Localized Realized Volatility Modeling Ying Chen Wolfgang Karl Härdle Uta Pigorsch National University of Singapore Humboldt-Universität zu Berlin University of Mannheim
Motivation 2 An important realized volatility fact sacf (1985-Feb.2005) 0.8 0.6 0.4 0.2 0.0 -0.2 20 40 60 80 100 120 lag 0.8 0.6 sacf (1995) 0.4 0.2 0.0 -0.2 20 40 60 80 100 120 lag Figure 1: Sample autocorrelations of log RV for different sample periods. Localized Realized Volatility Modeling
Motivation 3 A dual view ⊡ The long memory point of view: Volatility is generated by long memory processes, i.e. fractionally integrated, I ( d ) , processes. ⊡ The short memory point of view: Volatility may equally well be generated by a short memory process with structural changes. Example: GARCH model with changing parameters. Localized Realized Volatility Modeling
Motivation 4 Realized volatility ⊡ Volatility forecasts are important for an adequate risk management and derivative pricing. ⊡ Realized volatility is based on high-frequency information. ⊡ It is a more precise volatility estimator than daily squared or absolute returns. ⊡ Exhibits better forecast properties, Andersen, Bollerslev, Diebold and Labys (2001). Localized Realized Volatility Modeling
Motivation 5 Localized realized volatility ⊡ For a point τ in time, find a past time interval for which a local volatility model is a good approximator. ⊡ The time interval is determined by adaptive statistical methods. ⊡ Represents a local analysis, i.e. changes are detected close to the forecasting time point. Localized Realized Volatility Modeling
Motivation 6 Outline 1. Motivation � 2. Realized volatility 3. Localized realized volatility 4. Long memory models 5. Empirical analysis 6. Conclusion Localized Realized Volatility Modeling
Realized volatility 7 Realized volatility Daily realized volatility M � � r 2 RV t = t , j , j = 1 with r t , j = p t , n j − p t , n j − 1 , j = 1 , . . . , M , and p t , n j the log price observed at time point n j of trading day t . It converges to the quadratic variation for M → ∞ (Andersen and Bollerslev, 1998; Barndorff-Nielsen and Shephard, 2002b). Localized Realized Volatility Modeling
Realized volatility 8 Realized volatility (smoothed) Tukey-Hanning kernel � h − 1 � H ∗ � RV t = � RV t , 1 + ( γ t , h + γ t , − h ) k H ∗ h = 1 k ( x ) = sin 2 � π 2 ( 1 − x ) 2 � , γ t , h = � M j = 1 r t , j r t , j − h (one-minute returns), √ � H ∗ = 5 . 74 RV t , 1 / 2 M M with RV t , i the realized variance estimator � RV t , 15 based on i minute returns. (Barndorff-Nielsen, Hansen, Lunde and Shephard, 2008) Localized Realized Volatility Modeling
Realized volatility 9 Data S&P500 index futures from January 2, 1985 to February 4, 2005. LB(21) ( 1 ) Series Mean Std.Dev. Skewness Kurtosis 1.07 8.16 59.08 3861 1375 RV t log ( RV t ) -0.51 0.87 0.43 4.99 46809 ( 1 ) The critical value of this Ljung-Box test is 32.671. Table 1: Descriptive statistics. Localized Realized Volatility Modeling
