Univariate Volatility Modeling

Transcription

1 Univariate Volatility Modeling Kevin Sheppard Oxford MFE This version: January 10, 2013 January 14, 2013

2 Financial Econometrics (Finally) This term Volatility measurement and modeling Value-at-Risk measurement Multiple time-series: Vector Autoregressions Correlation measurement and modeling Modeling non-linear dependence 2 / 31

3 Lectures Change in the lecture schedule Lectures Monday and Tuesday Classes Monday From week 3 Computing/Modeling Classes 9:00-11:00 Tuesday From week 2 Weekly PS due on Thursdays now Week after distributed, still 11 days Will take feedback on this schedule after week 2 3 / 31

4 Volatility Overview What is volatility? Why does it change? What are ARCH, GARCH, TARCH, EGARCH, SWARCH, ZARCH, APARCH, STARCH, etc. models? What does time-varying volatility look like? What are the basic properties of ARCH and GARCH models? What is the news impact curve? How are the parameters of ARCH models estimated? What about inference? Twists on the standard model Forecasting conditional variance Realized Variance Implied Volatility 4 / 31

5 What is volatility? Volatility Standard deviation Realized Volatility T σ = T 1 (r t µ) 2 t=1 Other meaning: variance computed from ultra-high frequency (UHF) data Conditional Volatility E t [σ t+1 ] Implied Volatility Annualized Volatility ( 252 daily, 12 monthly) Variance 5 / 31

6 Why does volatility change? Possible explanations: News Announcements Leverage Volatility Feedback Illiquidity State Uncertainty None can explain all of the time-variation Most theoretical models have none 6 / 31

7 This arch? 7 / 31

8 No, this arch 8 / 31

9 A basic volatility model: the ARCH(1) model r t = ε t σ 2 t = ω + α 1 ε 2 t 1 ε t = σ t e t i.i.d. e t N(0, 1) Autoregressive Conditional Heteroskedasticity Key model parameters ω sets the long run level α determines both the persistence and volatility of volatility (VoVo or VolVol) 9 / 31

10 Key Properties Conditional Mean: E t 1 [r t ] = E t 1 [ε t ] = 0 More on this later Unconditional Mean: E[ε t ] = 0 Follows directly from the conditional mean and the LIE Conditional Variance: E t 1 [rt 2 ] = E t 1 [ε 2 t ] = σt 2 σ 2 t and et 2 are independent Et 1 [et 2 ] = E[et 2 ] = 1 1 α1 > 0 : Required for stationarity, also α 1 0 ω > 0 is also required for stationarity (technical, but obvious) 10 / 31

11 Unconditional Variance Unconditional Variance E[ε 2 t ] = ω 1 α 1 Unconditional relates the dynamic parameters to average variance E[σt 2 ] = E[ω + α 1 ε 2 t 1] = ω + α 1 E[ε 2 t 1] = ω + α 1 E[σt 1]E[e 2 t 1] 2 (Why?) = ω + α 1 E[σt 1] 2 E[σt 2 ] α 1 E[σt 1] 2 = ω E[σt 2 ](1 α 1 ) = ω σ 2 = ω 1 α 1 11 / 31

12 More properties of the ARCH(1) ARCH models are really AR s in disguise Add ε 2 t σ 2 t to both sides σt 2 = ω + α 1 ε 2 t 1 σt 2 + ε 2 t σt 2 = ω + α 1 ε 2 t 1 + ε 2 t σt 2 ε 2 t = ω + α 1 ε 2 t 1 + ε 2 t σt 2 ε 2 t = ω + α 1 ε 2 t 1 + ν t y t = φ 0 + φ 1 y t 1 + ν t AR(1) in ε 2 t ν t = ε 2 t σt 2 is a mean 0 white noise (WN) process ν t Captures variance surprise : ε 2 t σt 2 = σt 2 (et 2 1) 12 / 31

13 Autocovariance/Autocorrelations Autocovariance E[(ε 2 t σ 2 )(ε 2 t 1 σ 2 )] = α 1 V[ε 2 t ] Same as in AR(1) j th Autocovariance is j th Autocorrelation is α j 1 V[ε2 t ] Corr(ε 2 t, ε 2 t j ) = αj 1 V[ε2 t ] V[ε 2 t ] = α j 1 Again, same as AR(1) ARCH(P) is AR(P) Just apply results from AR models 13 / 31

