High Frequency data and Realized Volatility Models

Transcription

1 High Frequency data and Realized Volatility Models Fulvio Corsi SNS Pisa 7 Dec 2011 Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

2 High Frequency (HF) data Data: S&P 500 futures from Price-data.com DATE, TIME, NAME, PRICE, , 13:22:05, SP 03U, , , 13:22:08, SP 03U, , , 13:22:11, SP 03U, , , 13:22:21, SP 03U, , , 13:22:24, SP 03U, , , 13:22:27, SP 03U, , , 13:22:35, SP 03U, , , 13:22:36, SP 03U, , , 13:22:40, SP 03U, , , 13:22:45, SP 03U, , , 13:23:55, SP 03U, , , 13:23:15, SP 03U, , , 13:23:35, SP 03U, , , 13:24:10, SP 03U, , , 13:24:22, SP 03U, , , 13:24:41, SP 03U, , Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

3 HF data: Why are they interesting? Allow to study in detail various phenomena: how information gets impounded in asset prices how the trading mechanism affects asset prices liquidity, market frictions, etc., to better design the market trading strategies, execution risk, etc. spillover effects, contagions, etc. construct better volatility and covariance measures (realized variance and covariance)... Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

4 HF data: Characteristics Huge amount of data (with data errors) Irregular temporal spacing of transactions Discreteness of price changes (price rounding): one-tick, two-tick, etc. Diurnal pattern (also for FX markets) of volumes, volatilities, etc. Temporal dependence of HF returns, volumes, volatilities, etc. Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

5 HF data: Bid - Ask and Transaction Prices Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

6 HF data: Autocorrelation function of r Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

7 HF data: volatility U shape Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

8 Realized Volatility - The Basic Idea Given dp(t) = µ(t)dt + σ(t)dw(t) the Integrated (ex-post) Volatility of day t is t IV t σ 2 (ω)dω t 1 Realized Volatility idea: integrated volatility approximated by the sum of many intraday squared returns: RV ( ) t 1/ rt+j 2 j=0 p 0 t σ 2 (ω)dω IV t t 1 where r t+j p(t + j ) p(t + (j 1) ). Andersen, Bollerslev, Diebold and Labys (2001) and Barndorff-Nielsen and Shephard (2001) formalize this intuition within the class of special (finite mean) semimartingales using Quadratic Variation theory. Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

9 Why RV works, the intuition 1/ E[rt+j 2 F t+(j 1) ] j=1 j=1 1/ 1/ Var[r t+j 2 F t+(j 1) ] 2 σ(t+j ) 4 2 1/ t+1 σ 2 (t+j ) σ 2 (s)ds IV t+1 j=1 t j=1 t+1 2 σ 4 (s)ds 2 IQ t+1 t Consistency (for 0) lim 0 RV t+1 ( ) IV t+1 Asymptotic normality (for 0) 1/ RV t+1( ) IV t+1 2IQ 1/2 t+1 N(0, 1) Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

10 Realized Volatility - Advantages Model free Volatility become an observable quantity! characterize distribution of volatility (conditional & unconditional) direct fit & forecast time series models. Much less noisy target function better optimization better evaluation of forecasting models better extraction of underlying signal better forecasts. Insensitive to intradaily seasonalities. Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

11 Realized Volatility - In practice Definition of RV involve 2 time parameters: 1 Time horizon over which compute RV (typically one day) 2 Returns interval: For Gaussian RW, statistical error as function of returns frequency can be computed: RMSE [ = 1d] = 141.4% σ 2 RMSE [ = 1h] = 28.8% σ 2 RMSE [ = 1 ] = 3.7% σ 2 RMSE [ = 1 ] = 0.5% σ 2 In theory δt 0. In practice HF Data. BUT when return interval shrinks market microstructure effects arise BIG BIAS. Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

12 Realized Volatility - In practice 1.8 USD/CHF E[ r[ δt] 2 ] 1/ δt [day] Scaling of normalized RV for δt ranging from 1 min to 1 week. Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

