Quantitative Finance II Lecture 10

Transcription

1 Quantitative Finance II Lecture 10 Wavelets III - Applications Lukas Vacha IES FSV UK May 2016

2 Outline Discrete wavelet transformations: MODWT Wavelet coherence: daily financial data FTSE, DAX, PX Wavelet variance decomposition Wavelet-based realized variance The wavelet realized covariance estimator Unit Root and Cointegration Tests with Wavelets

3 Maximal Overlap Discrete Wavelet Transform (MODWT) The MODWT of level J performs a decomposition of a given signal X (as the DWT). All wavelet and scaling coefficient vectors yielded by the MODWT are in R N, with N the length of the decomposed time series X. The MODWT is associated with zero phase filters, that is, its filtered output detail and approximation series are shift-invariant, i.e., it is not sensitive to the choice of a starting point. The MODWT has advantages over DWT in wavelet variance estimation (Percival 1995)

4 MODWT filters There are three basic properties that both the MODWT wavelet filter must fulfill: L 1 h l = 0, l=0 L 1 h l 2 = 1/2, l=0 l= h l h l+2n = 0, n Z N, (1) for the MODWT scaling filter: L 1 l=0 g l = 1, L 1 l=0 g 2 l = 1/2, l= g l g l+2n = 0, n Z N. (2) where L is the length of the filter. For example Daubechies D(4) wavelet filter has length L = 4.

5 MODWT pyramid algorithm Circular filtering time series X using the MODWT filters without subsampling. For a time series X of length N we obtain the MODWT wavelet and scaling coefficients on k = 0,..., N 1 via circular filtering with the MODWT wavelet and scaling filters ( h, g): L 1 W 1,k h l X k lmodn, (3) Ṽ 1,k l=0 L 1 l=0 g l X k lmodn (4)

6 MODWT pyramid algorithm II For the second stage, we replace X with the scaling coefficients Ṽ1 L 1 W 2,k h l Ṽ 1,k lmodn, (5) Ṽ 2,k l=0 L 1 l=0 g l Ṽ 1,k lmodn (6) After two stages we have: W = ( W 1, W 2, Ṽ2)

7 Inverse MODWT Inverse of the MODWT from the first level decomposition can be obtained as: L 1 L 1 X k = h l W1,k+lmodN + g l Ṽ 1,k+lmodN, (7) l=0 l=0 for k = 0, 1,..., N 1.

8 Quadratic variation Quadratic variation: t QV t,h = σs 2 ds + Js 2. (8) t h }{{} t h s t }{{} IV t,h Jump Variation The estimator of realized variance over [t h, t], for 0 h t T, is defined by RV t,h = N r t h+( 2 i N )h, (9) i=1 where N is the number of observations in [t h, t].

9 Variance decomposition using wavelets The variance (energy) of the time series X k, k = 0,..., N 1 can be decomposed on a scale-by-scale bases J log 2 N so that X 2 = J W j 2 + ṼJ 2 (10) j=1 where X 2 = N 1 k=0 X k 2, W j 2 = N 1 k=0 W j,k 2, ṼJ 2 = N 1 k=0 V J,k 2 and W j and Ṽj are N dimensional vectors of the j-th level wavelet and scaling MODWT coefficients. (Percival and Mofjeld JASA 1997).

10 Wavelet-based realized variance The wavelet-based realized variance over [t h, t], for 0 h t T, is defined by RV (WRV ) J m +1 t,h = j=1 N k=1 W 2 j,t h+ k h, (11) N where N is the number of intraday observations in [t h, t] and J m is the number of scales we consider. W j,t h+ k N h are the MODWT coefficients on returns data r t,h on scales j = 1,..., J m + 1, where J m log 2 N. Example: Barunik, J., Vacha, L. (2015). Realized wavelet-based estimation of integrated variance and jumps in the presence of noise. Application of TSRV and jumps detection.

11 Table: Bias (variance in parenthesis) 10 4 of all estimators from 10,000 simulations of jump-diffusion model with ɛ 1 = 0, ɛ 2 = , ɛ 3 = 0.001, ɛ 4 = RV 5 min. realized variance estimator, BV 5 min. bipower variation estimator, TSRV 5 min. two-scale realized volatility, JWTSRV 5 min. jump wavelet two-scale realized variance. TSRV and JWTSRV are minimum variance estimators, and RK is Realized Kernel. RV BV TSRV TSRV RK JWTSRV No Jumps ɛ (0.65) (0.82) (0.43) (0.02) (2.51) (0.43) ɛ (0.93) (1.18) (0.45) 0.98 (0.51) (2.63) (0.45) ɛ (2.10) (2.87) (0.45) (0.90) (2.91) 0.19 (0.48) ɛ (5.40) (8.00) (0.43) (1.34) (3.13) 7.71 (0.58) One Jump ɛ (19.31) (1.85) (18.64) (18.09) (23.19) (0.44) ɛ (20.91) (2.77) (19.67) (19.61) (23.10) (0.48) ɛ (23.12) (5.15) (19.79) (20.44) (25.62) (0.64) ɛ (27.54) (10.79) (20.30) (21.02) (25.25) (1.41) Two Jumps ɛ (41.12) (3.84) (39.47) (38.99) (47.36) (0.43) ɛ (41.99) (4.56) (39.51) (39.69) (45.65) 3.43 (0.49) ɛ (44.71) (7.67) (39.83) (39.52) (48.27) (0.81) ɛ (47.55) (15.04) (39.15) (38.93) (47.75) (2.34) 1 Three Jumps ɛ (62.38) (6.58) (60.11) (58.86) (72.17) (0.46) ɛ (61.60) (7.34) (58.62) (58.56) (68.89) 6.04 (0.51) ɛ (68.71) (10.71) (61.89) (60.72) (74.86) (0.95) ɛ (69.52) (18.55) (58.93) (59.37) (71.48) (3.19) 1