14 Kurtosis Kurtosis Important Variance is not constant Volatility of Volatility > 0 κ = E [ ε 4 ] t E [ ] ε 2 2 = E [ [ ]] E t 1 ε 4 t t E [ [ ]] E t 1 e 2 t σt 2 2 = E [ [ ] ] E t 1 e 4 t σ 4 t E [ [ ] ] E t 1 e 2 t σ 2 2 t = E [ 3σt 4 ] E [ ] σt 2 2 = 3 E [ σt 4 ] E [ ] σt Alternative: E[σ 4 t ] = V[σ 2 t ] + E[σ 2 t ] 2 Law of Iterated Expectations In ARCH(1): Finite if α 1 < κ = 3(1 α2 1 ) (1 3α 2 1 ) > / 31

15 The ARCH(P) model Definition (P th Order ARCH) An Autoregressive Conditional Heteroskedasticity process or order P is given by r t = µ t + ε t µ t = φ 0 + φ 1 r t φ s r t S σ 2 t = ω + α 1 ε 2 t 1 + α 2 ε 2 t α P ε 2 t P ε t = σ t e t i.i.d. e t N(0, 1). Mean µ t can be an appropriate form - AR, MA, ARMA, ARMAX, etc. E t [r t µ t ] = 0 et is the standardized residual, often assumed normal σ 2 t is the conditional variance 15 / 31

16 Alternative expression of an ARCH(P) Model where both mean and variance are time varying Natural extension of model definition for time varying mean model r t F t 1 N(µ t, σt 2 ) µ t = φ 0 + φ 1 r t φ s r t S σt 2 = ω + α 1 ε 2 t 1 + α 2 ε 2 t α P ε 2 t P ε t = r t µ t r t given the information set at time t 1 is conditionally normal with mean µ t and variance σ 2 t 16 / 31

17 The data S&P 500 Source: Yahoo! Finance Daily January 1, December 31, ,515 observations WTI Spot Prices Source: EIA Daily January December 31, ,815 observations All represented as 100 returns 17 / 31

18 Graphical Evidence of ARCH 10 S&P 500 Returns WTI Returns / 31

19 Graphical Evidence: Squared Data Plot Squared S&P 500 Returns Squared WTI Returns / 31

20 Graphical Evidence: Absolute Data Plot Absolute S&P 500 Returns Absolute WTI Returns / 31

21 A simple GARCH(1,1) r t = ε t σ 2 t = ω + α 1 ε 2 t 1 + β 1 σ 2 t 1 ε t = σ t e t i.i.d. e t N(0, 1) Adds lagged variance to the ARCH model ARCH( ) in disguise σt 2 = ω + α 1 ε 2 t 1 + β 1 σt 1 2 = ω + α 1 ε 2 t 1 + β 1 (ω + α 1 ε 2 t 2 + β 1 σt 2) 2 = ω + β 1 ω + α 1 ε 2 t 1 + β 1 α 1 ε 2 t 2 + β1 2 σt 2 2 More backward subbing in = β1 i ω + β1 i α 1ε 2 t i 1 i=0 i=0 21 / 31

22 Important Properties σ 2 t = ω + α 1 ε 2 t 1 + β 1 σ 2 t 1 Unconditional Variance Kurtosis Stationarity α1 + β 1 < 1 ω > 0, α1 0, β 1 0 ARMA in disguise σ 2 = E[σt 2 ω ] = 1 α 1 β 1 κ = 3(1 + α 1 + β 1 )(1 α 1 β 1 ) 1 2α 1 β 1 3α 2 1 β 2 1 > 3 σt 2 + ε 2 t σt 2 = ω + α 1 ε 2 t 1 + β 1 σt ε 2 t σt 2 ε 2 t = ω + α 1 ε 2 t 1 + β 1 σt ε 2 t σt 2 ε 2 t = ω + α 1 ε 2 t 1 + β 1 ε 2 t 1 β 1 ν t 1 + ν t ε 2 t = ω + (α 1 + β 1 )ε 2 t 1 β 1 ν t 1 + ν t 22 / 31

23 The Complete GARCH model Definition (GARCH(P,Q) process) A Generalized Autoregressive Conditional Heteroskedasticity (GARCH) process of orders P and Q is defined as r t = µ t + ε t µ t = φ 0 + φ 1 r t φ s r t S P Q σt 2 = ω + α p ε 2 t p + β q σt q 2 ε t = σ t e t p=1 i.i.d. e t N(0, 1) q=1 Mean model can be altered to fit data AR(S) here Adds lagged variance to ARCH 23 / 31