13 Observed price and microstructure noise General model for microstructure effects p n = p n + u n a martingale fundamental price, p n a stationary market microstructure noise, u n IID ( 0,η 2) Bid-Ask spread Price discreteness (rounding) Thus the observed tick-by-tick return r n is r n = p n p n 1 + u n u n 1 r n +ɛ n Hence the obs tick-by-tick return r n - process is MA(1) with E(r n) = 0 and σ 2 + 2η 2 for k = 0 E(r nr n k) = η 2 for k = 1 0 for k 2 Hence N N N N RV (N) = ( r i +ɛ i ) 2 = r i 2 + ɛ 2 i + 2 r i ɛ i i=1 i=1 i=1 i=1 Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

14 Market microstructure fixes Sample sparsely: Andersen, Bollerslev, Diebold and Labys (2000, Risk) & (2001, JASA) Optimal sampling schemes Aït-Sahalia, Mykland and Zhang (2005, RFS) Bandi and Russell (2006, J.Fin.Eco.) Bandi and Russell (2005, wp) Pre-filtering and kernel methods: Zhou (1996, JBES) Corsi, Zumbach, Müller, and Dacorogna (2001, Eco.Notes) Andersen, Bollerslev, Diebold and Ebens (2001, J.Fin.Eco.) Hansen and Lunde (2006, JBES) Barndorff-Nielsen, Hansen, Lunde and Shephard (2008, wp) Sub-sampling Zhang, Aït-Sahalia and Mykland (2005, JASA) Zhang (2006, Bernoulli) Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

15 Kernel Estimators Define the realised autocovariance γ h = N r i r (i+h) Realized kernel estimators: H ( ) h 1 RV K = γ 0 + K (γ h +γ h ) H h=1 ( ) where K h 1 are the kernel weights. H i=1 Special cases: RV = γ 0 Realized Variance RV AC1 = γ 0 +(γ 1 +γ 1) Zhou (1996) RV NW = γ 0 + ) H h=1 (γh +γ h) Barlett ( h 1 H Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

16 The Multi-Scales Estimators Zhang, Mykland & Aït-Sahalia (2004) propose the Two-Scales estimator (TS) : TS is based on subsampling and averaging RV at a slower time scale (RV (avg) ) and bias-correcting it taking a linear combination with the fastest one (RV (all) ). Its rate of convergence is N 1/6. RV TS = RV }{{ (avg) N(avg) } N (all) RV }{{ (all) } slow time scale fast time scale Zhang (2004) generalize to Multi-Scales estimators (MS) having rate N 1/4 RV MS = K α j RV (k j) - RV (k j) : the daily RV obtained by sampling every k j -th data point - α j : weights to be determined, according to some (asymptotic) criteria. j=1 Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

17 Regression Based Multi-Scales Estimator Curci and Corsi (2006) present a simpler approach to the construction of Multi-Scales estimators. Defining: - RV (k j) the RV computed with observed returns at frequency k j (with full overlap). - N (k j) the number of k j -returns in the day. Then, under i.i.d. noise (or for an appropriate choice of k j s) we have: [ ] E RV (k j) = IV + 2N (kj) η 2 IV and η 2 can be estimated by a simple linear regression of RV (k j) on N (k j) Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

18 Realized Covariance Idea: as for Realized Volatility, employing high frequency data for the computation of covariances between two assets. Standard way: choose a time interval, construct an artificially regularly spaced time series (say 5-min) and take the sum of contemporaneous cross products. But, as for the RV, the presence of market microstructure can induce significant bias in standard RCov. The so called Epps effect. The microstructure effects responsible for this bias is the so called non-synchronous trading effect (Lo and MacKinlay 1990) which induces a bias toward zero in the standard interpolated measure: Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

19 Realized Covariance - the Epps effect lost portion of covariance Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

20 Realized Covariance - Hayashi & Yoshida Hayashi and Yoshida (2005): does not resort on construction of regular grid. Intuition i.e. summing all cross products of tick-by-tick returns with non zero overlap M i,t M j,t HY t = r i,s r j,q I(λ q,s > 0) s=1 q=1 with I( ) the indicator function and λ q,s = max(0, min(n i,s+1, n j,q+1 max(n i,s, n j,q)) being the overlap between any two tick-by-tick returns r i,s and r j,q. Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

