SLIDE 1
Rough and Smooth: Measuring, Modeling and Forecasting Financial Market Volatility
Tim Bollerslev Duke University and NBER
International Conference on Finance
Copenhagen, September 2-4, 2005
SLIDE 2 Some Related Realized Volatility Papers:
"Roughing It Up: Including Jump Components in the Measurement, Modeling and Forecasting of Return Volatility" (with Torben G. Andersen and Francis X. Diebold), unpublished manuscript. "Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts" (with T.G. Andersen), International Economic Review, Vol.39, No.4, pp.885-905, 1998. "The Distribution of Realized Exchange Rate Volatility" (with T.G. Andersen, F.X. Diebold and P. Labys), Journal of the American Statistical Association, Vol.96, pp.42-55, 2001. "Modeling and Forecasting Realized Volatility" (with T.G. Andersen, F.X. Diebold, and P. Labys), Econometrica, Vol.71, No.2, pp.579-625, 2003. "Parametric and Nonparametric Volatility Measurements" (with T.G. Andersen and F.X. Diebold), in Handbook of Financial Econometrics (Y. Aït-Sahalia and L.P. Hansen, eds.) forthcoming, 2005. "The Distribution of Realized Stock Return Volatility" (with T.G. Andersen, F.X. Diebold and H. Ebens), Journal of Financial Economics, Vol.61, pp.43-76, 2001. "Analytic Evaluation of Volatility Forecasts" (with T.G. Andersen and N. Meddahi), International Economic Review, Vol.45, No.4, pp.1079-1110, 2004. "Correcting the Errors: Volatility Forecast Evaluation Using High-Frequency Data and Realized Volatilities" (with T.G. Andersen and N. Meddahi), Econometrica, Vol.73, No.1, pp.279-296, 2005. "A Framework for Exploring the Macroeconomic Determinants of Systematic Risk" (with T.G. Andersen, F.X. Diebold and G. Wu), American Economic Review, Vol.95, No.2, pp.398-404, 2005. "Realized Beta: Persistence and Predictability" (with T.G. Andersen, F.X. Diebold and G. Wu), in Advances in Econometrics (T. Fomby, ed.) forthcoming, 2005. "Estimating Stochastic Volatility Diffusions Using Conditional Moments of Integrated Volatility" (with H. Zhou), Journal of Econometrics, Vol.109, pp.33- 65, 2002. "Bridging the Gap Between the Distribution of Realized Volatility and ARCH Modeling: The GARCH-NIG Model" (with L. Forsberg), Journal of Applied Econometrics, Vol.17, pp.535-548, 2002. "Exchange Rate Returns Standardized by Realized Volatility are (Nearly) Gaussian" (with T.G. Andersen, F.X. Diebold, and P. Labys), Multinational Finance Journal, Vol.4, pp.159-179, 2000. "Great Realisations" (with T.G. Andersen, F.X. Diebold, and P. Labys), Risk, pp.105-108, 2000.
SLIDE 3
· Financial Market Volatility Central · Asset Pricing · Asset Allocation · Risk Management · Modeling and Forecasting Volatility · ARCH and Stochastic Volatility Models · Implied Volatilities · High-Frequency Data · Theory / Practice · Realized Volatility
SLIDE 4
· Outline · Realized Volatility · Modeling and Forecasting Realized Volatility · Jumps and Bi-Power Variation Measures · “Significant” Jumps · Market Microstructure “Noise” · The HAR-RV-CJ Model · Conclusion and Extensions
SLIDE 5 · Continuous Time Diffusion dp(t) = µ(t) dt + (t) dW(t) · One Period Notional (Actual) Volatility [ p , p ]t1 [ p , p ]t
t