24 Glosten Jagannathan Runkle-GARCH Named after the three authors who first described the process Extends GARCH(1,1) to include an asymmetric term Definition (Glosten-Jagannathan-Runkle (GJR)-GARCH process) A GJR-GARCH(P,O,Q) process is defined as r t = µ t + ε t µ t = φ 0 + φ 1 r t φ s r t S P O σt 2 = ω + α p ε 2 t p + γ o ε 2 t oi [εt o<0] + ε t = σ t e t p=1 i.i.d. e t N(0, 1) o=1 Q β q σt q 2 where I [εt o<0] is an indicator function that takes the value 1 if ε t o < 0 and 0 otherwise. q=1 24 / 31

25 GJR-GARCH(1,1,1) example GJR(1,1,1) model σt 2 = ω+α 1 ε 2 t 1 + γ 1 ε 2 t 1I [εt 1 <0] + β 1 σt 1 2 α 1 + γ 1 0 α 1 0 β 1 0 ω > 0 γ1 ε 2 t 1 I [ε t 1 <0]: Variances are larger after negative shocks than after positive shocks Leverage Effect 25 / 31

26 Threshold ARCH Threshold ARCH is similar to GJR-GARCH Also known as ZARCH (Zakoain (1994)) or AVGARCH when symmetric Definition (Threshold ARCH (TARCH) process) A TARCH(P,O,Q) process is defined r t = µ t + ε t µ t = φ 0 + φ 1 r t φ s r t S P O σ t = ω + α p ε t p + γ o ε t o I [εt o<0] + ε t = σ t e t p=1 i.i.d. e t N(0, 1) o=1 Q β q σ t q where I [εt o<0] is an indicator function that takes the value 1 if ε t o < 0 and 0 otherwise. q=1 26 / 31

27 TARCH(1,1,1) example TARCH(1,1,1) model σ t = ω+α 1 ε t 1 + γ 1 ε t 1 I [εt 1 <0] + β 1 σ t 1 α 1 + γ 1 0 α 1 0 β 1 0 ω > 0 Note the different power: σ t and ε t 1 Model for conditional standard deviation Nonlinear variance models complicate some things Forecasting Memory of volatility News impact curves GARCH(P,Q) becomes TARCH(P,O,Q) or GJR-GARCH(P,O,Q) TARCH and GJR-GARCH are sometimes (wrongly) used interchangeably. 27 / 31

28 EGARCH Definition (EGARCH(P,O,Q) process) An Exponential Generalized Autoregressive Conditional Heteroskedasticity (EGARCH) process of order P, O and Q is defined r t = µ t + ε t µ t = φ 0 + φ 1 r t φ s r t S ( P ln(σt 2 ε ) t p 2 ) = ω + α p + π ε t = σ t e t p=1 i.i.d. e t N(0, 1) σ t p O o=1 γ o ε t o σ t o + Q β q ln(σt q) 2 In the original parameterization of Nelson (1991), P and O were required to be identical. q=1 28 / 31

29 EGARCH(1,1,1) EGARCH(1,1,1) r t = µ + ε t ( ) ln(σt 2 ε t 1 ) = ω + α 1 2 π ε t = σ t e t, σ t 1 i.i.d. e t N(0, 1) + γ 1 ε t 1 σ t 1 + β 1 ln(σ 2 t 1) Modeling using ln removes any parameter restrictions ( β 1 < 1) AR(1) with two shocks ln(σ 2 t ) = ω + α 1 ( e t 1 Symmetric shock ( e t 1 2 π ) 2 + γ 1 e t 1 + β 1 ln(σ 2 π t 1) ) and asymmetric shock e t 1 Note, shocks are standardized residuals (unit variance) Often provides a better fit that GARCH(P,Q) 29 / 31

30 Asymmetric Power ARCH Nests ARCH, GARCH, TARCH, GJR-GARCH, EGARCH (almost) and other specifications Only present the APARCH(1,1,1): σ δ t = ω+α 1 ( εt 1 + γ 1 ε t 1 ) δ + β1 σ δ t 1 α 1 > 0, 1 γ 1 1, δ > 0, β 1 0, ω > 0 Parameterizes the power parameter Different values for δ affect the persistence. Lower values higher persistence of shocks ARCH: γ = 0, β = 0, δ = 2 GARCH: γ = 0, δ = 2 GJR-GARCH: δ = 2 AVGARCH: γ = 0, δ = 1 TARCH: δ = 1 EGARCH: (almost) lim δ 0 30 / 31

