Measuring price and volatility from high-frequency stock prices Siem - PowerPoint PPT Presentation

Measuring price and volatility from high-frequency stock prices Siem Jan Koopman , Free University Amsterdam, Netherlands, with Borus Jungbacker (Amsterdam) and Eugenie Hol (ING) Presentation at Mathematisches Forschungsinstitut Oberwolfach , ”Statistics and Finance” workshop, January 15, 2004

Outline of presentation Motivation is to forecast stock index volatility. • Empirical results of a forecast study • Forecasts based on high frequency data are superior • Methods for measuring volatility using high frequency data • Model for measuring price and volatility for tick-by-tick data

Data Standard & Poor’s 100 (S&P 100) stock index transaction prices during the period from 6 January 1997 to 14 November 2003. Daily return R n is first difference between closing prices, R n = 100(ln P n − ln P n − 1 ) , n = 1 , . . . , N, where P n is the closing asset price at trading day n . Intraday return (5-minute) is taken between successive log prices, R n,d = 100(ln P n,d − ln P n,d − 1 ) , n = 1 , . . . , N, d = 1 , . . . , D, where P n,d is the asset price at trading day n and 5-minute period d . Overnight return is R n, 0 = 100(ln P n, 0 − ln P n − 1 ,D ).

Data Realised volatility is computed as D σ 2 n = R 2 R 2 � ˜ n, 0 + n = 1 , . . . , N, n,d , d =1 but overnight return is special so it is better to take account of this: D σ 2 σ 2 n = ˆ oc + ˆ σ 2 co R 2 � ˜ n,d , σ 2 ˆ oc d =1 where 10 , 000 σ 2 � N n =1 (log P n,D − log P n, 0 ) 2 , ˆ = oc N 10 , 000 σ 2 � N n =1 (log P n, 0 − log P n − 1 ,D ) 2 . ˆ = co N

Implied volatility s 2 n is obtained from Chicago Board Options Exchange Market Volatility Index (VIX), a highly liquid options market. The VIX index is calculated from midpoint bid-ask option prices using a binomial method that takes into account the level and timing of dividend payments. Black-Scholes model assumption of constant volatility intro- duces bias into the implied volatility measure but magnitude of the bias is small for near-the-money and close-to-maturity options.

Data is based on S&P 100 stock index for the period between 6 January 1997 and 15 November 2003 (1725 observations) Summary Statistics of return and volatility time series daily return realised vol. implied vol. R 2 σ 2 σ 2 s 2 log s 2 R n ˜ log ˜ n n n n n Mean 0 . 020 1 . 889 0 . 920 − 0 . 612 26 . 46 3 . 253 Stand.Dev. 1 . 374 4 . 058 1 . 359 0 . 981 5 . 998 0 . 208 Skewness − 0 . 122 7 . 918 5 . 109 0 . 245 1 . 266 0 . 744 Exc.Kurt. 5 . 621 110 . 8 39 . 80 0 . 524 1 . 482 0 . 135 Minimum − 8 . 994 0 0 . 004 − 5 . 484 16 . 84 2 . 834 Maximum 5 . 702 80 . 89 15 . 38 2 . 733 50 . 48 3 . 922

Data : R n , R 2 σ 2 σ 2 n , s 2 n , log s 2 n , ˜ n , log ˜ n (row-wise) 10 0.4 1 0 0.2 0 1997 1999 2001 2003 −10 −5 0 5 0 20 40 100 0.50 1 50 0.25 0 1997 1999 2001 2003 0 25 50 75 0 20 40 20 1.0 1 10 0.5 0 1997 1999 2001 2003 0 5 10 15 0 20 40 5 0.50 1.0 0 0.25 0.5 −5 1997 1999 2001 2003 −5 0 0 20 40 60 1.0 0.10 40 0.5 0.05 20 1997 1999 2001 2003 20 40 0 20 40 60 1.0 0.10 40 0.5 0.05 20 1997 1999 2001 2003 20 40 0 20 40

V olatility modelling Consider spot price P ( t ) with return defined as R ( t ) = log P ( t ) − log P (0) , t > 0 . which follows the continuous time process d R ( t ) = µ ( t )d t + σ ( t )d W ( t ) , t > 0 , where µ ( t ) is drift process, σ ( t ) is spot volatility and W ( t ) is standard Brownian motion. Mean and variance of spot volatility are given by � σ 2 ( t ) � � σ 2 ( t ) � = ω 2 . E = ξ, var The actual volatility for the n -th day interval of length h is then defined as � t n = σ ∗ ( hn ) − σ ∗ (( n − 1) h ) , where σ ∗ ( t ) = σ 2 0 σ 2 ( s )d s.

OU type models for SV σ 2 n is accurate estimator of av σ 2 It is established that rv ˜ n . Barndorff-Nielsen and Shephard (2002) have studied the statis- tical properties of this estimator and its error σ 2 σ 2 n − ˜ n . Also they conclude that a model for spot volatility σ 2 ( t ) can significantly improve estimation of actual volatility. A candidate model for σ 2 ( t ) is based on the superposition of OU processes τ j ( t ), that is J σ 2 ( t ) = τ j ( t ) , d τ j ( t ) = − λ j τ j ( t )d t + d z j ( λ j t ) , � (1) j =1 where z j ( t ) is independent L´ evy process (with non-negative in- crements, known as a subordinator) and λ j is unknown.