Realized volatility 10 0.50 0.45 0.40 0.35 0.30 density 0.25 0.20 0.15 0.10 0.05 0.00 -4 -3 -2 -1 0 1 2 3 4 log. RV Figure 2: Kernel density estimates (solid line: log RV , shaded area: point- wise 95% confidence intervals, dashed line: normal distribution). Localized Realized Volatility Modeling
Realized volatility 11 8 6 4 2 log. RV 0 -2 -4 -6 -8 86 88 90 92 94 96 98 00 02 04 time 0.80 0.70 0.60 0.50 sacf 0.40 0.30 0.20 0.10 0.00 20 40 60 80 100 120 lag Figure 3: log RV and its sample autocorrelation function. Localized Realized Volatility Modeling
Localized realized volatility 12 Localized realized volatility LAR(1) model with parameter set θ t = ( θ 1 t , θ 2 t , θ 3 t ) ⊤ : ε t ∼ N ( 0 , θ 2 log RV t = θ 1 t + θ 2 t log RV t − 1 + ε t , 3 t ) . (1) Suppose θ t ≡ θ ∗ for t ∈ I = [ 1 , T ] ˜ θ t = argmax θ ∈ Θ L ( log RV ; θ ) � − T = argmax θ ∈ Θ 2 log 2 π − T log θ 3 � T � − 1 ( log RV t − θ 1 − θ 2 log RV t − 1 ) 2 . 2 θ 2 3 t = 1 Goal: identify a local homogeneous interval I τ for time point τ . Localized Realized Volatility Modeling
Localized realized volatility 13 Identify local homogeneity At time point τ , choose a local homogeneous interval from { I k τ } K k = 1 = { I 1 τ , I 2 τ , · · · , I K τ } where I k τ = [ τ − s k , τ ) with 0 < s k < τ , which leads to the best possible accuracy of estimation. ⊡ Under local homogeneity θ τ ≡ θ ∗ τ within I k τ = [ τ − s k , τ ) : τ at rate 1 / √ s k θ ( k ) ˜ estimates θ ∗ τ ⊡ The modeling bias of approximating LAR(1) increases w.r.t. k . The optimal choice ˆ I τ : balances the bias and variation. Localized Realized Volatility Modeling
Localized realized volatility 14 Estimation under local homogeneity Given I τ = [ τ − s , τ ) , the local MLE is: ˜ θ τ = argmax θ ∈ Θ L ( log RV ; I τ , θ ) Under local homogeneity : θ τ ≡ θ ∗ τ , the fitted likelihood ratio measures the estimation risk: LR ( I τ , ˜ L ( I τ , ˜ θ τ , θ ∗ θ τ ) − L ( I τ , θ ∗ τ ) = τ ) . (2) Localized Realized Volatility Modeling
Localized realized volatility 15 Estimation under local homogeneity The estimation risk LR ( I τ , ˜ θ τ , θ ∗ τ ) is stochastically bounded: � � � r ≤ ξ r � LR ( I τ , ˜ θ τ , θ ∗ E θ ∗ τ ) τ � ξ ≥ 0 ξ r − 1 e − ξ d ξ = 2 r Γ( r ) . with ξ r = 2 r It leads to the confidence set : E ( ε ) = { θ : LR ( I τ , ˜ θ τ , θ ∗ τ ) ≤ ε } in the sense that P θ ∗ � E ( ε ) �∋ θ ∗ � ≤ α, Polzehl and Spokoiny (2006). Localized Realized Volatility Modeling
Localized realized volatility 16 Localized AR(1) model Interval set { I k τ } for k = 1 , · · · , K : I 1 I 2 I K τ = [ τ − 1w , τ ) τ = [ τ − 1m , τ ) · · · τ = [ τ − 5y , τ ) ↓ ↓ ↓ ↓ θ ( 1 ) θ ( 2 ) θ ( K ) ˜ ˜ ˜ · · · τ τ τ ⊡ The interval is growing in length. ⊡ Local homogeneity is assumed at I 1 τ . ⊡ Final estimate ˆ θ τ is based on a sequential testing. Localized Realized Volatility Modeling