31 S&P Results Coeff (0.000) Coeff (0.023) Coeff (0.138) ARCH(5) ω α 1 α 2 α 3 α 4 α 5 LL (0.063) (0.000) (0.000) (0.000) (0.000) GARCH(1,1) ω α 1 β 1 LL (0.000) (0.000) EGARCH(1,1,1) ω α 1 γ 1 β 1 LL (0.000) (0.000) (0.000) / 31

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37 Univariate Volatility Modeling Kevin Sheppard Oxford MFE This version: January 18, 2013 January 21 22, 2013

38 This week Comparing alternative ARCH specifications Parameter estimation Alternative distributions for the standardized residuals Testing for ARCH Forecasting Forecast Evaluation Realized Variance and Volatility 2 / 44

39 Comparing different models Comparing models which are not nested can be difficult The News Impact Curve provides one method Defined: n(e t ) = σt+1(e 2 t σt 2 = σ 2 ) NIC(e t ) = n(e t ) n(0) Measures the effect of a shock starting at the unconditional variance Allows for asymmetric shapes GARCH(1,1) NIC(e t ) = α 1 σ 2 et 2 GJR-GARCH(1,1,1) TARCH(1,1,1) NIC(e t ) = (α 1 + γ 1 I [et<0]) σ 2 e 2 t NIC(e t ) = (α 1 + γ 1 I [εt<0]) 2 σ 2 e 2 t + (2ω + 2β 1 σ)(α 1 + γ 1 I [εt<0]) ε t 3 / 44

40 S&P 500 News Impact Curves 4 3 S&P 500 News Impact Curve ARCH(1) GARCH(1,1) GJR GARCH(1,1,1) TARCH(1,1,1) APARCH(1,1,1) / 44

41 Estimation r t = µ t + ε t σ 2 t = ω + α 1 ε 2 t 1 + β 1 σ 2 t 1 ε t = σ t e t i.i.d. e t N(0, 1) So: r t F t 1 N(µ t, σ 2 t ) Need initial values for σ0 2 and ε2 0 to start recursion Normal Maximum Likelihood is a natural choice ( ) T f (r; θ ) = (2πσt 2 ) 1 2 exp (r t µ t ) 2 2σ 2 t=1 t l(θ ; r) = T 1 2 log(2π) 1 2 log(σ2 t ) (r t µ t ) 2. t=1 2σ 2 t 5 / 44

42 Inference MLE are asymptotically normal T( ˆθ θ 0 ) d N(0, I 1 ) I = E [ ] 2 l(θ 0 ; r t ) θ θ If data are not conditionally normal, Quasi MLE (QMLE) T( ˆθ θ 0 ) d N(0, I 1 J I 1 ) [ l(θ 0 ; r t ) J = E θ ] l(θ 0 ; r t ) θ Known as Bollerslev-Wooldridge Covariance estimator in GARCH models Also known as a sandwich covariance estimator Default vcvrobust in MFE GARCH code White and Newey-West Covariance estimators are also sandwich estimators 6 / 44

43 Independence of the mean and variance The mean and the variance can be estimated consistently using 2-stages. Standard errors are also correct as long as a robust VCV estimator is used. Use LS to estimate mean parameters, then use estimated residuals in GARCH Efficient estimates one of two ways Joint estimation of mean and variance parameters using MLE GLS estimation Estimate mean and variance in 2-steps as above Re-estimate mean using GLS Re-estimate variance using new set of residuals 7 / 44

44 Independence of the mean and variance The mean and the variance can be estimated consistently using 2-stages. Standard errors are also correct as long as a robust VCV estimator is used. First Order Condition: The second order condition is E l(r t θ ) = 1 2 log(2π) 1 2 log(σ2 t ) (r t µ t ) 2 2 l(r θ ) µ t σt 2 µ t σ t φ ψ = ] [ 2 l(r θ ) µ t µ t σ t φ E σ 2 t ψ [ 2 l(r θ ) µ t µ t σ t φ E T t=1 2σ 2 t (r t µ t) µ t σt 2 σt 4 φ ψ [ T E t [r t µ t] = E σ 2 t ψ ] t=1 = E [ T [ 2 l(r θ ) µ t σt 2 µ t σ t φ ψ σ 4 t 0 σ 4 t=1 t ] = 0 µ t σt 2 φ ψ ] µ t σt 2 φ ψ Use LS to estimate mean parameters, then use estimated residuals in GARCH ] 8 / 44