21 Stylized Facts I: Heteroskedasticity - Fat Tail 6 USD/CHF std deviations Gaussian noise 4 2 std deviation S&P std deviations Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

22 Stylized Facts II: Volatility Persistence Sample autocorrelation coefficients S&P500 Brownian Motion USD/CHF conf interval conf interval Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

23 Long Memory Models In std GARCH and SV models volatility shocks decay with exponential rate: ρ h γ h with 0 < γ < 1 while empirical data show an hyperbolic decay rate: ρ h h γ with 0 < γ < 1 Fractional Integration: generalize the usual differencing of I(1) series y t to get an I(0) ɛ t as: (1 L) d y t = ɛ t I(d) gives an infinite MA representation for y t = k=0 a k(d)ɛ t k with a k (d) = Γ(k+d) Γ(k+1)Γ(d) which displays long memory with γ = 2d 1 Fractional Integration + ARMA ARFIMA Fractional Integration + GARCH FIGARH Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

24 Stylized Facts III: Fractality & Multifractality Self-Similar, Fractal or Scaling Process: (Y t1, Y t2, Y t3,...) d = c H (Y ct1, Y ct2, Y ct3,...) which in terms of generalized volatilities implies: E[ r( t) q ] t H(q) if H(q) is linear i.e. H(q) = H q Unifractal or Monofractal process: - Brownian Motion (H = 0.5) - Fractionally Integrated processes (H = d 0.5) if H(q) is nonlinear Multifractal process: different scaling of different generalized volatility (Ding et al. 1993, Lux 1996, Andersen and Bollerslev econophysicists ). - Multifractal Model of Asset Returns (MMAR), Mandelbrot, Calvet and Fisher (1997): X(t) B[θ(t)] where θ(t) = c.d.f. of multifractal measure Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

25 Stylized Facts IV: Volatility Cascade and Scaling Asymmetric propagation of volatility: volatility over longer time intervals have stronger influence on those at shorter time intervals than conversely. (Müller et al. 1997, Arneodo et al. 1998, Lynch 2000, Gençay et al. 2002). Possible economic explanation: long term volatility matters for short-term traders while, short-term volatility does not affect long-term trading strategies. induced some authors to propose analogies with energy cascades of turbulent fluids, borrowing from Kolmogorov model of hydrodynamic turbulence: the so called Stochastic Multiplicative Cascade (SMC) Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

26 Stylized Facts and Volatility Models Standard volatility models are not able to reproduce all the stylized facts: GARCH and SV (one factor): no long memory no scaling no volatility cascade Fractionally Integrated models: no multi scaling no volatility cascade Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

27 Models Summary F I A Desired Properties G G R A A F M of Volatility Models R R I M S C S C M A M H V H A R C Fat tail Tail cross-over Long memory Self-similarity Multifractality Volatility Cascade Economic Interpretation Simplicity of estimation Multivariate Extendibility Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

28 A different approach: Heterogeneity Additive processes with heterogeneous components can generate those stylized facts! Es, Le Baron (2001): combination of only 3 AR(1) can display apparent long memory. If the aggregation level is not than the lowest frequency component asymptotically short memory models can be mistaken for long memory i.e. they are empirically indistinguishable. Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

29 Model Ingredients 1 Heterogeneous Market Hypothesis (Müller et al. 1993): Main heterogeneity: difference in time horizons agents perceive, react and cause different volatility components σ ( ) t 2 Volatility Cascade: hierarchical process from Low to High Frequency. 3 Realized Volatility Measures: makes volatility observable. Cascade of Few Heterogeneous Realized Volatility Components we consider only 3 partial volatility components: daily σ (d) t, weekly σ (w) t, monthly σ (m) t Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

