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

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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.

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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) + dLJ(t), where LJ(t) − LJ(s) =

s≤τ≤t κ(τ).

• Within-day geometric returns

rt,j = p(t−1+j/M)−p(t−1+(j−1)/M), j = 1, 2, . . . , M.

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Jumps and Price Variance June 25, 2005

• Two realized measures:

– Realized Variance RVt =

M

• j=1

r2

t,j

– Realized Bipower Variation BVt = µ−2

1 (

M M − 1)

M

• j=2

|rt,j||rt,j−1|, where µa = E(|Z|a),

Z ∼ N(0, 1), a > 0.

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Jumps and Price Variance June 25, 2005

Key properties lim

M→∞ RVt =

t

t−1

σ2(s)ds +

Nt

• j=1

κ2

t,j,

lim

M→∞ BVt =

t

t−1

σ2(s)ds where N(t) is the number of jumps from t − 1 to t.

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Jumps and Price Variance June 25, 2005

Thus the difference RVt − BVt is a measure of the jump component. The jump test statistics are of the form RVt − BVt

• Avar(RVt − BVt)

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Jumps and Price Variance June 25, 2005

The joint asymptotic distribution under the null of no jumps is given by Barndorff-Nielsen and Shephard

M

1 2

t

t−1

σ4(s)ds −1

2

×

• RVt −

t

t−1 σ2(s)ds

BVt − t

t−1 σ2(s)ds

• D

→N(0, vqq vqb vqb vbb

• )

vqq = 2 vqb = 2 vbb = (π

2)2 + π − 3

Generalizations available based on earlier work by Jacod (1994) and Jacod and Protter (1998).

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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 RVt is the most efficient estimator of the integrated

• variance. BVt is a less efficient (but more robust)

estimate and RVt − BVt is independent of RVt conditional on the instantaneous volatility process.

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Jumps and Price Variance June 25, 2005

Thus the Relative Jump RJt = RVt − BVt RVt 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).

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Jumps and Price Variance June 25, 2005

In order to studentize RVt − BVt or RJt one needs to estimate the integrated quarticity t

t−1

σ4(s)ds 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).

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Jumps and Price Variance June 25, 2005

Daily Statistics

Raw form zTP,t = RVt − BVt

• (vbb − vqq) 1

MTPt

zQP,t = RVt − BVt

• (vbb − vqq) 1

MQPt

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Jumps and Price Variance June 25, 2005

Ratio adjusted zTP,r,t =

RVt−BVt RVt

• (vbb − vqq) 1

M TPt BV 2

t

zQP,r,t =

RVt−BVt RVt

• (vbb − vqq) 1

M QPt BV 2

t

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Jumps and Price Variance June 25, 2005

Max version zTP,rm,t =

RVt−BVt RVt

• (vbb − vqq) 1

M max(1, TPt BV 2

t )

zQP,rm,t =

RVt−BVt RVt

• (vbb − vqq) 1

M max(1, QPt BV 2

t )

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Jumps and Price Variance June 25, 2005

• 3. Monte Carlo Analysis

Data generation processes:

• SV1F

dp(t) = µ dt + exp [β0 + β1v(t)] dwp(t) dv(t) = αvv(t) dt + dwv(t)

• SV1FJ

dp(t) = µ dt + exp [β0 + β1v(t)] dwp(t) + dLJ(t) dv(t) = αvv(t) dt + dwv(t)

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Jumps and Price Variance June 25, 2005

SV2F A two-factor continuous model: dp(t) = µ dt + s-exp [β0 + β1v1(t) + β2v2(t)] dwp(t) dv1(t) = αv1v1(t) dt + dwv1(t) dv2(t) = αv2v2(t) dt + [1 + βv2v2(t)]dwv2(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.

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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.

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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 TPt versus QPt matters little.

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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.

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Jumps and Price Variance June 25, 2005

• Power

– Jump intensity increases power. – Jump size increases power. – Sampling frequency increases power. – The tests are inconsistent.