45 GARCH-in-mean models Your finance professor would like to believe there is a risk-return tradeoff In a GARCH model this can be expressed r t = µ + δσ 2 t + ε t σ 2 t = ω + α 1 ε 2 t 1 + β 1 σ 2 t 1 ε t = σ t e t i.i.d. e t N(0, 1) δ measures the reward per unit variance risk σ 2 t. For GIM, ignore my previous rant about estimation Must be estimated together Alternative forms or r t = µ + δσ t + ε t r t = µ + δ ln(σ 2 t ) + ε t 9 / 44

46 GARCH-in-mean on the S&P 500 Standard GIM model with GARCH(1,1) variance r t = µ + δg(σ 2 t ) + ε t ε t N(0, σ 2 t ) σ 2 t = ω + α 1 ε 2 t 1 + β 1 σ 2 t 1 Parameter estimates S&P Garch-in-Mean Estimates Mean Specification µ δ ω α 1 β 1 Log Lik. σ t (0.694) σ 2 t (0.203) ln(σ t ) (0.020) (0.034) (0.457) (0.418) (0.020) (0.019) (0.027) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) / 44

47 Alternative Distributional Assumptions Equity returns are not conditionally normal Can replace the normal likelihood with a more realistic one Common choices: Standardized Student s T Nests the normal as ν Generalized error distribution Nests the normal when ν = 2 Hansen s Skew-T Captures both skewness and heavy tails Use hyperparameters to control shape (ν and λ) All can have heavy tails Only Skew-T is skewed Dozens more in academic research But for what gain? 11 / 44

48 Empirical vs. Normal 0.4 S&P IBM 0.3 Empirical Density Normal / 44

49 Effect of dist. choice on estimated volatility S&P 500 Returns 80 Normal Students T 60 GED IBM Returns Normal Students T GED / 44

50 What are fat tails? Fat-tailed and Thin-tailed Definition (Leptokurtosis) A random variable x t is said to be leptokurtotic if its kurtosis, κ = E[(xt E[xt])4 ] E[(x t E[x t]) 2 ] 2 is greater than that of a normal (κ > 3). Leptokurtotic variables are also known as heavy tailed or fat tailed. Definition (Platykurtosis) A random variable x t is said to be platykurtotic if its kurtosis, κ = E[(xt E[xt])4 ] E[(x t E[x t]) 2 ] 2 is less than that of a normal (κ < 3). Platykurtotic variables are also known as thin tailed. 14 / 44

51 Model Building ARCH and GARCH models are essentially ARMA models Steps: Box-Jenkins Methodology Parsimony principle 1. Inspect the ACF and PACF of ε 2 t ε 2 t = ω + (α + β)ε 2 t 1 βν t 1 + ν t ACF indicates α (or ARCH of any kind) PACF indicates β 2. Build initial model based on these observation 3. Iterate between model and ACF/PACF of ê 2 t = ε2 t ˆσ 2 t 15 / 44

52 S&P 500 ε 2 t ACF/PACF Squared Residuals ACF Squared Residuals PACF Standardized Squared Residuals ACF Standardized Squared Residuals PACF / 44

53 IBM ε 2 t ACF/PACF Squared Residuals ACF Squared Residuals PACF Standardized Squared Residuals ACF Standardized Squared Residuals PACF / 44

54 How I built a model for the S&P 500 ω α 1 α 2 γ 1 γ 2 β 1 β 2 LL GARCH(1,1) (.556) GARCH(2,1) (.169) GARCH(1,2) (.600) GJR-GARCH(1,1,1) (.493) GJR-GARCH(1,2,1) (.004) TARCH(1,1,1) (.000) EGARCH(1,1,1) (.981) EGARCH(2,1,1)** (.973) EGARCH(1,2,1) (.988) EGARCH(1,1,2) (.991) (.000) (.999) (.000) (.999) (.999) (.999) (.000) (.000) (.000) (.545) (.000) (.000) (.000) (.000) (.000) (.000) (.000) (.000) (.701) (.999) (.174) (.000) (.000) (.000) (.000) (.000) (.000) (.000) (.000) (.000) (.000) (.999) (.938) / 44