30 The HAR-RV Model we work with logs to avoid negativity issues and get approximately Normal distributions. Consider the log RV t aggregated, as follow: log RV (n) t = 1 n (log RVt +...+log RV t n+1) at the 3 different horizons: daily d = 1, weekly w = 5, monthly m = 22 Hence the model reads: r t log σ (m) t+m log σ (w) t+w log σ (d) t+d = σ (d) t z t = c (m) +φ (m) log RV (m) t + ω (m) t+m = c (w) +φ (w) log RV (w) t = c (d) +φ (d) log RV (d) t +γ (w) E t[log σ (m) t+m ]+ ω(w) t+w +γ (d) E t[log σ (w) t+w ]+ ω(d) t+d A possible economic interpretation each market components forms expectation for the next period volatility based on: the current RV experienced at the same time scale ( AR(1) part ) the expectation for the next longer horizon partial volatility (Hierarchical part) Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

31 HAR-RV Model By straightforward recursive substitution logσ (d) t+1d = c+β (d) log RV (d) t +β (w) log RV (w) t +β (m) log RV (m) t +ɛ (d) t+1d A three factors Stochastic Volatility model where the factors are directly the past RV Moreover, being: logσ (d) t+1d = log RV(d) t+1d + ɛ t+1d where ɛ t is the measurement errors of log RV, we get log RV (d) t+1d = c+β (d) log RV (d) t +β (w) log RV (w) t +β (m) log RV (m) t +ɛ t+1d a simple AR-type model in the RV with the feature of considering volatilities realized over different interval sizes. Heterogeneous AR model in the RV (HAR-RV). Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

32 Simulation Results - daily returns 1 USD/CHF Simulation Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

33 Simulation Results - daily RV 0.5 USD/CHF Simulation Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

34 Simulation Results - return pdf daily weekly monthly Kurtosis daily returns weekly returns monthly returns USD/CHF HAR-RV model Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

35 Simulation Results - RV acf 0.7 Sample autocorrelation coefficients sacf values k values Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

36 Simul. Results - RV partial-acf 0.8 Sample partial autocorrelation coefficients USD/CHF 0.6 spacf values Simulated process spacf values Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

37 HAR-GARCH In HAR models, the innovations of the log RV are assumed to be i.i.d. However, volatility clustering in the residuals are often observed. Corsi Mittnik Pigorsch and Pigorsch (2008) extend the HAR model by explicitly modelling the volatility of RV: adding GARCH type innovations to the standard HAR: log V (1) t+1 = c+β(1) log V (1) t +β (5) log V (5) t +β (22) log V (22) t + h tε t q p h t = ω + a j u 2 t j + b j h t j j=1 j=1 ε t Ω t 1 iid(0, 1), where Ω t 1 is the filtration at time t 1 and u t = h tε t. The HAR-GARCH model has been further extended by Bollerslev Pigorsch Pigorsch and Tauchen (2009), Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38

38 Extending HAR: Leveraged HAR (LHAR) idea: extend the heterogeneous structure to Leverage effect With r t = X t X t 1 define aggregated negative and positive returns as: r (n)+ t r (n) t = 1 n (rt +...+r t n) I {(rt+...+r t n) 0} = 1 n (rt +...+r t n) I {(rt+...+r t n)<0} Suppose that now latent volatility is determined by the cascade: log σ (n 1) t+n 1 = c (n1) +β (n1) log RV (n 1) t +γ (n1) r (n 1) t +γ (n1)+ r (n 1)+ t +ε (n 1) t+n 1 [ ] log σ (n 2) t+n 2 = c (n2) +β (n2) log RV (n 2) t +γ (n2) r (n 2) t +γ (n2)+ r (n 2)+ t +δ (n2) E t log σ (n 1) t+n 1 +ε (n 2) t+n 2 which now gives: log RV (n 2) t+n 2 = c + β (n2) log RV (n 2) t +β (n1) log RV (n 1) t + γ (n 2) r (n 2) t +γ (n 2)+ r (n 2)+ t +γ (n 1) r (n 1) t +γ (n 1)+ r (n 1)+ t + ε (n 2) t+n 2 leverage effects influence each market component separately, and then they appear aggregated at different horizons in the realized volatility dynamics. Fulvio Corsi High Frequency data and () Realized Volatility Models SNS Pisa 7 Dec / 38