1/32 : Variance Measures and Jump Testing George Tauchen Duke University January 21
1. Introduction and Motivation 2/32 Stochastic volatility models account for most of the anomalies in financial price series. Pure diffusive models are very convenient in financial economics. Might need jumps to account for the sometimes violent price movements. Barndorff-Nielsen and Shephard in a series of papers develop a very powerful toolkit for detecting the presence of jumps in higher frequency financial time series.
3/32 The following material is an update of Huang and Tauchen (25). Goals: Examine the Barndorff-Nielsen and Shephard jump detection tests at the daily level. Also look at updated versions. Example of empirical work using five minute returns on S&P Index, 1982 22. Review market microstructure noise in the price process.
Variance Measures and Jumps 4/32 Dynamics of the log price process: Continuous dp(t) = µ(t)dt + σ(t)dw(t). With jumps dp(t) = µ(t)dt + σ(t)dw(t) + dl J (t), where L J (t) L J (s) = s τ t κ(τ).
Log Returns 5/32 Within-day geometric returns r t,j = p(t 1 + jδ) p(t 1 + (j 1)δ), for j = 1, 2,...,M, integer t.
Two Realized Measures, RV & BV 6/32 Realized Variance RV t = Realized Bipower Variation M j=1 r 2 t,j BV t = µ 2 1 ( M M M 1 ) r t,j r t,j 1, j=2 where µ a = E( Z a ), Z N(, 1), a >.
Key properties 7/32 t lim RV t = M t 1 N t σ 2 (s)ds + κ 2 t,j, j=1 t lim BV t = σ 2 (s)ds M t 1 where N(t) is the number of jumps from t 1 to t.
Studentize RV t BV t 8/32 If there are no jumps, then we expect RV BV. We can test by forming the z statistic: z t = RV t BV t N(, 1) Var(RVt BV t ) One sided test: z t > z 1 α indicates a jump on day t In order to do this we must know the probability distribution of RV t and BV t. Barndorff-Nielsen and Shephard work this out in their papers:
The joint asymptotic distribution of RV and BV 9/32 M 1 2 [ t σ 4 (s)ds t 1 ] 1 2 [ RV t t t 1 σ2 (s)ds BV t t t 1 σ2 (s)ds v qq = 2 v qb = 2 v bb = ( π 2 )2 + π 3 where M is the number of within-day returns. ] D N(, [ vqq v qb v qb v bb ] )
Estimate the Quarticity 1/32 In order to studentize RV t BV t one needs to estimate the integrated quarticity t t 1 σ4 (s)ds. In their work Andersen, Bollerslev, and Diebold suggest using the jump-robust realized Tri-Power Quarticity statistic.
Tri-Power Quarticity 11/32 M TP t = M( M 2 )µ 3 4/3 j=3 t TP t even in the presence of jumps. M r t,j 2 4/3 r t,j 1 4/3 r t,j 4/3 t 1 σ 4 (s)ds
Quad-Power Quarticity of Barndorff-Nielsen and Shephard 12/32 M M QP t = M( M 3 )µ 4 1 r t,j 3 r t,j 2 r t,j 1 r t,j QP t j=4 t t 1 in the presence of jumps as well. σ 4 (s)ds
Raw Form 13/32 There are many different ways to compare RV and BV via a z statistic. In the raw form RV t BV t z TP,t = (v bb v qq ) 1 M TP t z QP,t = RV t BV t (v bb v qq ) 1 M QP t
Ratio max-adjusted form 14/32 z TP,rm,t = z QP,rm,t = RV t BV t RV t (v bb v qq ) 1 M max(1, TP t BV 2 t RV t BV t RV t (v bb v qq ) 1 M max(1, QP t BV 2 t ) )
Application: Are there jumps in the S&P Index? 15/32 Huang and Tauchen (25) apply these statistics to the S&P Index at the 5-minute level. There is a z t -type statistic for each day. The decision rule is that if z t > z 1 α, then reject the null hypothesis of no jumps on day t at the significance level α. Note that z.5 = 1.64, z.1 = 2.33, and z.1 = 3.9, so we look for z-statistics exceeding these values. Here is the outcome:
z Statistics, S&P Index 1982 22 16/32 15 1 5 1982 1984 1986 1988 199 1992 1994 1996 1998 2 22 15 1 5 1982 1984 1986 1988 199 1992 1994 1996 1998 2 22 15 1 5 1982 1984 1986 1988 199 1992 1994 1996 1998 2 22 15 1 5 1982 1984 1986 1988 199 1992 1994 1996 1998 2 22 15 1 5 1982 1984 1986 1988 199 1992 1994 1996 1998 2 22
Evidence for Jumps in Stocks, Bonds, and Foreign Exchange? 17/32 Evidence just presented above for S&P Index. We also have Andersen, Bollerslev, Diebold (27): Andersen, T., Bollerslev, T., and Diebold, F. (27) Roughing It Up: Including Jump Components In The Measurement, Modeling, And Forecasting Of Return Volatility, The Review of Economics and Statistics 89(4), pp. 71 72.