55 Testing for (G)ARCH ARCH is autocorrelation in ε 2 t All ARCH processes have this, whether GARCH or EGARCH or other ARCH-LM test Directly test for autocorrelation: ε 2 t = φ 0 + φ 1 ε 2 t φ P ε 2 t P + η t H 0 : φ 1 = φ 2 =... = φ P = 0 T R 2 d χp 2 Standard LM test from a regression. More powerful test: Fit an ARCH(P) model The forbidden hypothesis σ 2 t = ω + α 1 ε 2 t 1 + β 1 σ 2 t 1 H 0 : α 1 = 0, H 1 α > 0 19 / 44

56 Forecasting: ARCH(1) Simple ARCH model ε t N(0, σ 2 t ) σ 2 t = ω + α 1 ε 2 t 1 1-step ahead forecast is known today All ARCH-family models have this property ε t N(0, σt 2 ) σt 2 =ω + α 1 ε 2 t 1 E t [σt+1] 2 =E t [ω + α 1 ε 2 t ] =ω + α 1 ε 2 t Note: E t [ε 2 t+1 ] = E t[et+1 2 σ2 t+1 ] = σ2 t+1 E t[et+1 2 ] = σ2 t+1 Further: E t [ε 2 t+h ] = E t[e t+h 1 [et+h 2 σ2 t+h ]] = E t[e t+h 1 [et+h 2 ]σ2 t+h ] = E t[σt+h 2 ] 20 / 44

57 Forecasting: ARCH(1) 2-step ahead E t [σt+2] 2 = E t [ω + α 1 ε 2 t+1] =ω + α 1 E t [ε 2 t+1] =ω + α 1 (ω + α 1 ε 2 t ) =ω + α 1 ω + α 2 1ε 2 t h-step ahead forecast h 1 E t [σt+h 2 ] = α i 1ω + α h 1ε 2 t i=0 Just the AR(1) forecasting formula Why? 21 / 44

58 Forecasting: GARCH(1,1) 1-step ahead E t [σ 2 t+1] = E t [ω + α 1 ε 2 t + β 1 σ 2 t ] = ω + α 1 ε 2 t + β 1 σ 2 t 2-step ahead E t [σt+2] 2 = E t [ω + α 1 ε 2 t+1 + β 1 σt+1] 2 = ω + α 1 E t [ε 2 t+1] + β 1 E t [σt+1] 2 = ω + α 1 E t [et+1σ 2 t+1] 2 + β 1 E t [σt+1] 2 = ω + α 1 E t [et+1]σ 2 t β 1 E t [σt+1] 2 = ω + α 1 1 σt β 1 E t [σt+1] 2 = ω + α 1 σt β 1 E t [σt+1] 2 = ω + (α 1 + β 1 )σt / 44

59 Forecasting: GARCH(1,1) h-step ahead h 1 E t [σt+h 2 ] = (α 1 + β 1 ) i ω + (α 1 + β 1 ) h 1 (α 1 ε 2 t + β 1 σt 2 ) i=0 Also essentially an AR(1), technically ARMA(1,1) 23 / 44

60 Forecasting: TARCH(1,0,0) This one is a mess Nonlinearities cause problems All ARCH-family models are nonlinear, but some are linearity in ε 2 t Others are not σ t = ω + α 1 ε t 1 Forecast for t + 1 is known at time t Always, always, always,... E t[σt+1] 2 = E t[(ω + α 1 ε t ) 2 ] = E t[ω 2 + 2ωα 1 ε t + α 2 1ε 2 t ] = ω 2 + 2ωα 1E t[ ε t ] + α 2 1E t[ε 2 t ] = ω 2 + 2ωα 1 ε t + α 2 1ε 2 t 24 / 44

61 TARCH(1,0,0) continued... Multi-step is less straightforward E t[σ 2 t+2] = E t[(ω + α 1 ε t+1 ) 2 ] = E t[ω 2 + 2ωα 1 ε t+1 + α 2 1ε 2 t+1] = ω 2 + 2ωα 1E t[ ε t+1 ] + α 2 1E t[ε 2 t+1] = ω 2 + 2ωα 1E t[ e t+1 σ t+1] + α 2 1E t[e 2 t σ 2 t+1] = ω 2 + 2ωα 1E t[ e t+1 ]E t[σ t+1] + α 2 1E t[e 2 t ]E t[σ 2 t+1] = ω 2 + 2ωα 1E t[ e t+1 ](ω + α 1 ε t ) + α (ω 2 + 2ωα 1 ε t + α 2 1ε 2 t ) 2 If e t+1 N(0, 1), E[ e t+1 ] = π E t [σ 2 t+2] = ω 2 + 2ωα 1 2 π (ω + α 1 ε t ) + α 2 1(ω 2 + 2ωα 1 ε t + α 2 1ε 2 t ) 25 / 44

