The Relative Contribution of Jumps to Total Price Variance Xin - PowerPoint PPT Presentation

The Relative Contribution of Jumps to Total Price Variance Xin Huang, Duke University George Tauchen, Duke University

Jumps and Price Variance June 25, 2005 Outline 1. Introduction 2. Jump Detection Tests 3. Monte Carlo Analysis 4. Empirical Results 5. Jump Tests under Market Microstructure Noise 6. Conclusions 2

Jumps and Price Variance June 25, 2005 • Examine the finite sample properties of the new jump detection tests at the daily level. • Consider a relative jump measure. • Empirical work on five minute returns on S&P Index, cash 1997-2002, and futures 1982-2002. • Introduce market microstructure noise into the price process and study the performance of the jump tests. 3

Jumps and Price Variance June 25, 2005 2. Jump Detection Tests • Dynamics of the log price process dp ( t ) = µ ( t ) dt + σ ( t ) dw ( t ) + d L J ( t ) , where L J ( t ) − L J ( s ) = � s ≤ τ ≤ t κ ( τ ) . • Within-day geometric returns r t,j = p ( t − 1+ j/M ) − p ( t − 1+( j − 1) /M ) , j = 1 , 2 , . . . , M. 4

Jumps and Price Variance June 25, 2005 • Two realized measures: – Realized Variance M � r 2 RV t = t,j j =1 – Realized Bipower Variation M M � BV t = µ − 2 1 ( M − 1) | r t,j || r t,j − 1 | , j =2 where µ a = E ( | Z | a ) , Z ∼ N (0 , 1) , a > 0 . 5

Jumps and Price Variance June 25, 2005 Key properties � t N t � σ 2 ( s ) ds + κ 2 M →∞ RV t = lim t,j , t − 1 j =1 � t σ 2 ( s ) ds M →∞ BV t = lim t − 1 where N ( t ) is the number of jumps from t − 1 to t . 6

Jumps and Price Variance June 25, 2005 Thus the difference RV t − BV t is a measure of the jump component. The jump test statistics are of the form RV t − BV t � Avar ( RV t − BV t ) 7

Jumps and Price Variance June 25, 2005 The joint asymptotic distribution under the null of no jumps is given by Barndorff-Nielsen and Shephard � v qq � − 1 � t �� t � � t − 1 σ 2 ( s ) ds 2 RV t − � v qb 1 D σ 4 ( s ) ds × → N (0 , ) M 2 � t t − 1 σ 2 ( s ) ds v qb v bb BV t − t − 1 = 2 v qq = 2 v qb 2 ) 2 + π − 3 ( π = v bb Generalizations available based on earlier work by Jacod (1994) and Jacod and Protter (1998). 8

Jumps and Price Variance June 25, 2005 Observation: The situation is that of the Hausman test: Under the maintained assumption of no jumps, the returns are asymptotically Gaussian and RV t is the most efficient estimator of the integrated variance. BV t is a less efficient (but more robust) estimate and RV t − BV t is independent of RV t conditional on the instantaneous volatility process. 9

Jumps and Price Variance June 25, 2005 Thus the Relative Jump RJ t = RV t − BV t RV t is asymptotically the ratio of two conditionally independent random variables. The Relative Jump measure is equivalent to the ratio statistic studied by Barndorff-Nielsen and Shephard (2005). 10

Jumps and Price Variance June 25, 2005 In order to studentize RV t − BV t or RJ t one needs to estimate the integrated quarticity � t σ 4 ( s ) ds t − 1 Andersen, Bollerslev, and Diebold (2004) suggest using the jump-robust realized Tri-Power Quarticity statistic, a special case of the multipower variation studied by Barndorff-Nielsen and Shephard (2004a). 11

Jumps and Price Variance June 25, 2005 Daily Statistics Raw form RV t − BV t z TP,t = � ( v bb − v qq ) 1 M TP t RV t − BV t z QP,t = � ( v bb − v qq ) 1 M QP t 12

Jumps and Price Variance June 25, 2005 Ratio adjusted RV t − BV t RV t z TP,r,t = � ( v bb − v qq ) 1 TP t BV 2 M t RV t − BV t RV t z QP,r,t = � QP t ( v bb − v qq ) 1 BV 2 M t 13

Jumps and Price Variance June 25, 2005 Max version RV t − BV t RV t z TP,rm,t = � ( v bb − v qq ) 1 M max(1 , TP t t ) BV 2 RV t − BV t RV t z QP,rm,t = � M max(1 , QP t ( v bb − v qq ) 1 t ) BV 2 14

Jumps and Price Variance June 25, 2005 3. Monte Carlo Analysis Data generation processes: • SV1F dp ( t ) = µ dt + exp [ β 0 + β 1 v ( t )] dw p ( t ) dv ( t ) = α v v ( t ) dt + dw v ( t ) • SV1FJ dp ( t ) = µ dt + exp [ β 0 + β 1 v ( t )] dw p ( t ) + d L J ( t ) dv ( t ) = α v v ( t ) dt + dw v ( t ) 15

Jumps and Price Variance June 25, 2005 SV2F A two-factor continuous model: dp ( t ) = µ dt + s-exp [ β 0 + β 1 v 1 ( t ) + β 2 v 2 ( t )] dw p ( t ) dv 1 ( t ) = α v 1 v 1 ( t ) dt + dw v 1 ( t ) dv 2 ( t ) = α v 2 v 2 ( t ) dt + [1 + β v 2 v 2 ( t )] dw v 2 ( t ) Need to handle regularity conditions by splicing the growth condition onto the exponential. Use Chernov et al (2003) and some fiddling to determine parameter values. 16

