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If you have any questions, please contact: Consumer Affairs Hasbm, Inc. 1027 Newport Avenue. Pawtucket, RI Tel: 1-800-844-3733 Fax: 1-401-431-8082 ELECTRONICS Product and colors may vary. O 2006 Hasbro. All Rights Reserved. TM & 0 denote U.S. Trademarks. PN 6581780000
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a CAUTION: TO AVOID BATTERY LEAKAGE
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IMPORTANT: BATTERY INFORMATION
Please retain this information for future reference. Batteries should be replaced by an adult.

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1. Press h e
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2.To st the resolution of your picture, press e 2048 x 1536, 1600 x 1200 or 640 x 480. 3. Press the Zoom button, picture.

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2. Press the T or W on the Zoom buwon t preview one picture o o r multiple pictures on the LCD/Viewfinder.

6. to switch between

T or W, to adjust the frame of your

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or @) to select the picture that you want to view.
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NOTE: If ou have an MMC/SD Card inserted (not included), your picture wii be automalically stored to theMMC/SD Card, not the internal memory and will be reviewed from his location. If you want to take ictures and review them from the internal memory only, remove the C/SD Card.
into the default DCIM11@@Media folder o f this camera o r t r y t o rename the files in this folder. If you do, the camera may haue a problem reading the memory card and i n some cases the camera may not operate.normally. Therefore w e strongly recommend you t o moue your images t o your hard disk before any change.

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The equipment is t o be supplied f r o m a n idemtified USB p o r t complying w i t h t h e requirementsofLimitedPowerSource.

82-000-02010

Financial Market Volatility:
From ARCH and GARCH to High-Frequency Data and Realized Volatility

2006 Zeuthen Lectures

University of Copenhagen December 6-8, 2006 Tim Bollerslev Duke University and NBER (i) Volatility Modeling: A Brief Historical Perspective (ii) High-Frequency Data and Realized Volatility (iii) Current Themes and New Directions
Volatility Modeling: A Brief Historical Perspective
C Volatility Clustering C Why Care? C ARCH and GARCH Models C Stochastic Volatility Models C Two Fallacies in Volatility Measurement and Modeling C High-Frequency Data and Realized Volatility

Zeuthen I, 6/12/06 - 1

Volatility Clustering
C Long history of seeing speculative returns as approximately serially uncorrelated over short (daily or weekly) horizons
C Random walk (Martingale) model
C Weak form efficiency C Old view Variance of (and rt ) constant

Zeuthen I, 6/12/06 - 2

S&P500

0 -4 -03 04

S & P 0 , D a ily R e t u r n s
C Old view of constant variance/volatility obviously violated C Volatility clusters in time
. large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes.
Mandelbrot (1963, J. Business)

Zeuthen I, 6/12/06 - 3

0 --04
M S F T , D a ily R e t u rn s

0 -2 -2004

E u ro / D o lla r, D a ily R e t u r n s

Zeuthen I, 6/12/06 - 4

U.K. consol prices, 1729-1957
Brown, Burdekin and Weidenmeir (2006, Journal of Financial Economics)

Zeuthen I, 6/12/06 - 5

C Old literature mainly concerned with rationalizing (fat-tailed) unconditional return distributions Stable Paretian distribution
Mandelbrot (1963, J. Business) Fama (1965, J. Business)

Student-t distribution

Praetz (1972, J. Business) Blattberg and Gonedes (1974, J. Business)
Mixture-of-Distributions Hypothesis (MDH)
Clark (1973, Econometrica) Tauchen and Pitts (1983, Econometrica)
C But, volatility clustering suggests that the conditional return distributions are timevarying

C Why do/should we care?

Zeuthen I, 6/12/06 - 6

Why Care?

C Volatility/risk is central to finance Sign forecasting and market timing Risk measurement and management Asset and option pricing Portfolio allocation .

Zeuthen I, 6/12/06 - 7

C Sign Forecasting and Market Timing
Andersen, Bollerslev, Christoffersen and Diebold (2006, Handbook of Economic Forecasting)

Same mean but

differs
Bankruptcy (assets < liabilities) Credit risk

Zeuthen I, 6/12/06 - 8

C Risk Measurement and Management Value-at-Risk (VaR) Specific quantile in loss/return distribution Basel Accord I and II

1% One-Day VaRs

Standard historical VaR correct on average, but.

Zeuthen I, 6/12/06 - 9

C Option Pricing
Bollerslev and Mikkelsen (1999, Journal of Econometrics)
Black-Scholes assumes constant

Zeuthen I, 6/12/06 - 10

C Asset Pricing CAPM

Standard

Conditional
Consumption-based CAPM Multi-factor (APT) models

Zeuthen I, 6/12/06 - 11

C Portfolio Allocation Mean-variance efficient portfolios
18.0% 16.0% 14.0% 12.0% 10.0% 8.0% 6.0% 4.0% 2.0% 0.0% 0.0%

Expected Return

Stocks Bonds Real Estate Commodities

Standard Deviation

Zeuthen I, 6/12/06 - 12
C Volatility clusters in time C Volatility crucially important for finance C But how to predict as opposed to merely measure volatility? Need a statistical model ARCH/GARCH and Stochastic Volatility models

Zeuthen I, 6/12/06 - 13

ARCH/GARCH Models
C Standard time series model
Conditional mean Conditional variance

time-varying constant

C Allow the conditional mean and the conditional variance to both depended non-trivially on t-1 But how?