62 Assessing forecasts: Augmented MZ Start from E t [r 2 t+h ] σ2 t+h t Standard Augmented MZ regression: ε 2 t+h ˆσ2 t+h t = γ 0 + γ 1 ˆσ 2 t+h t + γ 2z 1t γ K +1 z Kt + η t ηt is heteroskedastic in proportion to σt 2 : Use GLS. An improved GMZ regression (GMZ-GLS) ε 2 t+h ˆσ2 t+h t ˆσ 2 t+h t 1 z 1t z Kt = γ 0 ˆσ t+h t 2 + γ γ 2 ˆσ t+h t γ K +1 ˆσ t+h t 2 + ν t Better to use Realized Variance to evaluate forecasts RV t+h ˆσ 2 t+h t = γ 0 + γ 1 ˆσ 2 t+h t + γ 2z 1t γ K +1 z Kt + η t Also can use GLS version Both RV t+h and ε 2 t+h are proxies for the variance at t + h RV is just better, often 10 + more precise 26 / 44

63 Assessing forecasts: Diebold-Mariano Relative forecast performance MSE loss δ t = ( ) 2 ( ) 2 ε 2 t+h ˆσ2 A,t+h t ε 2 t+h ˆσ2 B,t+h t H0 : E[δ t ] = 0, H A 1 : E[δ t] < 0, H B 1 : E[δ t] > 0 ˆ δ = R 1 R r=1 δ r Standard t-test, 2-sided alternative Newey-West covariance always needed Better DM using Q-Like loss (Normal log-likelihood Kernel ) ( ) ( ) δ t = ln( ˆσ A,t+h t 2 ) + ε2 t+h ˆσ A,t+h t 2 ln( ˆσ B,t+h t 2 ) + ε2 t+h ˆσ B,t+h t 2 Patton & Sheppard (2008) 27 / 44

64 Realized Variance Variance measure computed using ultra-high-frequency data (UHF) Uses all available information to estimate the variance over some period Usually 1 day Variance estimates from RV can be treated as observable Standard ARMA modeling Variance estimates are consistent Asymptotically unbiased Variance converges to 0 as the number of samples increases Problems arise when applied to market data Noise (bid-ask bounce) Market closure Prices discrete Prices not continuously observable Data quality 28 / 44

65 Realized Variance Assumptions Log-prices are generated by an arbitrage-free semi-martingale Prices are observable Prices can be sampled often Defined RV (m) t = m ( ) 2 m pi,t p i 1,t = i=1 i=1 r 2 i,t. m-sample Realized Variance p i,t is the ith log-price on day t r i,t is the ith return on day t Only uses information on day t to estimate the variance on day t Consistent estimator of the integrated variance Total variance on day t ˆ t+1 t σ 2 s ds 29 / 44

66 Why Realized Variance Works Consider a simple Brownian motion m-sample Realized Variance Returns are IID normal Nearly unbiased Variance close to 0 dp t = µ dt + σ dw t [ V RV (m) t = m i=1 r 2 i,t ( ) i.i.d. µ r i,t N m, σ2 m [ E RV (m) t RV (m) t ] = µ2 m + σ2 ] = 4 µ2 σ 2 m σ4 m 30 / 44

67 Why Realized Variance Works Works for models with time-varying drift and stochastic volatility dp t = µ t dt + σ t dw t No arbitrage imposes some restrictions on µ t Works with price processes with jumps In the general case: RV (m) t p ˆ 1 0 σ 2 s ds + N n=1 J 2 n J n are jumps 31 / 44

68 Why Realized Variance Doesn t Work Multiple prices at the same time Define the price as the average share price (volume weighted price) Use simple average or median Not a problem Prices only observed on a discrete grid $.01 or.0025 Nothing can be done Small problem Data quality UHF price data is generally messy Typos Wrong time-stamps Pre-filter to remove obvious errors Often remove round trips Cross-fingers No price available at some point in time Use the last observed price: last price interpolation Averaging prices before and after leads to bias 32 / 44

69 Solutions to bid-ask bounce type noise Bid-ask bounce is a big problem Simple model with pure noise p i,t = p i,t + ν i,t p i,t is the observed price with noise p i,t is the unobserved efficient price ν i,t is the noise Easy to show r i,t = r i,t + η i,t ri,t is the unobserved efficient return η i,t = ν i,t ν i 1,t is a MA(1) error RV is badly biased RV (m) t RV t + mτ 2 Bias is increasing in m Variance is also increasing in m 33 / 44