2(s) ds · Return Variance Conditional on {(s), t s<t+1} · Option Pricing and Stochastic Volatility Hull and White (1987) · Realized Volatility RVt1()
j1
(p(tj) p(t(j1)))2
t
2(s) ds
SLIDE 6
· Logarithmic Price Process / Special Semimartingale p(t) - p(0) r(t) = µ(t) + m(t) µ - Predictable, Finite Variation m - Local Martingale · Quadratic Variation [ p , p ]t = p(t)2 - 20
t p-()dp() = [m c,m c ]t + 0t (p())2
p2(t) - [p , p ]t Local Martingale plimn { p2(0) + j1 [ p(tn,j) - p(tn,j-1) ]2 } [ p , p ]t · Notional (Actual )Volatility from t-h to t [ p , p ]t - [ p , p ]t-h
Andersen, Bollerslev and Diebold (2004) Andersen, Bollerslev, Diebold and Labys (2001, 2003) Barndorff-Nielsen and Shephard (2002, 2003, 2004)
SLIDE 7 · Realized Volatility from t-h to t RVth,t()
j1
(p(thj) p(th(j1)))2 · Theory of Quadratic Variation - 0 RVth,t()
· High-Frequency Data - Notional Volatility (Almost) Observable
SLIDE 8
· Measuring, Modeling, and Forecasting Realized (FX) Volatility
Andersen, Bollerslev, Diebold and Labys (2001, JASA) Andersen, Bollerslev, Diebold and Labys (2003, Econometrica)
· Data · DM/$ and Yen/$ Spot FX Quotations 12/1/86 - 06/30/99 · 4,500 DM/$ and 2,000 Yen/$ Quotes per Day · Theory: 0 · Market Microstructure Frictions: > 0 · Practical Measurements: = 48 - 288 · Unconditional Distribution of RV (approximately) Log-Normal · Long-Memory (type) Dynamic Dependencies in RV
SLIDE 9 · AR-RV Long-Memory Model A(L)(1 L)0.4(log(RVt ) µ ) t · Mincer-Zarnowitz Style Regression
Mincer and Zarnowitz (1969) Chong and Hendry (1986)
RVt1 b0 b1ARRVt1t b2Othert1t ut1 b0 b1 b2 R2 AR-RV
1.06 (.04)
AR-ABS 0.23 (.02)
RiskMetric 0.11 (.02)
GARCH
- 0.07 (.03)
- 1.01 (.04) 0.27
HF-FIEGARCH
- 0.17 (.03)
- 1.23 (.05) 0.32
AR-RV + AR-ABS
1.02 (.05) 0.11 (.07) 0.36 AR-RV + RiskMet.
0.94 (.06) 0.12 (.05) 0.36 AR-RV + GARCH
0.94 (.06) 0.16 (.06) 0.36 AR-RV + HF-FIEGRC
0.81 (.07) 0.33 (.10) 0.36
Mincer-Zarnowitz Regressions, DM/$, In-Sample (1986-96), One-Day-Ahead Andersen, Bollerslev, Diebold and Labys (2003, Econometrica)
SLIDE 10 b0 b1 b2 R2 AR-RV 0.02 (.05) 0.99 (.09)
AR-ABS 0.44 (.03)
0.03 RiskMetric 0.22 (.04)
0.10 GARCH 0.05 (.06)
0.10 HF-FIEGARCH
0.26 AR-RV + AR-ABS 0.04 (.05) 1.02 (.11)
0.25 AR-RV + RiskMetric 0.02 (.05) 0.98 (.13) 0.01 (.11) 0.25 AR-RV + GARCH 0.02 (.06) 0.98 (.13) 0.02 (.16) 0.25 AR-RV + HF-FIEGRC -0.07 (.06) 0.40 (.19) 0.66 (.20) 0.27
Mincer-Zarnowitz Regressions, DM/$, Out-of-Sample (1996-99), One-Day-Ahead Andersen, Bollerslev, Diebold and Labys (2003, Econometrica)
· R2
AR-RV R2 AR-RV + Other
· Other Markets
Areal and Taylor (2002), Deo, Hurvich and Lu (2005) Hol and Koopman (2002), Martens, van Dijk and Pooter (2004) Koopman, Jungbacker and Hol (2005), Oomen (2002) Pong, Shackleton, Taylor and Xu (2004), Thomakos and Wang (2003)
SLIDE 11 · Continuous Time Diffusion dp(t) = µ(t) dt + (t) dW(t) · One Period Notional (Actual) Volatility [ p , p ]t1 [ p , p ]t
t
2(s) ds · Realized Volatility RVt1()
j1
(p(tj) p(t(j1)))2
j1
r 2
tj,
t
2(s) ds
SLIDE 12
· Continuous Time Jump Diffusion dp(t) = µ(t) dt + (t) dW(t) + (t) dq(t) q(t): Counting Process (t): Time-Varying Intensity P[dq(t) = 1] = (t)dt (t): Size of Jumps (t) = p(t) - p(t-)
