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and Francis X. Diebold), unpublished manuscript.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)