70 Simple solution Do not sample frequently 5-30 minutes Better than daily but still inefficient Remove MA(1) by filtering η i,t is an MA(1) Fit an MA(1) to observed returns r i,t = θ ε i 1,t + ε i,t Use fit residuals ˆε i,t to compute RV Generally biased downward Use mid-quotes A little noise My usual solution 34 / 44

71 A modified Realized Variance estimator: RV AC1 Best solution is to use a modified RV estimator RV AC1 RV AC1 t (m) m m = ri,t r i,t r i 1,t i=1 Adds a term to RV to capture the MA(1) noise Looks like a simple Newey-West estimator Unbiased in pure noise model Not consistent Realized Kernel Estimator Adds more weighted cross-products Consistent in the presence of many realistic noise processes Fairly easy to implement i=2 35 / 44

72 One final problem Market closure Markets do not operate 24 hours a day (in general) Add in close-to-open return squared r CtO,t = p Open,t p Close,t 1 RV (m) t = r 2 CtO,t + Compute a modified RV by weighting the overnight and open hour estimates differently RV (m) t m i=1 r 2 i,t = λ 1 r 2 CtO,t + λ 2 RV (m) t 36 / 44

73 The volatility signature plot Hard to know how often to sample Visual inspection may be useful Definition (Volatility Signature Plot) The volatility signature plot displays the time-series average of Realized Variance T RV (m) t = T 1 RV (m) t t=1 as a function of the number of samples, m. An equivalent representation displays the amount of time, whether in calendar time or tick time (number of trades between observations) along the X-axis. 37 / 44

74 Some empirical results S&P 500 Depository Receipts SPiDeRs AMEX: SPY Exchange Traded Fund Ultra-liquid 100M shares per day Over 100,000 trades per day 23,400 seconds in a typical trading day January 1, July 31, 2008 Filtered by daily High-Low data Some cleaning of outliers 38 / 44

75 SPDR Realized Variance (RV ) RV, 15 seconds RV, 1 minute RV, 5 minutes 0.8 RV, 30 minutes / 44

76 SPDR Realized Variance (RV AC1 ) 0.8 RV AC1, 15 seconds 0.8 RV AC1, 1 minute RV AC1, 5 minutes RV AC1, 30 minutes / 44

77 Volatility Signature Plots Average RV Average RV AC x 10 5 Volatility Signature Plot for SPDR RV 5s 15s 30s45s Sampling Interval, minutes unless otherwise noted x Volatility Signature Plot for SPDR RV AC1 5s 15s 30s45s Sampling Interval, minutes unless otherwise noted 41 / 44

78 Modeling Realized Variance Two choices Treat volatility as observable and model as ARMA Really simply to do Forecasts are equally simple Theoretical motivation why RV may be well modeled by an ARMA(P,1) Continue to treat volatility as latent and use ARCH-type model Realized Variance is still measured with error A more precise measure of conditional variance that daily returns squared, rt 2, but otherwise similar 42 / 44

79 Treating RV as observable Main model used is a Heterogeneous Autoregression Restricted AR(22) in levels RV t = φ 0 + φ 1 RV t 1 + φ 5 RV 5,t 1 + φ 22 RV 22,t 1 + ε t Or in logs ln RV t = φ 0 + φ 1 ln RV t 1 + φ 5 ln RV 5,t 1 + φ 22 ln RV 22,t 1 + ε t where RV j,t 1 = j 1 j i=1 RV t i is a j lag moving average Model picks up volatility changes at the daily, weekly, and monthly scale Fits and forecasts RV fairly well MA term may still be needed 43 / 44

80 Leaving RV latent Alternative if to treat RV as latent and use a non-negative multiplicative error model (MEM) MEMs specify the mean of a process as µ t ψ t where ψ t is a mean 1 shock. A χ 2 1 is a natural choice here ARCH models are special cases of a non-negative MEM model Easy to model RV using existing ARCH mdoes 1. Construct r t = sign (r t ) RV t 2. Use standard ARCH model building to construct a model for r t σ 2 t = ω + α 1 r 2 t 1 + γ 1 r 2 t 1I [ rt 1<0] + β 1 σ 2 t 1 becomes σ 2 t = ω + α 1 RV t 1 + γ 1 RV t 1 I [rt 1<0] + β 1 σ 2 t 1 44 / 44

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