Andersen, Benzoni and Lund (2002) Bates (2000), Chan and Maheu (2002) Chernov, Gallant, Ghysels, and Tauchen (2003) Drost, Nijman and Werker (1998) Eraker (2004), Eraker, Johannes and Polson (2003) Johannes (2004), Johannes, Kumar and Polson (1999) Maheu and McCurdy (2004), Pan (2002)
SLIDE 13 · One Period Notional (Actual) Volatility
Andersen, Bollerslev and Diebold (2004) Andersen, Bollerslev and Diebold and Labys (2003) Barndorff-Nielsen and Shephard (2002, 2003)
[ p , p ]t1 [ p , p ]t
t
2(s) ds
2(s) · Realized Volatility RVt1()
j1
r 2
tj,
t
2(s)ds
2(s)
SLIDE 14 · Power Variation
Aït-Sahalia (2003) Barndorff-Nielsen and Shephard (2003a)
RPVt1()
p 1 p/2 1/ j1
rtj, p
t
2(s)ds
2(s) p 2
t
p(s)ds 0 < p < 2 p > 2
SLIDE 15 · (Standardized) Bi-Power Variation
Barndorff-Nielsen and Shephard (2004a, 2005)
BVt1()
1 1/ j2
rtj, rt(j1),
t
2(s)ds · Jump Component RVt1() BVt1()
2(s) · Non-Negativity Truncation Jt1() max[ RVt1() BVt1() , 0 ]
SLIDE 16
· Data · DM/$ Spot FX Rates, 12/1986 - 6/1999, 3,045 Days (O&A) · S&P500 Futures, 1/1990 - 12/2002, 3,213 Days (CME) · 30-Year T-Bond Futures, 1/1990 - 12/2002, 3,213 Days (CBOT) · Sampling Frequency - 0 · Market Microstructure Frictions · Discreteness · Bid-Ask Spread Positioning · Unevenly Spaced Observations · Linearly Interpolated Five-Minute Returns · = 1/288 DM/$ · = 1/97 S&P500 and T-Bond
Aït-Sahalia, Mykland and Zhang (2005) Andersen, Bollerslev, Diebold and Labys (2000) Andreou and Ghysels (2002), Areal and Taylor (2002) Bandi and Russell (2004a,b) Barndorff-Nielsen, Hansen, Lunde and Shephard (2004) Barucci and Reno (2002), Bollen and Inder (2002) Corsi, Zumbach, Müller, and Dacorogna (2001) Curci and Corsi (2003), Hansen and Lunde (2006) Malliavin and Mancino (2002), Oomen (2002, 2004) Zhang (2004), Zhang, Aït-Sahalia and Mykland (2005), Zhou (1996)
SLIDE 17 Table 1A Summary Statistics for Daily DM/$ Realized Volatilities and Jumps ____________________________________________________________________________________ RVt RVt
1/2
log(RVt ) Jt Jt
1/2
log(Jt+1) Mean 0.508 0.670
0.037 0.129 0.033 St.Dev. 0.453 0.245 0.657 0.110 0.142 0.072 Skewness 3.925 1.784 0.408 16.52 2.496 7.787 Kurtosis 26.88 8.516 3.475 434.2 18.20 108.5 Min. 0.052 0.227
0.000 0.000 0.000 Max. 5.245 2.290 1.657 3.566 1.889 1.519 LB10 3786 5714 7060 16.58 119.4 63.19 ____________________________________________________________________________________
SLIDE 18 Table 1B Summary Statistics for Daily S&P500 Realized Volatilities and Jumps ____________________________________________________________________________________ RVt RVt
1/2
log(RVt ) Jt Jt
1/2
log(Jt+1) Mean 1.137 0.927
0.164 0.232 0.097 St.Dev. 1.848 0.527 0.965 0.964 0.332 0.237 Skewness 7.672 2.545 0.375 20.68 5.585 6.386 Kurtosis 95.79 14.93 3.125 551.9 59.69 59.27 Min. 0.058 0.240
0.000 0.000 0.000 Max. 36.42 6.035 3.595 31.88 5.646 3.493 LB10 5750 12184 15992 558.0 1868 2295 ____________________________________________________________________________________
SLIDE 19 Table 1C Summary Statistics for Daily U.S. T-Bond Realized Volatilities and Jumps ____________________________________________________________________________________ RVt RVt
1/2
log(RVt ) Jt Jt
1/2
log(Jt+1) Mean 0.286 0.506
0.036 0.146 0.033 St.Dev. 0.222 0.173 0.638 0.069 0.120 0.055 Skewness 3.051 1.352 0.262 8.732 1.667 5.662 Kurtosis 20.05 6.129 3.081 144.6 10.02 57.42 Min. 0.026 0.163