Jumps and Price Variance June 25, 2005 Monte Carlo Simulation Details • Euler Clock: – 1 tick per second, or 60 ticks per minute. – 390 minutes (6.5 trading hours) per day. • Returns: – Simulate log price process. – Compute the 1-minute, 3-minute, 5-minute and 30-minute geometric returns. • Sample Size: 45,000 simulation days. 17

Jumps and Price Variance June 25, 2005 Monte Carlo Findings: Daily Statistics • Size – The raw z statistic over rejects in the 2.00–3.00 range. – Lower sampling frequency increases size distortion. – The log-adjustment improves over the raw z . – The ratio-adjustment and the max adjustment properly corrects the size. – The choice of TP t versus QP t matters little. 18

Jumps and Price Variance June 25, 2005 • Jump detection – The log and ratio-adjusted z statistics detect jumps. – Mean reversion has little impact. – Jump size positively affects the rejection frequency. – Increase in sampling interval decreases jump detection rate. – Increase in jump intensity above one increases jump detection rate. 19

Jumps and Price Variance June 25, 2005 • Power – Jump intensity increases power. – Jump size increases power. – Sampling frequency increases power. – The tests are inconsistent. 20

Jumps and Price Variance June 25, 2005 z TP,t Quantiles of Input Sample 10 5 0 −5 −5 −4 −3 −2 −1 0 1 2 3 4 5 z TP−lm,t Quantiles of Input Sample 10 5 0 −5 −5 −4 −3 −2 −1 0 1 2 3 4 5 z TP−rm,t Quantiles of Input Sample 5 0 −5 −5 −4 −3 −2 −1 0 1 2 3 4 5 Standard Normal Quantiles Figure 1: QQ Plots, z TP Daily Statistics 21

Jumps and Price Variance June 25, 2005 15 10 5 0 200 400 600 800 1000 1200 1400 15 10 5 0 200 400 600 800 1000 1200 1400 15 10 5 0 200 400 600 800 1000 1200 1400 15 10 5 0 200 400 600 800 1000 1200 1400 15 10 5 0 200 400 600 800 1000 1200 1400 5 0 −5 200 400 600 800 1000 1200 1400 Figure 2: Simulated z TP,t statistics under SV1FJ with σ jmp = 1 . 50 22

Jumps and Price Variance June 25, 2005 Table 1: Confusion Matrices Interval λ = 0 . 014 λ = 0 . 118 λ = 1 . 000 λ = 2 . 000 (NJ) (J) (NJ) (J) (NJ) (J) (NJ) (J) 5-minute (NJ) 0.960 0.040 0.963 0.037 0.980 0.020 0.991 0.009 z T P,t (J) 0.302 0.698 0.314 0.686 0.239 0.761 0.160 0.840 (NJ) 0.977 0.023 0.978 0.022 0.988 0.012 0.994 0.006 z T P,lm,t (J) 0.347 0.653 0.337 0.663 0.257 0.743 0.175 0.825 (NJ) 0.986 0.014 0.987 0.013 0.993 0.007 0.997 0.003 z T P,rm,t (J) 0.360 0.640 0.358 0.642 0.274 0.726 0.191 0.809 30-minute (NJ) 0.894 0.106 0.899 0.101 0.943 0.057 0.974 0.026 z T P,t (J) 0.558 0.442 0.543 0.457 0.489 0.511 0.432 0.568 (NJ) 0.954 0.047 0.957 0.043 0.975 0.025 0.988 0.012 z T P,lm,t (J) 0.620 0.380 0.641 0.359 0.590 0.410 0.540 0.460 (NJ) 0.986 0.014 0.987 0.013 0.992 0.008 0.996 0.004 z T P,rm,t (J) 0.744 0.257 0.749 0.251 0.706 0.294 0.671 0.329 23

Jumps and Price Variance June 25, 2005 4. Empirical Application • Data – 5-minute S&P 500 Index cash, 1997-2002. – 5-minute S&P 500 Index futures, 1982-2002. • Examine – Five daily statistics. – Full sample statistics. 24

Jumps and Price Variance June 25, 2005 15 10 5 0 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 15 10 5 0 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 15 10 5 0 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 15 10 5 0 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 15 10 5 0 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 Figure 3: z TP,t statistics, S&P Index futures 25

Jumps and Price Variance June 25, 2005 5. Jump Tests under Market Microstructure Noise • Question : Are the many jumps found in this paper and Andersen, Bollerslev and Diebold (2004) due to the presence of market microstructure noise? 26

Jumps and Price Variance June 25, 2005 • Some recent (ongoing) papers for realized variance measures and market microstructure noise: – A¨ ıt-Sahalia, Mykland, and Zhang (2004) – Zhang, Mykland, and A¨ ıt-Sahalia (2004) – Hansen and Lunde (2004a,b) – Bandi and Russell (2004a,b) – papers referenced by those above – other papers at this conference ... 27

Jumps and Price Variance June 25, 2005 • Assumptions – Observed log price process: p t,j = p ∗ t,j + u t,j where ∗ p ∗ t,j is the efficient log price process. ∗ u t,j is the microstructure noise, distributed as i.i.d. N (0 , σ 2 mn ) , and independent of p ∗ t,j . ∗ j = 1 , 2 , . . . , M . 28