Upload
others
View
5
Download
0
Embed Size (px)
Citation preview
Rough and Smooth: Measuring, Modeling and
Forecasting Financial Market Volatility
Tim Bollerslev
Duke University and NBER
International Conference on FinanceCopenhagen, September 2-4, 2005
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and Francis X. Diebold), unpublished manuscript.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.
· 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
· 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
· Continuous Time Diffusion
dp(t) = µ(t) dt + σ(t) dW(t)
· One Period Notional (Actual) Volatility
[ p , p ]t%1
& [ p , p ]t
' mt%1
t
σ2(s) ds
· Return Variance Conditional on {F(s), t# s<t+1}
· Option Pricing and Stochastic Volatility Hull and White (1987)
· Realized Volatility
RVt%1
(∆ ) / j1/∆
j'1
(p( t% j∆ ) & p( t% (j&1)∆ ) )2 6 mt%1
t
σ2(s) ds
· Logarithmic Price Process / Special Semimartingale
p(t) - p(0) / r(t) = µ(t) + m(t)
µ - Predictable, Finite Variationm - Local Martingale
· Quadratic Variation
[ p , p ]t = p(t)2 - 2I0t p-(τ)dp(τ) = [m c,m c
]t + Σ0#τ#t (∆p(τ))2
p2(t) - [p , p ]t Local Martingale
plimn64 { p2(0) + Σj$1 [ p(tvτn,j) - p(tvτn,j-1) ]
2 } 6 [ p , p ]t
· Notional (Actual )Volatility from t-h to t
[ p , p ]t - [ p , p ]t-h Andersen, Bollerslev and Diebold (2004)
Andersen, Bollerslev, Diebold and Labys (2001, 2003)
Barndorff-Nielsen and Shephard (2002, 2003, 2004)
· Realized Volatility from t-h to t
RVt&h , t
(∆ ) / jh/∆
j'1
(p( t&h% j∆ ) & p( t&h% (j&1)∆ ) )2
· Theory of Quadratic Variation - )60
RVt&h, t
(∆ ) 6 [ p , p ]t& [ p , p ]
t&h
· High-Frequency Data - Notional Volatility (Almost) Observable
· Measuring, Modeling, and Forecasting Realized (FX) VolatilityAndersen, Bollerslev, Diebold and Labys (2001, JASA)
Andersen, Bollerslev, Diebold and Labys (2003, Econometrica)
· Data· DM/$ and Yen/$ Spot FX Quotations 12/1/86 - 06/30/99· 4,500 DM/$ and 2,000 Yen/$ Quotes per Day
· Theory: ) 6 0
· Market Microstructure Frictions: ) > 0
· Practical Measurements: ) = 48 - 288
· Unconditional Distribution of RV (approximately) Log-Normal
· Long-Memory (type) Dynamic Dependencies in RV
· AR-RV Long-Memory Model
A(L)(1 & L)0.4 ( log(RVt) & µ ) ' g
t
· Mincer-Zarnowitz Style RegressionMincer and Zarnowitz (1969)
Chong and Hendry (1986)
RVt%1
' b0% b
1AR&RV
t%1*t % b2Other
t%1*t % ut%1
b0 b1 b2 R2
AR-RV -0.01 (.02) 1.06 (.04) - 0.36
AR-ABS 0.23 (.02) - 1.21 (.06) 0.16
RiskMetric 0.11 (.02) - 0.77 (.03) 0.26
GARCH -0.07 (.03) - 1.01 (.04) 0.27
HF-FIEGARCH -0.17 (.03) - 1.23 (.05) 0.32
AR-RV + AR-ABS -0.02 (.02) 1.02 (.05) 0.11 (.07) 0.36
AR-RV + RiskMet. -0.02 (.02) 0.94 (.06) 0.12 (.05) 0.36
AR-RV + GARCH -0.05 (.02) 0.94 (.06) 0.16 (.06) 0.36
AR-RV + HF-FIEGRC -0.07 (.03) 0.81 (.07) 0.33 (.10) 0.36Mincer-Zarnowitz Regressions, DM/$, In-Sample (1986-96), One-Day-Ahead
Andersen, Bollerslev, Diebold and Labys (2003, Econometrica)
b0 b1 b2 R2
AR-RV 0.02 (.05) 0.99 (.09) - 0.25AR-ABS 0.44 (.03) - 0.45 (.09) 0.03RiskMetric 0.22 (.04) - 0.62 (.08) 0.10GARCH 0.05 (.06) - 0.85 (.11) 0.10HF-FIEGARCH -0.07 (.06) - 1.01 (.10) 0.26AR-RV + AR-ABS 0.04 (.05) 1.02 (.11) -0.11 (.10) 0.25AR-RV + RiskMetric 0.02 (.05) 0.98 (.13) 0.01 (.11) 0.25AR-RV + GARCH 0.02 (.06) 0.98 (.13) 0.02 (.16) 0.25AR-RV + HF-FIEGRC -0.07 (.06) 0.40 (.19) 0.66 (.20) 0.27