0.000 0.000 0.000 Max. 2.968 1.723 1.088 1.714 1.309 0.998 LB10 1022 1718 2238 20.53 34.10 26.95 ____________________________________________________________________________________
SLIDE 20
SLIDE 21
SLIDE 22
SLIDE 23 · Consistent Jump Measurements Jt1() max[ RVt1() BVt1() , 0 ] · Empirically “Too Many” Small Jumps · Significant Jumps · Asymptotic (0) Distribution in the Absence of Jumps
Barndorff-Nielsen and Shephard (2004a, 2005)
1/2 RVt1() BVt1() [ (µ4
1 2µ2 1 5) t1 t
4(s)ds ]1/2
SLIDE 24 · Realized Quarticity
Barndorff-Nielsen and Shephard (2002a) Andersen, Bollerslev and Meddahi (2005)
RQt1()
4 1/ j1
r 4
tj,
t
4(s)ds
- Diverges in the Presence of Jumps
· Realized Tri-Power Quarticity TQt1() 1 µ3
4/3 1/ j3
rtj,4/3rt(j1),4/3rt(j2), 4/3
t
4(s)ds
- Even in the Presence of Jumps
SLIDE 25 · Feasible Test Statistic for Jumps Wt1()
RVt1() BVt1() [(µ4
1 2µ2 1 5) TQt1() ]1/2
· Variance Stabilizing Ratio Transformation and Max. Adjustment Zt1()
[ RVt1() BVt1()] RVt1()1 [ (µ4
1 2µ2 1 5) max{ 1 , TQt1() BVt1()2 }]1/2
- Better Behaved in Finite (>0) Samples
Huang and Tauchen (2005)
SLIDE 26 · (“Significant”) Jumps Jt1,()
- I [ Zt1() > ] [ RVt1() BVt1() ]
- Depends upon choice of (and )
- Previous Jt+1 corresponds to = 0.5
- Shrinkage type estimator
· Continuous Sample Path Variation Ct1,()
I [ Zt1() > ] BVt1() · Realized Variation RVt1()
SLIDE 27
· Theory - 0 · Market Microstructure Frictions · Discreteness · Bid-Ask Spread Positioning · Unevenly Spaced Observations · Fixed >> 0 ; e.g. Five-Minute Returns
Andersen, Bollerslev, Diebold and Labys (2000, 2001)
· Pre-Filtering, Kernel and Fourier Methods
Andreou and Ghysels (2002), Areal and Taylor (2002) Barndorff-Nielsen, Hansen, Lunde and Shephard (2004) Barucci and Reno (2002) Corsi, Zumbach, Müller, and Dacorogna (2001) Hansen and Lunde (2006), Malliavin and Mancino (2002) Oomen (2002, 2004), Zhou (1996)
· “Optimal” and Sub-Sampling Schemes
Aït-Sahalia, Mykland and Zhang (2005) Bandi and Russell (2004a,b), Müller (1993) Zhang (2004), Zhang, Aït-Sahalia and Mykland (2005)
SLIDE 28 · High-Frequency Returns rt,
t, t,
p*(t): “Fundamental” Price (t): Market Microstructure “Noise” · Realized Variation
t,] E[(r t, t,)2] E[(r t,)2]
- Noise Term Dominates the Variation for 0
- RVt() Formally Inconsistent as 0
- Motivates >> 0 and other Techniques
SLIDE 29 · High-Frequency Returns rt,
t, t,
p*(t): “Fundamental” Price (t): Market Microstructure “Noise” · Realized Bi-Power Variation
t, t,] E[r t,]
tj,r t(j1),]
- RVt() - BVt() too Small and TQt() too Large
- Test for Jumps Under-Rejects, too Few Jumps
- Staggering, or Skip-One, Breaks Dependence
SLIDE 30 · “Standard” Bi-Power Variation BVt1()
1 1/ j2
rtj, rt(j1), · Staggered Bi-Power Variation BV1,t1()
1 (1 2)1 1/ j3
rtj, rt(j2), · Staggered Tri-Power Quarticity may be Defined Similarly
SLIDE 31 · Test for Jumps based on , and BV1,t1() TQ1,t1() Z1,t1()
- = 0.99, variance of noise 0-50 percent over 5-minute intervals
Huang and Tauchen (2005) "The Monte Carlo evidence suggests that, under the arguably realistic scenarios considered here, the recently developed tests for jumps perform impressively and are not easily fooled."
