 1 6x 2 +6x 3 (0 x 1/2) 2(1 x) 3 (1/2 <x 1) Tukey-Hanning (1 + cos(πx))/2 Mod. Tukey-Hanning (1 cos π(1 x) 2 )/2 Tukey-Hanning p sin 2 (π(1 x) p /2) 16

17 An example of WRC: Hayashi-Yoshida estimator Hayashi-Yoshida (2005) estimator: HY = n 1,n 2 r o 1 (t n 1 )r o 2 (t n 2 )I{(t n1 1,t n1 ] (t n2 1,t n2 ] }, where I{ } is indicator function. Hayashi-Yoshida weight w n1 n 2 = { 1 if (tn1 1,t n1 ] (t n2 1,t n2 ] 0 otherwise. 17

18 Griffin and Oomen (2006) estimator (Lower frequency version of HY): HY (k) = kn 1,kn 2 Δ k p o 1 (t kn 1 )Δ k p o 2 (t kn 2 )I{(t kn1 k,t kn1 ] (t kn2 k,t kn2 ] }, where k is a positive integer. Weight matrix: r o 1 (k):[n 1/k] 1 { }} { kn1 =1 r i(t n1 ) 2k n1 =k+1 r i(t n1 ). = where 1 k =(1 1), 0 k =(0 0). A 1 (k):[n 1 /k] N 1 {}}{ 1 k 0 N1 k 0 k 1 k 0 N1 2k r 1... HY (k) =(r o 1 (k)) D(k)r o 2 (k) =(ro 1 ) W {}}{ A 1 (k) D(k)A 2 (k) r o 2 18

19 An example of WRC: Subsampling estimator Subsampling HY: r o 1 (k 1) i { }}{ in1 =1 r i(t n1 ) i+k n 1 =i+1 r i(t n1 ). = A i 1 (k 1) { }}{ 1 i 0 N1 i 0 1 k 0 N1 k i r 1... HY ss (k1,k2) = (k 1 k 2 ) 1 i,j (r o 1 (k 1) i ) D(k 1,k 2 ) ij r o 2 (k 2) j =(r1 o ) i,j Ai 1 (k 1) D(k 1,k 2 ) ij A j 2 (k 2) r2 o } k 1 {{ k 2 } W D(k 1,k 2 ) W (k 1,k 2 ) can reduece the MSE? 19

20 Monte Carlo study Efficient price process: ( ) ( )( ) dp1 (t) σ11 (t) 0 dz1 (t) =, 0 t T dp 2 (t) σ 21 (t) σ 22 (t) dz 2 (t) dσ ij (t) =κ ( θ σ ij (t) ) dt + γdz ij (t), i,j =1, 2. where κ =0.1, θ =1, γ =0.1, T =1(day), Δ=1/ (one second precision for Japanese stock exchanges). Time differences are drawn from an exponential distribution: F ( t ni t ni 1) =1 exp { λi ( tni t ni 1)}, i =1, 2 where F ( ) denotes a cumulative distribution function, λ i =1/60Δ. (Average duration is 60 seconds) Independent noise: e 1 (t n1 ) NID(0, 0.025), e 2 (t n2 ) NID(0, 0.05) 20

21 Monte Carlo result 1 λ i Δ=1/60, e 1 (t n1 ) NID(0, 0.025), e 2 (t n2 ) NID(0, 0.05) Sample MSE Ave. of optimal parameter HY LHY (k ) k =7.37 MFE(n ) n =12.4 EF(h ) h = BAR(H ) H = EPA(H ) H = PAR(H ) H = TH(H ) H = MTH(H ) H = Sample MSE =(1/500) 500 r=1 (estimate (r) IC (r)) 2 21

22 Monte Carlo result 2 λ i Δ=1/60, e 1 (t n1 ) NID(0, 0.005), e 2 (t n2 ) NID(0, 0.01) Sample MSE Ave. of optimal parameter HY LHY (k ) k =1 MFE(n ) n =25.3 EF(h ) h = BAR(H ) H = EPA(H ) H = PAR(H ) H = TH(H ) H = MTH(H ) H = Sample MSE =(1/500) 500 r=1 (estimate (r) IC (r)) 2 22

23 Monte Carlo result 3 λ i Δ=1/15, e 1 (t n1 ) NID(0, 0.005), e 2 (t n2 ) NID(0, 0.01) Sample MSE Ave. of optimal parameter HY LHY (k ) k =6.05 MFE(n ) n =49.5 EF(h ) h = BAR(H ) H = EPA(H ) H = PAR(H ) H = TH(H ) H = MTH(H ) H = Sample MSE =(1/500) 500 r=1 (estimate (r) IC (r)) 2 23

24 MSE(-constant) a: λ i Δ=1/60, e 1 (t n1 ) NID(0, 0.025), e 2 (t n2 ) NID(0, 0.05) b: λ i Δ=1/60, e 1 (t n1 ) NID(0, 0.005), e 2 (t n2 ) NID(0, 0.01) c: λ i Δ=1/15, e 1 (t n1 ) NID(0, 0.005), e 2 (t n2 ) NID(0, 0.01) 24

25 MSE(-constant) a: λ i Δ=1/60, e 1 (t n1 ) NID(0, 0.025), e 2 (t n2 ) NID(0, 0.05) b: λ i Δ=1/60, e 1 (t n1 ) NID(0, 0.005), e 2 (t n2 ) NID(0, 0.01) c: λ i Δ=1/15, e 1 (t n1 ) NID(0, 0.005), e 2 (t n2 ) NID(0, 0.01) 25

26 MSE(-constant) a: λ i Δ=1/60, e 1 (t n1 ) NID(0, 0.025), e 2 (t n2 ) NID(0, 0.05) b: λ i Δ=1/60, e 1 (t n1 ) NID(0, 0.005), e 2 (t n2 ) NID(0, 0.01) c: λ i Δ=1/15, e 1 (t n1 ) NID(0, 0.005), e 2 (t n2 ) NID(0, 0.01) 26

27 MSE(-constant) κ 0.1κ, γ 10γ a: λ i Δ=1/60, e 1 (t n1 ) NID(0, 0.025), e 2 (t n2 ) NID(0, 0.05) b: λ i Δ=1/60, e 1 (t n1 ) NID(0, 0.005), e 2 (t n2 ) NID(0, 0.01) c: λ i Δ=1/15, e 1 (t n1 ) NID(0, 0.005), e 2 (t n2 ) NID(0, 0.01) 27

28 MSE(-constant) κ 0.1κ, γ 10γ a: λ i Δ=1/60, e 1 (t n1 ) NID(0, 0.025), e 2 (t n2 ) NID(0, 0.05) b: λ i Δ=1/60, e 1 (t n1 ) NID(0, 0.005), e 2 (t n2 ) NID(0, 0.01) c: λ i Δ=1/15, e 1 (t n1 ) NID(0, 0.005), e 2 (t n2 ) NID(0, 0.01) 28

29 MSE(-constant) κ 0.1κ, γ 10γ a: λ i Δ=1/60, e 1 (t n1 ) NID(0, 0.025), e 2 (t n2 ) NID(0, 0.05) b: λ i Δ=1/60, e 1 (t n1 ) NID(0, 0.005), e 2 (t n2 ) NID(0, 0.01) c: λ i Δ=1/15, e 1 (t n1 ) NID(0, 0.005), e 2 (t n2 ) NID(0, 0.01) 29

30 Summary Contribution: A framework for evaluating finite sample MSE in the presence of the noise more efficient examples than existing methods (LHY) Remaining works: More general weight Asymptotic theory 30