Mincer-Zarnowitz Regressions, DM/$, Out-of-Sample (1996-99), One-Day-Ahead
Andersen, Bollerslev, Diebold and Labys (2003, Econometrica)
· R2AR-RV . R2
AR-RV + Other
· Other MarketsAreal 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)
· Continuous Time Diffusion
dp(t) = µ(t) dt + σ(t) dW(t)
· One Period Notional (Actual) Volatility
[ p , p ]t%1
& [ p , p ]t
' mt%1
t
σ2(s) ds
· Realized Volatility
RVt%1
(∆ ) / j1/∆
j'1
(p( t% j∆ ) & p( t% (j&1)∆ ) )2 / j1/∆
j'1
r2
t% j@∆ ,∆
6 mt%1
t
σ2(s) ds
· Continuous Time Jump Diffusion
dp(t) = µ(t) dt + σ(t) dW(t) + κ(t) dq(t)
q(t): Counting Process8(t): Time-Varying Intensity
P[dq(t) = 1] = 8(t)dt
6(t): Size of Jumps6(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)
· 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 ]t%1
& [ p , p ]t
' mt%1
t
σ2(s) ds % j
t<s#t%1
κ2(s)
· Realized Volatility
RVt%1
(∆ ) / j1/∆
j'1
r2
t% j@∆ ,∆ 6 mt%1
t
σ2(s)ds % j
t<s#t%1
κ2(s)
· Power VariationAït-Sahalia (2003)
Barndorff-Nielsen and Shephard (2003a)
RPVt%1
(∆ ) / µ&1
p ∆1& p/2 j
1/∆
j'1
* rt% j@∆ ,∆
*p
6 mt%1
t
σ2(s)ds % j
t<s#t%1
κ2(s) p ' 2
6 mt%1
t
σp(s)ds 0 < p < 2
6 4 p > 2
· (Standardized) Bi-Power VariationBarndorff-Nielsen and Shephard (2004a, 2005)
BVt%1
(∆ ) / µ&2
1 j1/∆
j'2
* rt% j@∆ ,∆
* * rt% (j&1)@∆ ,∆
*
6 mt%1
t
σ2(s)ds
· Jump Component
RVt%1
(∆ ) & BVt%1
(∆ ) 6 jt<s#t%1
κ2(s)
· Non-Negativity Truncation
Jt%1
(∆ ) / max[ RVt%1
(∆ ) & BVt%1
(∆ ) , 0 ]
· 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 - ) 6 0
· Market Microstructure Frictions· Discreteness· Bid-Ask Spread Positioning· Unevenly Spaced Observations
· Linearly Interpolated Five-Minute Returns· ) = 1/288 DM/$· ) = 1/97 S&P500 and T-Bond
Aït-Sahalia, Mykland and Zhang (2005)
Andersen, Bollerslev, Diebold and Labys (2000)
Andreou and Ghysels (2002), Areal and Taylor (2002)
Bandi and Russell (2004a,b)
Barndorff-Nielsen, Hansen, Lunde and Shephard (2004)
Barucci and Reno (2002), Bollen and Inder (2002)
Corsi, Zumbach, Müller, and Dacorogna (2001)
Curci and Corsi (2003), Hansen and Lunde (2006)
Malliavin and Mancino (2002), Oomen (2002, 2004)
Zhang (2004), Zhang, Aït-Sahalia and Mykland (2005), Zhou (1996)
Table 1A
Summary Statistics for Daily DM/$ Realized Volatilities and Jumps
____________________________________________________________________________________
RVt RVt1/2 log(RVt ) Jt Jt
1/2 log(Jt+1)
Mean 0.508 0.670 -0.915 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 -2.961 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
____________________________________________________________________________________
Table 1B
Summary Statistics for Daily S&P500 Realized Volatilities and Jumps
____________________________________________________________________________________
RVt RVt1/2 log(RVt ) Jt Jt
1/2 log(Jt+1)
Mean 1.137 0.927 -0.400 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 -2.850 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
____________________________________________________________________________________
Table 1C
Summary Statistics for Daily U.S. T-Bond Realized Volatilities and Jumps
____________________________________________________________________________________
RVt RVt1/2 log(RVt ) Jt Jt
1/2 log(Jt+1)