SLIDE 32
· Case Studies
12/10/87: Z1,t1() 10.315 Swelling U.S. trade deficit ($17.6B) announced at 13:30GMT (8:30 EST). 9/17/92: (max. sample) Z1,t1() 0.326 C1,t1() 3.966 The day following the temporary withdrawal of the British Pound from the European Monetary System. WSJ: “The dollar sank more than 2% against the mark as nervousness persisted in the currency market.”
SLIDE 33
· Case Studies
6/30/99: Z1,t1() 7.659 FED raised short term rate by ¼ percent at 13:15 CST (14:15 EST), but indicated that it “might not raise rates again in the near term due to conflicting forces in the economy.” 7/24/02: (max. sample) Z1,t1() 0.704 C1,t1() 27.077 Record Big Board trading volume of 2.77 billion shares.
SLIDE 34
· Case Studies
8/1/96: Z1,t1() 6.877 National Association of Purchasing Managers (NAPM) index released at 9:00CST (10:00 EST). 12/7/01: Z1,t1() 0.915 WSJ: Rise in jobless claims increased the expectation that the FED would lower rates at its Board meeting the following day (which it did).
SLIDE 35
- Not all significant jumps map as nicely into specific news
- What causes financial prices to “jump”?
SLIDE 36
SLIDE 37
SLIDE 38
SLIDE 39 Table 3A Summary Statistics for Significant Daily DM/$ Jumps ____________________________________________________________________________________
0.950 0.990 0.999 0.9999 Prop. 0.859 0.409 0.254 0.137 0.083 Mean. 0.059 0.047 0.037 0.028 0.021 St.Dev. 0.136 0.137 0.135 0.131 0.127 LB10 , Jt, 65.49 26.30 6.197 3.129 2.414 LR , I(Jt, >0) 0.746 2.525 0.224 0.994 0.776 LB10 ,Dt, 10.78 9.900 7.821 6.230 19.95 LB10 ,Jt,
+
73.62 116.4 94.19 87.69 34.57 ____________________________________________________________________________________
SLIDE 40 Table 3B Summary Statistics for Significant Daily S&P500 Jumps ____________________________________________________________________________________
0.950 0.990 0.999 0.9999 Prop. 0.737 0.255 0.141 0.076 0.051 Mean. 0.163 0.132 0.111 0.095 0.086 St.Dev. 0.961 0.961 0.958 0.953 0.950 LB10 , Jt, 300.6 271.9 266.4 260.9 221.6 LR , I(Jt, >0) 2.415 1.483 12.83 8.418 7.824 LB10 , Dt, 50.83 31.47 22.67 36.18 49.25 LB10 , Jt,
+
320.8 146.0 77.06 35.11 25.49 ____________________________________________________________________________________
SLIDE 41 Table 3C Summary Statistics for Significant Daily U.S. T-Bond Jumps ____________________________________________________________________________________
0.950 0.990 0.999 0.9999 Prop. 0.860 0.418 0.254 0.132 0.076 Mean. 0.048 0.038 0.030 0.021 0.016 St.Dev. 0.094 0.096 0.096 0.090 0.085 LB10 , Jt, 30.34 30.37 27.85 19.80 18.85 LR , I(Jt, >0) 4.746 21.62 13.69 3.743 1.913 LB10 , Dt, 45.55 100.1 59.86 103.3 81.42 LB10 , Jt,
+
21.23 17.18 15.18 9.090 11.98 ____________________________________________________________________________________
SLIDE 42
· Significant ( = 0.999) Jump Dynamics
SLIDE 43
· Volatility Modeling and Forecasting · Strong Unit-Root Like Volatility Persistence