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Jumps and Price Variance June 25, 2005

−5 −4 −3 −2 −1 1 2 3 4 5 −5 5 10 Quantiles of Input Sample zTP,t −5 −4 −3 −2 −1 1 2 3 4 5 −5 5 10 Quantiles of Input Sample zTP−lm,t −5 −4 −3 −2 −1 1 2 3 4 5 −5 5 Standard Normal Quantiles Quantiles of Input Sample zTP−rm,t

Figure 1: QQ Plots, zTP Daily Statistics

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Jumps and Price Variance June 25, 2005

200 400 600 800 1000 1200 1400 5 10 15 200 400 600 800 1000 1200 1400 5 10 15 200 400 600 800 1000 1200 1400 5 10 15 200 400 600 800 1000 1200 1400 5 10 15 200 400 600 800 1000 1200 1400 5 10 15 200 400 600 800 1000 1200 1400 −5 5

Figure 2: Simulated zTP,t statistics under SV1FJ with σjmp = 1.50

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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 zT P,t (NJ) 0.960 0.040 0.963 0.037 0.980 0.020 0.991 0.009 (J) 0.302 0.698 0.314 0.686 0.239 0.761 0.160 0.840 zT P,lm,t (NJ) 0.977 0.023 0.978 0.022 0.988 0.012 0.994 0.006 (J) 0.347 0.653 0.337 0.663 0.257 0.743 0.175 0.825 zT P,rm,t (NJ) 0.986 0.014 0.987 0.013 0.993 0.007 0.997 0.003 (J) 0.360 0.640 0.358 0.642 0.274 0.726 0.191 0.809 30-minute zT P,t (NJ) 0.894 0.106 0.899 0.101 0.943 0.057 0.974 0.026 (J) 0.558 0.442 0.543 0.457 0.489 0.511 0.432 0.568 zT P,lm,t (NJ) 0.954 0.047 0.957 0.043 0.975 0.025 0.988 0.012 (J) 0.620 0.380 0.641 0.359 0.590 0.410 0.540 0.460 zT P,rm,t (NJ) 0.986 0.014 0.987 0.013 0.992 0.008 0.996 0.004 (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.

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Jumps and Price Variance June 25, 2005

1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 5 10 15 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 5 10 15 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 5 10 15 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 5 10 15 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 5 10 15

Figure 3: zTP,t statistics, S&P Index futures

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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?

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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 ...

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Jumps and Price Variance June 25, 2005

• Assumptions

– Observed log price process: pt,j = p∗

t,j + ut,j

where ∗ p∗

t,j is the efficient log price process.

∗ ut,j is the microstructure noise, distributed as i.i.d. N(0, σ2

mn), and independent of p∗ t,j.

∗ j = 1, 2, . . . , M.

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Jumps and Price Variance June 25, 2005

• From A¨

ıt-Sahalia et al (2004), the proportion of the expected total return variance attributable to market microstructure noise is πmn(δ) = 2σ2

mn

σ2

dδ + 2σ2 mn

Here δ is the observation interval, σ2

dδ denotes the

expected integrated variance over this interval, and σ2

d is the unconditional daily variance.

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Jumps and Price Variance June 25, 2005

• Noise contribution in Monte Carlo

Use implied noise contribution from the literature: – Set πmn for {0%, 10%, 20%, 30%, 40%, 50%} at the 5-minute level. – Correspondingly, σmn = {0.000, 0.027, 0.040, 0.052, 0.065, 0.080}.

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Jumps and Price Variance June 25, 2005

Bollerslev suggested staggering in BVt, TPt and QPt to account for the market microstructure noise. BVi,t = constant ×

M

• j=2+i

|rt,j−(1+i)||rt,j| where i ≥ 0. The staggering breaks (mitigates) the correlation between adjacent returns rt,j−1, rt,j.

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Jumps and Price Variance June 25, 2005

Theory for the impact of microstructure noise on jump test statistics. Suppress t, condition on the volatility process, and write the jth within-day return as rj

D

= (σ2

jδ + σ2 mn)

1 2 Zj

where the Zj’s are standard Gaussian random variables and the sampling interval is of width δ.

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Jumps and Price Variance June 25, 2005

The correlations of the Zj’s are ρj,k ≡ Corr(Zj, Zj−k) =   

−σ2

mn

• (σ2

jδ+σ2 mn)(σ2 j−kδ+σ2 mn) |k| = 1

|k| ≥ 1 The correlations are critical in what follows.

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Jumps and Price Variance June 25, 2005

All of the jump-test z-statistics have the common structure of a studentized measure of discrepancy (D): z = D (RV, BV )

• Avar [D (RV, BV )]

where Avar is given by the Barndorff-Nielsen and Shephard theory.