Localized realized volatility 17 Sequential testing θ ( k − 1 ) θ ( k − 1 ) is a homogeneous interval: ˆ = ˜ Suppose that I k − 1 . The τ τ τ null hypothesis at step k : H 0 : I k τ is an homogeneous interval. ✡ ✠ τ ✫ ✪ is homogeneous: ˆ = ˜ I k − 1 θ k − 1 θ k − 1 τ τ τ τ : ˆ τ = ˜ Test homogeneity of I k θ k θ k τ or terminates at I k − 1 τ � � r � � τ , ˜ τ , ˆ � LR ( I k θ k θ k − 1 Test: ) ≤ ζ k , where ζ k is critical value (CV). � τ Localized Realized Volatility Modeling
Localized realized volatility 18 Adaptive procedure 1. Initialization: ˆ τ = ˜ θ 1 θ 1 τ . 2. k = 1 � � r � � � LR ( I τ , ˜ , ˆ θ k + 1 θ k while τ ) ≤ ζ k + 1 and k < K , � τ k = k + 1 ˆ ˜ θ k θ k = τ τ 3. Final estimate: ˆ θ τ = ˆ θ k τ Localized Realized Volatility Modeling
Localized realized volatility 19 Parameter choice: Interval set ⊡ { I k } 13 k = 1 for every τ with the following interval lengths: { s k } 13 = { 1w , 1m , 3m , 6m , 1y , 1.5y , k = 1 2y , 2.5y , 3y , 3.5y , 4y , 4.5y , 5y } , where w denotes a week (5 days), m refers to one month (21 days) and y to one year (252 days). Localized Realized Volatility Modeling
Localized realized volatility 20 Parameter choice: CV ⊡ Monte Carlo simulation: generate AR(1) processes with θ t = θ ∗ = ( θ ∗ 3 ) ⊤ for all t 1 , θ ∗ 2 , θ ∗ y t = θ ∗ 1 + θ ∗ 2 y t − 1 + ε t , ε t ∼ N ( 0 , θ ∗ 2 3 ) . ◮ ◮ y 0 = θ ∗ 1 / ( 1 − θ ∗ 2 ) 100 000 paths, each including 1261 observations ◮ ⊡ Choice of critical values: Parametric case θ t ≡ θ ∗ : � � �� r � � I K , ˜ t , ˆ θ K θ K E θ ∗ � LR ≤ ξ r (3) � t ( ζ 1 ,...,ζ K ) � � �� k − 1 r � � I k , ˜ t , ˆ θ k θ k E θ ∗ � LR � ≤ K − 1 ξ r (4) t ( ζ 1 ,...,ζ k ) Localized Realized Volatility Modeling
Localized realized volatility 21 Parameter choice: CV Sequential choice of critical values ⊡ Choice of ζ 1 = ∞ initializing the procedure ˆ t ( ζ 1 ) = ˜ θ 1 θ 1 t ⊡ Choice of ζ 2 leading to ˆ θ k t ( ζ 1 , ζ 2 ) by setting ζ 3 = . . . = ζ K = ∞ : � � �� 1 � r ≤ � LR I k , ˜ t , ˆ θ k θ k E θ ∗ t ( ζ 1 , ζ 2 ) K − 1 ξ r , k = 2 , . . . , K . ⊡ Choice of ζ 3 leading to ˆ θ k t ( ζ 1 , ζ 2 , ζ 3 ) by setting ζ 4 = . . . = ζ K = ∞ : � �� � 2 � r ≤ � LR I k , ˜ t , ˆ θ k θ k E θ ∗ t ( ζ 1 , ζ 2 , ζ 3 ) K − 1 ξ r , k = 3 , . . . , K . Localized Realized Volatility Modeling
Localized realized volatility 22 Parameter choice: CV Sequential choice of critical values ⊡ Choice of ζ k leading to ˆ θ l t ( ζ 1 , · · · , ζ k ) by setting ζ k + 1 = . . . = ζ K = ∞ : � �� � � r ≤ k − 1 � LR I k , ˜ θ l t , ˆ θ l E θ ∗ t ( ζ 1 , . . . , ζ k ) K − 1 ξ r , l = k , . . . , K . Localized Realized Volatility Modeling
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