Engle and Bollerslev (1986), Bollerslev and Engle (1993)
· FIGARCH and Long-Memory SV Models
Baillie, Bollerslev and Mikkelsen (1996) Bollerslev and Mikkelsen (1996, 1999), Breidt, Crato and de Lima (1998) Ding, Granger and Engle (1993), Harvey (1998), Robinson (1991)
· ARFIMA-RV Models
Andersen, Bollerslev, Diebold and Labys (2003), Areal and Taylor (2002) Deo, Hurvich and Lu (2003), Koopman, Jungbacker and Hol (2005) Martens, van Dijk and Pooter (2004), Oomen (2002) Pong, Shackleton, Taylor and Xu (2004), Thomakos and Wang (2003)
· Multi-Factor and Component Structures
Andersen and Bollerslev (1997), Calvet and Fisher (2001, 2002) Chernov, Gallant, Ghysels and Tauchen (2003), Dacorogna et al. (2001) Engle and Lee (1999), Gallant, Hsu and Tauchen (1999), Müller et al. (1997)
SLIDE 44 · Heterogeneous AR Realized Volatility (HAR-RV) Model
Corsi (2003)
RVt+1 = 0 + D RVt + W RVt-5,t + M RVt-22,t + t+1 RVt,t+h h-1( RVt+1 + RVt+2 + ... + RVt+h ) h = 1, 5, 22 (Daily, Weekly, Monthly)
- “Poor Man’s” Long-Memory Model
· HAR-RV-CJ Model RVt+1 = 0 + CD Ct + CW Ct-5,t + CM Ct-22,t + JD Jt + JW Jt-5,t + JM Jt-22,t + t+1
- Jt Jt, , Ct Ct, , = 0.999
- Nests HAR-RV Model
SLIDE 45
· Multi-Period Horizons RVt,t+h = 0 + CD Ct + CW Ct-5,t + CM Ct-22,t + JD Jt + JW Jt-5,t + JM Jt-22,t + t,t+h MA(h-1) · Other Volatility Transforms (RVt,t+h )1/2 = 0 + CD Ct
1/2 + CW (Ct-5,t )1/2 + CM (Ct-22,t )1/2
+ JD Jt
1/2 + JW (Jt-5,t )1/2 + JM (Jt-22,t )1/2 + t,t+h
log(RVt,t+h ) = 0 + CD log(Ct ) + CW log(Ct-5,t ) + CM log(Ct-22,t ) + JD log(Jt +1) + JW log(Jt-5,t+1) + JM log(Jt-22,t+1) + t,t+h · MIDAS Regressions
Ghysels, Santa-Clara and Valkanov (2004, 2005)
· Multiplicative Error Models (MEM)
Engle (2002), Engle and Gallo (2005)
SLIDE 46 Table 4A Daily, Weekly, and Monthly DM/$ HAR-RV-CJ Regressions ____________________________________________________________________________________ RVt,t+h = 0 + CD Ct + CW Ct-5,t + CM Ct-22,t + JD Jt + JW Jt-5,t + JM Jt-22,t + t,t+h ____________________________________________________________________________________ RVt,t+h (RVt,t+h)1/2 log(RVt,t+h) _____________________ _____________________ ______________________ h 1 5 22 1 5 22 1 5 22 _____________________ _____________________ ______________________ 0.083 0.131 0.231 0.096 0.158 0.292
(0.015) (0.018) (0.025) (0.015) (0.021) (0.034) (0.024) (0.036) (0.057) CD 0.407 0.210 0.101 0.397 0.222 0.127 0.369 0.205 0.130 (0.044) (0.040) (0.021) (0.032) (0.029) (0.019) (0.026) (0.021) (0.016) CW 0.256 0.271 0.259 0.264 0.289 0.264 0.295 0.318 0.258 (0.077) (0.054) (0.046) (0.048) (0.051) (0.042) (0.039) (0.048) (0.040) CM 0.226 0.308 0.217 0.212 0.281 0.205 0.217 0.270 0.213 (0.072) (0.078) (0.074) (0.044) (0.060) (0.068) (0.036) (0.055) (0.071) JD 0.096 0.006
0.022 0.001 0.003 0.043 0.024
(0.089) (0.040) (0.017) (0.027) (0.017) (0.010) (0.111) (0.076) (0.044) JW
0.001 0.002
(0.168) (0.199) (0.125) (0.033) (0.044) (0.028) (0.239) (0.327) (0.242) JM
0.055
0.014