Mean 0.286 0.506 -1.468 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 -3.633 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
____________________________________________________________________________________
· Consistent Jump Measurements
Jt%1
(∆ ) / max[ RVt%1
(∆ ) & BVt%1
(∆ ) , 0 ]
· Empirically “Too Many” Small Jumps
· Significant Jumps
· Asymptotic ()60) Distribution in the Absence of Jumps
Barndorff-Nielsen and Shephard (2004a, 2005)
∆&1/2
RVt%1
(∆ ) & BVt%1
(∆ )
[ (µ&4
1 % 2µ&2
1 & 5) mt%1
t
σ4(s )ds ]1/2
Y N ( 0 , 1 )
- Non-Feasible
· Realized QuarticityBarndorff-Nielsen and Shephard (2002a)
Andersen, Bollerslev and Meddahi (2005)
RQt%1
(∆ ) / ∆&1 µ
&1
4 j1/∆
j'1
r4
t% j∆ ,∆ 6 mt%1
t
σ4(s)ds
- Diverges in the Presence of Jumps
· Realized Tri-Power Quarticity
TQt%1
(∆ ) / ∆&1 µ&3
4/3 j1/∆
j'3
*rt% j∆ ,∆
*4/3*rt% (j&1)∆ ,∆
*4/3*rt% (j&2)∆ ,∆
*4/3
6 mt%1
t
σ4(s)ds
- Even in the Presence of Jumps
· Feasible Test Statistic for Jumps
Wt%1
(∆ ) / ∆&1/2
RVt%1
(∆ ) & BVt%1
(∆ )
[ (µ&4
1 % 2µ&2
1 & 5) TQt%1
(∆ ) ]1/2
· Variance Stabilizing Ratio Transformation and Max. Adjustment
Zt%1
(∆ ) /
∆&1/2
[ RVt%1
(∆ ) & BVt%1
(∆ ) ] RVt%1
(∆ )&1
[ (µ&4
1 % 2µ&2
1 & 5) max{ 1 , TQt%1
(∆ ) BVt%1
(∆ )&2 }]1/2
- Better Behaved in Finite ()>0) SamplesHuang and Tauchen (2005)
· (“Significant”) Jumps
Jt%1,α
(∆ ) / I [ Zt%1
(∆ ) > Φα
] @ [ RVt%1
(∆ ) & BVt%1
(∆ ) ]
- Depends upon choice of " (and ) )
- Previous Jt+1 corresponds to " = 0.5
- Shrinkage type estimator
· Continuous Sample Path Variation
Ct%1,α
(∆ ) / I [ Zt%1
(∆ ) # Φα
] @ RVt%1
(∆ ) %
I [ Zt%1
(∆ ) > Φα
] @ BVt%1
(∆ )
· Realized Variation
RVt%1
(∆ ) ' Ct%1,α
(∆ ) % Jt%1,α
(∆ )
· Theory - ) 6 0
· Market Microstructure Frictions· Discreteness· Bid-Ask Spread Positioning· Unevenly Spaced Observations
· Fixed ) $ * >> 0 ; e.g. Five-Minute ReturnsAndersen, Bollerslev, Diebold and Labys (2000, 2001)
· Pre-Filtering, Kernel and Fourier MethodsAndreou 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 SchemesAït-Sahalia, Mykland and Zhang (2005)
Bandi and Russell (2004a,b), Müller (1993)
Zhang (2004), Zhang, Aït-Sahalia and Mykland (2005)
· High-Frequency Returns
rt,∆
/ p ((t) % ν(t) & p ((t&∆) & ν(t&∆) / r(
t,∆ % ηt,∆
p*(t): “Fundamental” Price
<(t): Market Microstructure “Noise”
· Realized Variation
- E[r2
t,) ] ' E[ (r(
t,) % 0t,) )2 ] … E[ (r
(
t,) )2 ]
- Noise Term Dominates the Variation for )60
- RVt()) Formally Inconsistent as )60
- Motivates )$* >> 0 and other Techniques
· High-Frequency Returns
rt,∆
/ p ((t) % ν(t) & p ((t&∆) & ν(t&∆) / r(
t,∆ % ηt,∆
p*(t): “Fundamental” Price
<(t): Market Microstructure “Noise”
· Realized Bi-Power Variation
- E[*rt ,)* ] ' E[*r
(
t ,) % 0t ,)* ] … E[*r
(
t ,)* ]
- E[*rt% j@) ,)**r
t% (j&1)@) ,)* ] … E[*r(
t% j@) ,)**r(
t% (j&1)@) ,)* ]
- RVt()) - BVt()) too Small and TQt()) too Large
- Test for Jumps Under-Rejects, too Few Jumps
- Staggering, or Skip-One, Breaks Dependence
· “Standard” Bi-Power Variation
BVt%1
(∆ ) / µ&2
1 j1/∆
j'2
* rt% j@∆ ,∆
* * rt% (j&1)@∆ ,∆
*
· Staggered Bi-Power Variation
BV1, t%1
(∆ ) / µ&2
1 (1 & 2∆ )&1 j1/∆
j'3
* rt% j@∆ ,∆
* * rt% (j&2)@∆ ,∆
*
· Staggered Tri-Power Quarticity may be Defined Similarly
· Test for Jumps based on , and BV1, t%1
() ) TQ1, t%1
() ) Z1, t%1
() )
- α = 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."