ABD (27) Evidence 18/32 FX, Stocks, and Bonds: p. 713. Graphical evidence: p. 711.
Many Other Tests for Jumps 19/32 Theodosiou and Zikes (29) contains thorough review and exposition. Working paper is at the course web site. BN-S... Lee and Mykland (28)... Ait-Sahalia and Jacod (29)
Median-Based Test 2/32 Andersen, Dobrev, and Schaumburg (29): MedVar M,t = c M 1 j=2 [ med ( rt,j 1, r t,j, r t,j+1 )] 2 where c is a constant (see papers). Under the null hypothesis of no jumps ( ) lim RVM,t MedVar M,t = M
Median-Based z-test 21/32 z med,t = RV M,t MedVar M,t Var(RVM,t MedVar M,t ) Reject if z med,t > z crit,α. Just like BN-S except use MedVar M,t in place of BV M,t. Note the robustness to jumps and zero returns.
Truncated Variation 22/32 Some measurement problems and jump tests use a truncated variance measure of variance: Truncated Var = M r t,j 2 I( r t,j cut M,t ) j=1 where cut M,t is a threshold value. If cut M,t at a suitable rate then the truncated variance converges to the daily integrated variance (without squared jumps).
The Cutoff Value 23/32 One reasonable choice would be cut M,t = 2.5 VR t where VR t is a jump-robust estimate of day t s variance, say BV t 1. Some sensitivity to the selection of 2.5 standard deviations and VR t might be needed.
Is the Jump Test Accurate Under Controlled Conditions? 24/32 Huang and Tauchen simulate from the model: dp(t) = µ(t)dt + e f t dw(t) + dl J (t), where f t is the continuous volatility factor. They can assess the accuracy of the Barndorff-Nielsen and Shephard z-statistics, because the controlled conditions allow us to form simulated data with and without jumps. We can also control the size and frequency of the jumps.
Simulated realization from the SV1F model, daily 25/32 1 5 Level Simulation (Daily Values) of 1 Factor SV, No Jumps 5 1 2 3 4 5 6 7 8 9 1 1 Return 1 1 2 3 4 5 6 7 8 9 1 1 Volatility Factor 1 1 2 3 4 5 6 7 8 9 1 1 Jump 1 1 2 3 4 5 6 7 8 9 1
Simulated realization from the SV1FJ model, daily 26/32 1 5 Level Simulation (Daily Values) of 1 Factor SV with Jumps 5 1 2 3 4 5 6 7 8 9 1 1 Return 1 1 2 3 4 5 6 7 8 9 1 1 Volatility Factor 1 1 2 3 4 5 6 7 8 9 1 5 Jump 5 1 2 3 4 5 6 7 8 9 1
Simulated z-statistics under SV1FJ with σ jmp = 1.5 27/32 15 1 5 15 1 5 15 1 5 15 1 5 15 1 5 5 2 4 6 8 1 12 14 2 4 6 8 1 12 14 2 4 6 8 1 12 14 2 4 6 8 1 12 14 2 4 6 8 1 12 14 5 2 4 6 8 1 12 14
Conclusion from the Validity Check 28/32 The Barndorff-Nielsen and Shephard statistic does very well under controlled conditions and z TP,rm,t is the best of the group, but there is really not much difference across the various z s. QUESTION: Were the controlled simulations close enough to observed data to make the conclusion applicable in general practice?
29/32 We now consider the role of short-term trading frictions, which would occur in well functioning financial markets. The correct stock price under the dividend growth model is the expected present value of the future profit (or dividend) stream: S t = k=1 1 (1 + µ t ) k E t (Π t+k ) where Π t+k is the unknown future profit and µ t is the risk-adjusted discount rate.
3/32 If the profit (or dividend) is expected to grow at rate g t then the correct or fundamental value becomes S t = E t(π t+1 ) µ t g t where g t is the expected growth of the of profit (or dividend).
Price Movements and Noise 31/32 As new information comes in over time, the market is constantly changing the assessment of the E t (Π t+1 ), µ t, and g t. The market price P t cannot possibly be kept perfectly in line with S t : P t S t (exactly). due to trading frictions over very short intervals such as 1-second or 1-minutes.
32/32 Thus we think of the log observed price p t = log(p t ) as log(s t ) plus a little noise: p t = log(s t ) + ǫ t where ǫ t is the so-called market microstructure noise. PROBLEM: As we look at the log price movements over shorter and shorter intervals, 1-minutes, 5-minutes, 1-minute, 3-seconds, the noise begins to dominate the price movement: p t+δ p t = log(s t+δ ) log(s t ) + }{{} ǫ t+δ ǫ }{{} t smaller magitude same magnitude as the sampling interval δ. Var(ǫ t+δ ǫ t ) = 2σ 2 ǫ regardless of δ.