(0.329) (0.460) (0.604) (0.057) (0.087) (0.127) (0.408) (0.668) (0.990) R2
HAR-RV-CJ
0.368 0.427 0.361 0.443 0.486 0.397 0.485 0.514 0.415 ____________________________________________________________________________________
SLIDE 47 Table 4B Daily, Weekly, and Monthly S&P500 HAR-RV-CJ Regressions ____________________________________________________________________________________ RVt,t+h = 0 + CD Ct + CW Ct-5,t + CM Ct-22,t + JD Jt + JW Jt-5,t + JM Jt-22,t + t,t+h ____________________________________________________________________________________ RVt,t+h (RVt,t+h)1/2 log(RVt,t+h) _____________________ _____________________ ______________________ h 1 5 22 1 5 22 1 5 22 _____________________ _____________________ ______________________ 0.143 0.222 0.393 0.062 0.103 0.202
0.003 0.026 (0.040) (0.057) (0.075) (0.018) (0.028) (0.037) (0.013) (0.019) (0.036) CD 0.356 0.224 0.135 0.381 0.262 0.183 0.320 0.224 0.162 (0.067) (0.043) (0.023) (0.041) (0.031) (0.024) (0.028) (0.022) (0.020) CW 0.426 0.413 0.204 0.367 0.413 0.272 0.368 0.383 0.274 (0.120) (0.114) (0.070) (0.063) (0.072) (0.061) (0.043) (0.053) (0.049) CM 0.111 0.168 0.319 0.163 0.206 0.322 0.246 0.297 0.403 (0.063) (0.076) (0.070) (0.042) (0.062) (0.065) (0.032) (0.049) (0.056) JD
0.005
0.005
0.018 (0.063) (0.049) (0.022) (0.043) (0.027) (0.017) (0.066) (0.049) (0.031) JW 0.465 0.362 0.456 0.082 0.096 0.132 0.062 0.163 0.198 (0.233) (0.205) (0.287) (0.071) (0.075) (0.113) (0.105) (0.126) (0.176) JM 0.355 0.458 0.215 0.133 0.170 0.190 0.207 0.233 0.246 (0.304) (0.448) (0.202) (0.054) (0.084) (0.105) (0.085) (0.136) (0.201) R2
HAR-RV-CJ
0.421 0.574 0.478 0.613 0.700 0.639 0.696 0.763 0.722 ____________________________________________________________________________________
SLIDE 48 Table 4C Daily, Weekly, and Monthly U.S. T-Bond HAR-RV-CJ Regressions ____________________________________________________________________________________ RVt,t+h = 0 + CD Ct + CW Ct-5,t + CM Ct-22,t + JD Jt + JW Jt-5,t + JM Jt-22,t + t,t+h ____________________________________________________________________________________ RVt,t+h (RVt,t+h)1/2 log(RVt,t+h) _____________________ _____________________ ______________________ h 1 5 22 1 5 22 1 5 22 _____________________ _____________________ ______________________ 0.085 0.095 0.133 0.133 0.166 0.236
(0.011) (0.012) (0.017) (0.016) (0.019) (0.031) (0.040) (0.052) (0.079) CD 0.107 0.064 0.031 0.087 0.069 0.034 0.091 0.068 0.036 (0.031) (0.015) (0.006) (0.025) (0.013) (0.006) (0.022) (0.012) (0.007) CW 0.299 0.238 0.196 0.306 0.223 0.180 0.297 0.203 0.168 (0.051) (0.047) (0.037) (0.045) (0.042) (0.033) (0.043) (0.042) (0.030) CM 0.366 0.426 0.369 0.367 0.428 0.380 0.389 0.439 0.382 (0.062) (0.062) (0.068) (0.048) (0.055) (0.065) (0.046) (0.055) (0.064) JD
- 0.136
- 0.010
- 0.019
- 0.080
- 0.006
- 0.007
- 0.769
- 0.090
- 0.091
(0.055) (0.021) (0.008) (0.026) (0.012) (0.006) (0.185) (0.082) (0.041) JW 0.230 0.050
0.090 0.043
0.775 0.227