· Case Studies
12/10/87: Z1, t%1
(∆ ) ' 10.315
Swelling U.S. trade deficit ($17.6B) announced at 13:30GMT
(8:30 EST).
9/17/92: (max. sample)Z1, t%1
(∆ ) ' &0.326 C1, t%1
(∆ ) ' 3.966
The day following the temporary withdrawal of the British
Pound from the European Monetary System.
WSJ: “The dollar sank more than 2% against the mark as
nervousness persisted in the currency market.”
· Case Studies
6/30/99: Z1, t%1
(∆ ) ' 7.659
FED raised short term rate by ¼ percent at 13:15 CST (14:15
EST), but indicated that it “might not raise rates again in the
near term due to conflicting forces in the economy.”
7/24/02: (max. sample)Z1, t%1
(∆ ) ' &0.704 C1, t%1
(∆ ) ' 27.077
Record Big Board trading volume of 2.77 billion shares.
· Case Studies
8/1/96: Z1, t%1
(∆ ) ' 6.877
National Association of Purchasing Managers (NAPM) index
released at 9:00CST (10:00 EST).
12/7/01: Z1, t%1
(∆ ) ' 0.915
WSJ: Rise in jobless claims increased the expectation that the
FED would lower rates at its Board meeting the following day
(which it did).
- Not all significant jumps map as nicely into specific news
- What causes financial prices to “jump”?
Table 3A
Summary Statistics for Significant Daily DM/$ Jumps
____________________________________________________________________________________
α 0.500 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,α 10.78 9.900 7.821 6.230 19.95
LB10 ,Jt,α + 73.62 116.4 94.19 87.69 34.57
____________________________________________________________________________________
Table 3B
Summary Statistics for Significant Daily S&P500 Jumps
____________________________________________________________________________________
α 0.500 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,α 50.83 31.47 22.67 36.18 49.25
LB10 , Jt,α + 320.8 146.0 77.06 35.11 25.49
____________________________________________________________________________________
Table 3C
Summary Statistics for Significant Daily U.S. T-Bond Jumps
____________________________________________________________________________________
α 0.500 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,α 45.55 100.1 59.86 103.3 81.42
LB10 , Jt,α + 21.23 17.18 15.18 9.090 11.98
____________________________________________________________________________________
· Significant (" = 0.999) Jump Dynamics
· Volatility Modeling and Forecasting
· Strong Unit-Root Like Volatility PersistenceEngle and Bollerslev (1986), Bollerslev and Engle (1993)
· FIGARCH and Long-Memory SV ModelsBaillie, 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 ModelsAndersen, 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 StructuresAndersen 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)
· Heterogeneous AR Realized Volatility (HAR-RV) ModelCorsi (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
· 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 Jt1/2 + $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) + $JM log(Jt-22,t+1) + ,t,t+h
· MIDAS RegressionsGhysels, Santa-Clara and Valkanov (2004, 2005)
· Multiplicative Error Models (MEM)Engle (2002), Engle and Gallo (2005)
Table 4A
Daily, Weekly, and Monthly DM/$ 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 0.083 0.131 0.231 0.096 0.158 0.292 -0.095 -0.114 -0.249
(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 -0.002 0.022 0.001 0.003 0.043 0.024 -0.004
(0.089) (0.040) (0.017) (0.027) (0.017) (0.010) (0.111) (0.076) (0.044)
$JW -0.191 -0.179 -0.073 -0.006 0.001 0.002 -0.076 -0.317 -0.127
(0.168) (0.199) (0.125) (0.033) (0.044) (0.028) (0.239) (0.327) (0.242)
$JM -0.001 0.055 -0.014 -0.034 -0.011 0.014 -0.690 -0.301 -0.261