12 Estimation of integrated covariation The quadratic covariation of jump processes {X t } and {Y t } is defined as X, Y T = T 0 σ (X ) t σ (Y ) t d B (X ), B (Y ) t + 0 t T J (X ) t J (Y ) t. (12) Thus, the quadratic covariation is composed from the continuous part (integrated covariance) IC T and sum of co-jumps. The term J (X ) t J (Y ) t is non-zero only if co-jump occurs, i.e., both J (X ) t and J (Y ) t are non-zero.

13 Integrated covariance The integrated covariance, IC T, over a time horizon [0 t T ] of the two price processes {X t } and {Y t } without jumps and noise, (Itô process). A limit theorem for stochastic processes states that realized covariance ÎC (RC) T = N (X ti X ti 1)(Y ti Y ti 1), t (0, T ) (13) i=1 is a consistent estimator for X, Y T as the numer of intraday observations, i = 1,..., N, goes to infinity.

14 The wavelet realized covariance estimator The wavelet realized covariance estimator return processes X t and Y t in L 2 (R) over a time horizon [0 t T ] as: ÎC (WRC) J m +1 T = j=1 N i=1 ÎC (WRC) T of the asset W (x) j,t i W (y) j,t i, (14) where N is the number of intraday observations and J m log 2 N is the number of scales considered. W (.) j,t i are the MODWT coefficients on scale j, unaffected by the boundary conditions.

15 Jump detection If for some W (X ) 1,t i obtained on the price process X t over a time horizon [0 t T ]: J (X ) t i = (J (X ) t i J (X ) t i 1 )1 { W (X ) 1,t i >d t 2 log N} t [0, T ] (15) then at the time position {t i } is the estimated jump location with the jump size J (X ) t i.

16 Jump variation d t denotes the intraday median absolute deviation estimator defined as: d t = 2 median{ W 1,t }, t [0, T ], (16) where W 1,t is a vector of wavelet coefficients at a day t. The jump variation over [0 t T ] in the discrete synchronized time is estimated as the sum of squares of all the estimated jump sizes: ĴV (X ) T = N i=1 ( J (X ) t i ) 2 t [0, T ]. (17)

17 Co-jumps The jump adjusted price process is defined as: X (J) (X ) = X ĴV T We detect all jumps in the X t and Y t price processes separately using wavelet decomposition, and then we estimate the co-jumps J (X ) t J (Y ) t. Let us define the co-jump sum simply as: N i=1 J (X ) t i J (Y ) t i t [0, T ], (18) therefore co-jump occurs only when at a time time position i in day t both jump at process X t and Y t occurs simultaneously. see Barunik, J., Vacha L. (2016) Do co-jumps impact correlations in currency markets?

18 Daily financial data Indices: DAX, FTSE and less liquid Czech PX Higher resolution, more information

19 Wavelet coherence daily returns

20 Unit Root and Cointegration Tests with Wavelets Fan, Y., Genay, R. (2006, April). Unit root and cointegration tests with wavelets. In Canadian Econometric Study Group Meeting, McGill University, Montreal, September (pp ). Wavelet approach is used to test the presence of a unit root in a stochastic process. The wavelet test is based directly on the different behavior of the spectra of a unit root process and that of a short memory stationary process. Test has substantial power against near unit root alternatives.

21 References Mallat, S. (1989) A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence 11, Ramsey, J.B. (2002) Wavelets in economics and finance: Past and future. Studies in Nonlinear Dynamics & Econometrics 3, 129. Daubechies, I. (1992) Ten Lectures on Wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM. Fan, Y. (2003) On the approximate decorrelation property of the discrete wavelet transform for fractionally differenced processes. IEEE Transactions on Information Theory 49, Percival, D.B. (1995) On estimation of the wavelet variance. Biometrica 82, Percival, D.B. and H.O. Mofjeld (1997) Analysis of subtidal coastal sea level fluctuations using wavelets. Journal of the American Statistical Association 92, Percival, D.B. and A.T. Walden (2000) Wavelet Methods for Time Series Analysis. Cambridge Press.