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
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in the Measurement, Modeling and Forecasting of Return Volatility" (with Torben G. Andersen and Francis X. Diebold ), unpublished manuscript.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 + � (t) dW(t) · One Period Notional ( Actual) Volatility t � 1 � � 2 ( s ) ds [ p , p ] t � 1 � [ p , p ] t � t · Return Variance Conditional on { � (s), t � s<t+1} · Option Pricing and Stochastic Volatility Hull and White (1987) · Realized Volatility t � 1 � 1/ � � � � RV t � 1 ( � ) ( p ( t � j � ) � p ( t � ( j � 1) � )) 2 � 2 ( s ) ds j � 1 t
· 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 - 2 � 0 t p - ( � )dp( � ) = [m c ,m c ] t + � 0 � � � t ( � p( � )) 2 p 2 (t) - [p , p ] t Local Martingale plim n �� { p 2 (0) + � j � 1 [ p(t � � n,j ) - p(t � � n,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)
· Realized Volatility from t-h to t � h / � � RV t � h , t ( � ) ( p ( t � h � j � ) � p ( t � h � ( j � 1) � )) 2 j � 1 Theory of Quadratic Variation - � � 0 · � RV t � h,t ( � ) [ p , p ] t � [ p , p ] t � h · High-Frequency Data - Notional Volatility (Almost) Observable
· 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
· AR-RV Long-Memory Model A(L)(1 � L) 0.4 (log(RV t ) � µ ) � � t · Mincer-Zarnowitz Style Regression Mincer and Zarnowitz (1969) Chong and Hendry (1986) RV t � 1 � b 0 � b 1 AR � RV t � 1 � t � b 2 Other t � 1 � t � u t � 1 R 2 b 0 b 1 b 2 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) 0.36 Mincer-Zarnowitz Regressions, DM/$, In-Sample (1986-96), One-Day-Ahead Andersen, Bollerslev, Diebold and Labys (2003, Econometrica )
R 2 b 0 b 1 b 2 AR-RV 0.02 (.05) 0.99 (.09) - 0.25 AR-ABS 0.44 (.03) - 0.45 (.09) 0.03 RiskMetric 0.22 (.04) - 0.62 (.08) 0.10 GARCH 0.05 (.06) - 0.85 (.11) 0.10 HF-FIEGARCH -0.07 (.06) - 1.01 (.10) 0.26 AR-RV + AR-ABS 0.04 (.05) 1.02 (.11) -0.11 (.10) 0.25 AR-RV + RiskMetric 0.02 (.05) 0.98 (.13) 0.01 (.11) 0.25 AR-RV + GARCH 0.02 (.06) 0.98 (.13) 0.02 (.16) 0.25 AR-RV + HF-FIEGRC -0.07 (.06) 0.40 (.19) 0.66 (.20) 0.27 Mincer-Zarnowitz Regressions, DM/$, Out-of-Sample (1996-99), One-Day-Ahead Andersen, Bollerslev, Diebold and Labys (2003, Econometrica ) AR-RV � R 2 R 2 · 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)
· Continuous Time Diffusion dp(t) = µ(t) dt + � (t) dW(t) · One Period Notional ( Actual) Volatility t � 1 � � 2 ( s ) ds [ p , p ] t � 1 � [ p , p ] t � t · Realized Volatility � � 1/ � 1/ � r 2 RV t � 1 ( � ) � ( p ( t � j � ) � p ( t � ( j � 1) � )) 2 � t � j � � , � j � 1 j � 1 t � 1 � � � 2 ( s ) ds t
· 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)
· One Period Notional ( Actual) Volatility Andersen, Bollerslev and Diebold (2004) Andersen, Bollerslev and Diebold and Labys (2003) Barndorff-Nielsen and Shephard (2002, 2003) t � 1 � � � 2 ( s ) ds � 2 ( s ) [ p , p ] t � 1 � [ p , p ] t � � t < s � t � 1 t · Realized Volatility � 1/ � t � 1 � � r 2 RV t � 1 ( � ) � � � 2 ( s ) ds � � 2 ( s ) t � j � � , � j � 1 t < s � t � 1 t
· Power Variation Aït-Sahalia (2003) Barndorff-Nielsen and Shephard (2003a) p � 1 � p /2 � 1/ � µ � 1 RPV t � 1 ( � ) � � r t � j � � , � � p j � 1 t � 1 � � � � 2 ( s ) ds � � 2 ( s ) p � 2 t < s � t � 1 t t � 1 � � � p ( s ) ds 0 < p < 2 t � � p > 2
· (Standardized) Bi-Power Variation Barndorff-Nielsen and Shephard (2004a, 2005) 1 � 1/ � µ � 2 � � r t � j � � , � � � r t � ( j � 1) � � , � � BV t � 1 ( � ) j � 2 t � 1 � � � 2 ( s ) ds t · Jump Component � RV t � 1 ( � ) � BV t � 1 ( � ) � � 2 ( s ) t < s � t � 1 · Non-Negativity Truncation J t � 1 ( � ) � max[ RV t � 1 ( � ) � BV t � 1 ( � ) , 0 ]
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