(0.122) (0.081) (0.067) (0.033) (0.029) (0.025) (0.390) (0.298) (0.271) JM
- 0.271
- 0.145
- 0.116
- 0.113
- 0.076
- 0.057
- 1.319
- 0.477
- 0.034
(0.177) (0.216) (0.245) (0.045) (0.058) (0.075) (0.589) (0.773) (0.918) R2
HAR-RV-CJ
0.144 0.325 0.377 0.192 0.353 0.393 0.222 0.365 0.400 ____________________________________________________________________________________
SLIDE 49
Figure 3A Daily, Weekly and Monthly DM/$ Realized Volatilities and HAR-RV-CJ Forecasts
SLIDE 50
Figure 3B Daily, Weekly and Monthly S&P500 Realized Volatilities and HAR-RV-CJ Forecasts
SLIDE 51
Figure 3C Daily, Weekly and Monthly T-Bond Realized Volatilities and HAR-RV-CJ Forecasts
SLIDE 52
Summary · Formal and effective framework for incorporating high- frequency financial data into volatility modeling and forecasting through easy-to-implement non-parametric lower frequency daily measurements · Estimation and distributional properties of “significant” and robust-to-market-microstructure “noise” jumps · Easy-to-implement and accurate HAR-RV-CJ “poor- man’s” long-memory volatility forecasting model
SLIDE 53
Extensions/Ongoing Work · Reduced form modeling and forecasting of Jt, and Ct,
Andersen, Bollerslev and Huang (2005)
· Effective score generator for EMM estimation
Bollerslev, Kretschmer, Pigorsch and Tauchen (2005)
· Pricing of jump and continuous sample path variability
Bollerslev, Huang and Zhou (2005)
· MDH, leverage effects and (signed) jumps
Andersen, Bollerslev and Dobrev (2005)
· Risk management, tails, and VaR · Market microstructure “noise” and jump measurements
Andersen, Bollerslev, Frederiksen and Nielsen (2005)
· Jumps and (macro) economic news arrivals
Andersen, Bollerslev, Diebold and Vega (2005)
· Multivariate jump measurements and co-jumping
SLIDE 54 100 200 300 400 500 600 700 800 900
0.0 2.5 5.0 7.5 Series: RET Sample 1765 5026 Observations 3262 Mean 0.020445 Median 0.053669 Maximum 7.156932 Minimum
1.024471 Skewness
Kurtosis 8.566824 Jarque-Bera 4244.720 Probability 0.000000
1 2 3 4
4 8 RET Normal Quantile
· Fat tailed (unconditional) return distributions rt Not N( , )
Fama (1965), Mandelbrot (1963)
SLIDE 55 50 100 150 200 250 300 350
1 2 3 Series: STRET Sample 1765 5026 Observations 3262 Mean 0.076853 Median 0.074362 Maximum 3.658999 Minimum
0.983838 Skewness 0.100477 Kurtosis 2.808346 Jarque-Bera 10.48102 Probability 0.005298
1 2 3 4
1 2 3 4 STRET Normal Quantile
· Approximate normality rt / RVt
1/2 N( 0 , 1 )
Andersen, Bollerslev, Diebold and Ebens (2001) Andersen, Bollerslev, Diebold and Labys (2000, 2001)
SLIDE 56 50 100 150 200 250 300 350
0.00 1.25 2.50 3.75 Series: STRETMJ Sample 1765 5026 Observations 3262 Mean 0.072899 Median 0.056310 Maximum 4.031369 Minimum
1.001778 Skewness 0.128175 Kurtosis 2.950615 Jarque-Bera 9.263268 Probability 0.009739
1 2 3 4
1 2 3 4 STRETMJ Normal Quantile
· Even closer approximate normality ( rt ± Jt
½ ) / BVt 1/2 N( 0 , 1 )
Andersen, Bollerslev and Dobrev (2005)