Zeuthen I, 6/12/06 - 14

C AutoRegressive Conditional Heteroskedasticity - ARCH
Engle (1982, Econometrica)
C ARCH(q) Model Rolling sample variance
More weight to recent observations
But, we need a well-defined statistical model as opposed to merely a measure of volatility
In practice, large q and (too) many alphas

Zeuthen I, 6/12/06 - 15

C Generalized ARCH, or GARCH
Bollerslev (1986, J. of Econometrics)

GARCH(p,q)

Simple GARCH(1,1) typically works well
GARCH(1,1) implies ARMA(1,1) for
Squared returns serially correlated

Zeuthen I, 6/12/06 - 16

Daily S&P500 returns and one-day-ahead GARCH(1,1) 95% confidence bands, or 2.5% VaR

-2 -4 -6 -2004

-2 -4 -2004

Zeuthen I, 6/12/06 - 17

C Extensions and Refinements Volatility persistence Conditional error distributions Variance-in-mean effects Volatility asymmetries Multivariate models .

Zeuthen I, 6/12/06 - 18

C Incomplete list of Models and Acronyms
ARCH GARCH IGARCH GARCH-t Log-GARCH TS-GARCH ARCH-M AGARCH EGARCH A-GARCH NGARCH QARCH STARCH TGARCH GJR-GARCH QTARCH NA-GARCH V-GARCH A-PARCH SWARCH PGARCH H-GARCH FIGARCH FIEGARCH Aug-GARCH HYGARCH. Engle (1982) Bollerslev (1986) Bollerslev and Engle (1986) Bollerslev (1987) Geweke (1986), Milhj (1987), Pantula (1986) Taylor (1986), Schwert (1989) Engle, Lilien and Robins (1987) Engle (1990) Nelson (1991) Bera, Higgins and Lee (1992) Higgins and Bera (1992) Sentana (1992) Harvey, Ruiz and Sentana (1992) Zakoian (1994) Glosten, Jagannathan and Runkle (1993) Gourieroux and Monfort (1992) Engle and Lee (1993) Engle and Lee (1993) Ding, Granger and Engle (1993) Hamilton and Susmel (1994) Bollerslev and Ghysels (1996) Hentschel (1995) Baillie, Bollerslev and Mikkelsen (1996) Bollerslev and Mikkelsen (1996) Duan (1997) Davidson (2004)

Zeuthen I, 6/12/06 - 19

Zeuthen I, 6/12/06 - 20
Stochastic Volatility Models
C GARCH is a discrete-time model Largely empirically motivated and ad hoc
C Finance theory often cast in continuous-time Continuous-time random walk
Black-Scholes and many other pricing formula But, of course, F is not constant

Zeuthen I, 6/12/06 - 21

C Time-varying diffusive volatility
C First generation models Cox-Ingersoll-Ross (CIR) model: Constant Elasticity of Variance (CEV): Soundly rejected empirically
C Allow F(t) to follow a separate stochastic process But how?

Zeuthen I, 6/12/06 - 22

C GARCH diffusion
Nelson (1990, Journal of Econometrics)
Diffusion limit of GARCH(1,1)

C Heston model

Heston (1993, Review of Financial Studies)
Like an AR(1) model for Closed form options prices

C Log-volatility model

Like an EGARCH model

C Multi-factor models

Zeuthen I, 6/12/06 - 23
C Estimation and inference for SV models generally much harder than for ARCH/GARCH models is latent not available in closed form
Simulated methods of moments, EMM, Bayesian MCMC, characteristic function approach,. Reprojection methods, particle filters,.

Zeuthen I, 6/12/06 - 24

C In spite of the empirical successes of the ARCH/GARCH/SV class of models mainstream academic finance has been relatively slow to adapt the ideas. Why? C Two related fallacies It is trivial to accurately measure volatility given high-frequency data Existing volatility models generally provide poor forecasts, so why bother?

Zeuthen I, 6/12/06 - 25

It is trivial to measure volatility
C Continuous-time random walk
Merton (1980, Journal of Financial Economics)
Over discrete time-intervals, )
Sample mean (MLE) based on n=h/ returns over (t-h,t]
Estimation of mean return/drift is hard and depends primarily on the span of the data

Zeuthen I, 6/12/06 - 26

Sample variance (uncentered) based on n=h/ returns over (t-h,t]
Estimation of volatility is easy given high-frequency data