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Jumps and Price Variance June 25, 2005

We analyze the expectations of the numerator and the denominator separately for the simplest case where D (RV, BV ) = RV − BV Can show that in presence

• f

the market microstructure noise E(RV )(↑) − E(BV )(↑↑)

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Jumps and Price Variance June 25, 2005

Can also show that E [Avar (RV − BV )] (↑) Hence z = RV (↑) − BV (↑↑)

• Avar (RV − BV ) (↑)

⇒ z(↓)

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Jumps and Price Variance June 25, 2005

Under staggering (i ≥ 1) the biases are much smaller and we expect that z = D (RV, BV )

• Avar [D (RV, BV )]

≈ N(0, 1) i.e, the Barndorff-Nielsen and Shephard approximations to work much better. We need to verify by Monte Carlo.

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Jumps and Price Variance June 25, 2005

Table 2: Size of zTP under noise (i=0)

πmn: 0% 10% 20% 30% 40% 50% Interval σmn: 0.000 0.027 0.040 0.052 0.065 0.080 1-minute zT P,t 0.020 0.007 0.002 0.001 0.000 0.000 zT P,rm,t 0.012 0.004 0.001 0.000 0.000 0.000 3-minute zT P,t 0.032 0.027 0.020 0.014 0.009 0.005 zT P,rm,t 0.012 0.011 0.008 0.005 0.003 0.002 5-minute zT P,t 0.041 0.038 0.034 0.029 0.022 0.017 zT P,rm,t 0.014 0.014 0.011 0.009 0.007 0.005 30-minute zT P,t 0.107 0.107 0.107 0.107 0.107 0.107 zT P,rm,t 0.014 0.015 0.015 0.015 0.014 0.014

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Jumps and Price Variance June 25, 2005

Table 3: Size of zTP with staggering (i=1)

πmn: 0% 10% 20% 30% 40% 50% Interval σmn: 0.000 0.027 0.040 0.052 0.065 0.080 1-minute zT P,t 0.021 0.020 0.021 0.021 0.022 0.022 zT P,rm,t 0.012 0.012 0.011 0.012 0.012 0.012 3-minute zT P,t 0.031 0.034 0.033 0.032 0.032 0.032 zT P,rm,t 0.014 0.014 0.014 0.013 0.013 0.013 5-minute zT P,t 0.042 0.041 0.041 0.042 0.041 0.041 zT P,rm,t 0.014 0.014 0.015 0.014 0.014 0.014 30-minute zT P,t 0.129 0.128 0.128 0.127 0.127 0.126 zT P,rm,t 0.021 0.020 0.020 0.020 0.020 0.020

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Jumps and Price Variance June 25, 2005

Table 4: Confusion Matrices, zTP,rm,t, Large Rare Jumps and Microstructure Noise (i=0)

πmn = 0% πmn = 50% σmn = 0.000 σmn = 0.080 (NJ) (J) (NJ) (J) 1-minute (NJ) 0.988 0.012 1.000 0.000 (J) 0.214 0.786 0.558 0.442 5-minute (NJ) 0.986 0.014 0.995 0.005 (J) 0.360 0.640 0.503 0.497 30-minute (NJ) 0.986 0.014 0.986 0.014 (J) 0.744 0.257 0.740 0.260

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Jumps and Price Variance June 25, 2005

Table 5: Confusion Matrices, zTP,rm,t, Large Rare Jumps and Microstructure Noise (i=1)

πmn = 0% πmn = 50% σmn = 0.000 σmn = 0.080 (NJ) (J) (NJ) (J) 1-minute (NJ) 0.988 0.012 0.988 0.012 (J) 0.224 0.776 0.432 0.568 5-minute (NJ) 0.986 0.014 0.986 0.014 (J) 0.344 0.656 0.481 0.519 30-minute (NJ) 0.980 0.020 0.981 0.019 (J) 0.708 0.292 0.750 0.250

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Jumps and Price Variance June 25, 2005

• 6. Conclusions
• Monte Carlo evidence indicates that the recently

developed z-tests for jumps in ratio-form and max- adjusted perform impressively with excellent size and power properties.

• The daily test statistics do an outstanding job of

identifying the days on which jump(s) occur.

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Jumps and Price Variance June 25, 2005

• Effects of market microstructure noise

– Without staggering, the z statistics’ rejection frequency is downward biased. – Strengthens existing empirical evidence for jumps. – Staggering returns restores the nominal size and makes the rejection rate robust to the noise proportion. – Overall, the detected jumps are not likely induced by market microstructure noise. (Caveat)

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Jumps and Price Variance June 25, 2005

• There is strong empirical evidence for jumps with

the new tests: jumps account for about 4.5 to 7.3 percent of the total daily variance of the S&P Index.

• Economic importance of the relative contribution
• f jumps to total price variance can only be

addressed by examining the portfolio optimization behavior of an economic agent facing price series generated by the SV models enhanced with jumps.

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