(0.329) (0.460) (0.604) (0.057) (0.087) (0.127) (0.408) (0.668) (0.990)
R2HAR-RV-CJ 0.368 0.427 0.361 0.443 0.486 0.397 0.485 0.514 0.415
____________________________________________________________________________________
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 0.143 0.222 0.393 0.062 0.103 0.202 -0.063 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 -0.153 -0.016 0.005 -0.043 -0.013 0.005 -0.006 -0.027 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)
R2HAR-RV-CJ 0.421 0.574 0.478 0.613 0.700 0.639 0.696 0.763 0.722
____________________________________________________________________________________
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 0.085 0.095 0.133 0.133 0.166 0.236 -0.337 -0.335 -0.473
(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 -0.075 0.090 0.043 -0.004 0.775 0.227 -0.289
(0.122) (0.081) (0.067) (0.033) (0.029) (0.025) (0.390) (0.298) (0.271)
$JM -0.271 -0.145 -0.116 -0.113 -0.076 -0.057 -1.319 -0.477 -0.034
(0.177) (0.216) (0.245) (0.045) (0.058) (0.075) (0.589) (0.773) (0.918)
R2HAR-RV-CJ 0.144 0.325 0.377 0.192 0.353 0.393 0.222 0.365 0.400
____________________________________________________________________________________
Figure 3A
Daily, Weekly and Monthly DM/$ Realized Volatilities and HAR-RV-CJ Forecasts
Figure 3B
Daily, Weekly and Monthly S&P500 Realized Volatilities and HAR-RV-CJ Forecasts
Figure 3C
Daily, Weekly and Monthly T-Bond Realized Volatilities and HAR-RV-CJ Forecasts
Summary
· Formal and effective framework for incorporating high-frequency financial data into volatility modeling andforecasting through easy-to-implement non-parametriclower 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
Extensions/Ongoing Work
· Reduced form modeling and forecasting of Jt," and Ct,"
Andersen, Bollerslev and Huang (2005)
· Effective score generator for EMM estimationBollerslev, Kretschmer, Pigorsch and Tauchen (2005)
· Pricing of jump and continuous sample path variabilityBollerslev, Huang and Zhou (2005)
· MDH, leverage effects and (signed) jumpsAndersen, Bollerslev and Dobrev (2005)
· Risk management, tails, and VaR
· Market microstructure “noise” and jump measurementsAndersen, Bollerslev, Frederiksen and Nielsen (2005)
· Jumps and (macro) economic news arrivalsAndersen, Bollerslev, Diebold and Vega (2005)
· Multivariate jump measurements and co-jumping
0
100
200
300
400
500
600
700
800
900
-7.5 -5.0 -2.5 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 -8.696761
Std. Dev. 1.024471
Skewness -0.245350
Kurtosis 8.566824
Jarque-Bera 4244.720
Probability 0.000000
-4
-3
-2
-1
0
1
2
3
4
-12 -8 -4 0 4 8
RET
No
rma
l Qu
an
tile
· Fat tailed (unconditional) return distributions
rt Not N( @ , @ )Fama (1965), Mandelbrot (1963)
0
50
100
150
200
250
300
350
-2 -1 0 1 2 3
Series: STRET
Sample 1765 5026
Observations 3262
Mean 0.076853
Median 0.074362
Maximum 3.658999
Minimum -2.730636
Std. Dev. 0.983838
Skewness 0.100477
Kurtosis 2.808346
Jarque-Bera 10.48102
Probability 0.005298
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
STRET
No
rma
l Qu
an
tile
· Approximate normality
rt / RVt1/2 - N( 0 , 1 )
Andersen, Bollerslev, Diebold and Ebens (2001)
Andersen, Bollerslev, Diebold and Labys (2000, 2001)
0
50
100
150
200
250
300
350
-2.50 -1.25 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 -2.989198
Std. Dev. 1.001778
Skewness 0.128175
Kurtosis 2.950615
Jarque-Bera 9.263268
Probability 0.009739
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
STRETMJ
No
rma
l Qu
an
tile
· Even closer approximate normality
( rt ± Jt½ ) / BVt
1/2 - N( 0 , 1 )Andersen, Bollerslev and Dobrev (2005)