Zeuthen I, 6/12/06 - 27

C But, of course, F(t) is time-varying
Nelson (1992, Journal of Econometrics) Nelson and Foster (1994, Econometrica)
Sample variance estimator (uncentered) with n=h/ returns over (t-h,t]
Double asymptotic consistency (continuous sample paths of (t) )

h Bias

n Variance
Hard to mimic in practice

Zeuthen I, 6/12/06 - 28

Single asymptotic and efficiency improvements by weighting

GARCH(1,1) filter

Consistent for
Elegant theory, but doesnt work with actual high-frequency data

Zeuthen I, 6/12/06 - 29

C GARCH(1,1) estimates for different values of n
S&P500 returns, n = 2 (half-day),., 80 (5-minute)
n 2 T@n 79,280 39,640 19,820 15,856 9,910 7,928 4,955 3,964 1,982 "(n) 0.140 0.180 0.223 0.230 0.053 0.048 0.148 0.060 0.108 $(n) 0.837 0.765 0.664 0.630 0.935 0.940 0.764 0.890 0.798 "(n)+$(n) 0.977 0.945 0.887 0.861 0.988 0.988 0.912 0.951 0.906 Half-Life (minutes) 2,213 2,1,376 1,397
Andersen and Bollerslev (1997, Journal of Empirical Finance)
Estimates for "n and $n and implied half-lives vary erratically with n Why?

Zeuthen I, 6/12/06 - 30

C Practical market microstructure complications Price discreteness Bid-ask spreads Non-synchronous trades/quotes Intraday volatility patterns

Zeuthen I, 6/12/06 - 31

Intraday volatility patterns Intraday seasonality Intraday periodicities Diurnal patterns Circadian rhythm
The bull and bears are marking their territories they`re leading the blind with their international glories Hong Kong is present Taipei awakes all talk of circadian rhythms

Zeuthen I, 6/12/06 - 32

C Practical estimation or filtering of spot volatility from highfrequency data is not as easy as suggested by the MertonNelson theory/intuition

Zeuthen I, 6/12/06 - 33

Volatility models provide poor forecasts
C GARCH(1,1) (and other volatility models) typically provides a good in-sample fit
But, is it really a good model for volatility forecasting?
C Mincer-Zarnowitz style ex-post forecast evaluation regression Realization t = b0 + b1 Forecast t|t-1 + ut C Volatility forecasting
Typically very low R2.0.05 Seemingly suggests poor forecasts

Zeuthen I, 6/12/06 - 34

Cumby, Figlewski and Hasbrouch (1993, Journal of Derivatives)

Zeuthen I, 6/12/06 - 35

Jorion (1995, Journal of Finance)

Zeuthen I, 6/12/06 - 36

Pagan and Schwert (1990, Journal of Econometrics)

Zeuthen I, 6/12/06 - 37

C Low R2s from Mincer-Zarnowitz ex-post forecast regression
Simulated continuous-time GARCH(1,1) model
3.0 2.5 2.0 1.5 1.0 0.5 0.2000 2500

GARCH(1,1)

Squared Returns
is an extremely noisy measure of the true ex-post volatility
C So how do you measure volatility?

Zeuthen I, 6/12/06 - 38

C Stochastic volatility model
Variance of Integrated volatility/variation
3.0 2.5 2.0 1.5 1.0 0.5 0.IV 2000 2500

Zeuthen I, 6/12/06 - 39

C Standard volatility models are good at forecasting IVt+t

R2.0.05

R2.0.50
C But, how do you measure the integrated volatility in practice?

Zeuthen I, 6/12/06 - 40

Realized Volatility
C Latent integrated volatility
C Realized volatility from high-frequency data
C Theory of quadratic variation

Zeuthen I, 6/12/06 - 41

RVt(1)

2000 2500

RVt(1/3)

RVt(1/24)

RVt(1/288)

Zeuthen I, 6/12/06 - 42

C Nice theory, but does it work with actual high-frequency data? Yes!
C By measuring the variation over non-trivial daily (and longer) time-intervals the realized volatility avoids Intraday volatility patterns, or circadian rhythms Double asymptotic required for consistently estimating F(t)

Zeuthen I, 6/12/06 - 43

C Mincer-Zarnowitz forecast regression with daily squared returns
R2 = 0.047 Theory R2 = 0.063
Andersen and Bollerslev (1998, International Economic Review)

Zeuthen I, 6/12/06 - 44

C Mincer-Zarnowitz forecast regression with daily realized volatility
R2 = 0.479 Theory R2 = 0.495

Zeuthen I, 6/12/06 - 45

Same GARCH(1,1) forecasts, different ex-post volatility measures
C Yes, standard volatility models do provide good forecasts when properly evaluated

Zeuthen I, 6/12/06 - 46

Summary
C Volatility of speculative returns cluster in time C Volatility is crucially important for financial risk management and asset pricing C ARCH/GARCH models provide a simple discrete-time statistical framework for modeling and forecasting time-varying volatility C Continuous-time stochastic volatility models are often more convenient to work with from a theoretical asset pricing perspective C Continuous-time/spot volatility is not easily extracted from high-frequency data due to a host of practical market microstructure complications C The realized volatility concept provides simple model-free measures of volatility over non-trivial time intervals

Zeuthen I, 6/12/06 - 47

Tomorrow
C Theory of realized volatility Intuition Asymptotic distribution C Distribution of realized volatility Practical data considerations Empirical distributions C Modeling and forecasting realized volatility AR-RV model Why does it work? C Jumps Power and bi-power variation AR-RV-J model