Bandorff-Nielsen and Shephard (2001, 2002) : The SDE defining τ j ( t ) implies its acf to be � τ j ( t ) , τ j ( t + s ) � = e − λ j | s | . corr Assume E( τ j ( t )) = w j ξ and var( τ j ( t )) = w j ω 2 , acf for σ 2 ( t ) is J σ 2 ( t ) , σ 2 ( t + s ) � � w j e − λ j | s | . � corr = j =1 � nh It follows that acf of j -th component of av, τ j ( n − 1) h τ j ( t )d t , n ≡ is (1 − e − λ j h ) 2 n , τ j corr( τ j e − λ j h ( m − 1) , n + m ) = m = 1 , 2 . . . , 2( e − λ j h − 1 + λ j h ) where h is the length of the day interval.

These convenient ”BNS” results imply that τ j n have ARMA(1,1) representations: τ j n − w j ξ ) + θ j η j n +1 = w j ξ + φ j ( τ j n − 1 + η j η j n ∼ WN (0 , σ 2 n , η j ) , where WN (0 , σ 2 ) refers to a white noise process with zero mean and variance σ 2 . It follows that the autoregressive parameter φ j equals e − λ j h while Barndorff-Nielsen and Shephard (2003) show that corr( τ j n , τ j � 1 − 4 ϑ 2 1 − n +1 ) − φ j j θ j = , with ϑ j = . j ) − 2 φ j corr( τ j n , τ j (1 + φ 2 2 ϑ j n +1 ) Finally, the key to modelling realised volatility in this way is set of results in Barndorff-Nielsen and Shephard (2001), see next slide.

Define error u n = σ 2 σ 2 n − ˜ n , BNS establish it to be Gaussian with mean zero and variance   J 2 w j ω 2  ( ξh/D ) 2 + ( e − λ j h/D − 1 + λ j h/D ) σ 2  , � u = 2 D λ 2 j j =1 where D is the number of intra-daily intervals used to calculate σ 2 ˜ n . So rv model becomes (assuming av model is valid) J σ 2 τ j τ j u n ∼ NIID (0 , σ 2 � ˜ n = n + u n , n ∼ ARMA(1 , 1) , u ) , j =1 which is an unobserved ARMA components model. Model can be formulated in state space to be estimated and to compute forecasts. Note that model is not Gaussian due to η j n .

Long memory ARFIMA model for rv σ 2 Empirical work on realised volatility points out that ˜ n exhibits long memory features. This is more so when logs are taken. Suggestion is to model rv by ARFIMA model, see Andersen, Bollerslev, Diebold and Labys (2001, 2003). ARFIMA (1 , d, 1) model with mean µ is given by σ 2 (1 − φL )(1 − L ) d (˜ n − µ ) = (1 + θL ) ε n , where d , φ and θ are unknown and ε n is assumed Gaussian WN. Estimation is carried out by maximum likelihood and related procedures also provide forecasts, see Sowell (1992). For com- putational details, see Doornik and Ooms (2003).

Volatility models for daily returns The SV model is based on the ct process for returns. By dis- cretisation the return process at daily intervals and by assuming an AR for log-volatility, we obtain R n = µ + σ n ε n , ε n ∼ NID(0 , 1) , σ ∗ 2 exp( h n ) , σ 2 = n h n +1 = φh n + σ η η n , η n ∼ NID(0 , 1) , h 1 ∼ NID(0 , σ 2 η / { 1 − φ 2 } ) , for n = 1 , . . . , N and where µ is taken to be fixed and zero. The likelihood function can be constructed using simulation methods such as the ones developed by Shephard and Pitt (1997) and Durbin and Koopman (1997). For application to SV models, see also Sandmann and Koopman (1998). Note that similar methods can also be used to estimate SV models with leverage, see Koopman and Shephard (2003).

GARCH model can also be considered for daily returns and in its most simplest form is given by = ε n ∼ NID(0 , 1) , n = 1 , . . . , N, R n σ n ε n σ 2 ω + αR 2 n − 1 + βσ 2 = n − 1 , n with parameter restrictions ω > 0, α ≥ 0, β ≥ 0 and α + β < 1. The techniques of estimation and forecasting for this model are well established.

SV and GARCH models can be extended by including volatility measures in the volatility equation: SV with explanatory variable : h n = γ log s 2 n − 1 + η ∗ η ∗ n = φη ∗ n , n − 1 + σ η η n , GARCH with explanatory variable : σ 2 n = ω + αR 2 n − 1 + βσ 2 n − 1 + γs 2 n − 1 , Here s 2 n can be realised volatility (rv) or implied volatility (iv).

σ 2 Forecasting results : evaluated against ˜ m , for the Standard & Poor’s 100 with evaluation period from 17 October 2001 to 14 November 2003. Model Forecast loss functions R 2 MSE MAE HMSE HMAE UC 1 1 . 248 0 . 613 2 . 495 1 . 240 0 . 522 UC 2 0 . 996 0 . 505 1 . 546 0 . 792 0 . 593 ARFIMA 0 . 991 0 . 508 1 . 610 0 . 813 0 . 598 ARFIMA (log) 1 . 149 0 . 472 1 . 030 0 . 555 0 . 597 SV 2 . 433 1 . 240 5 . 080 2 . 948 0 . 386 SVX rv 2 . 256 1 . 037 3 . 368 2 . 063 0 . 437 SVX iv 3 . 132 1 . 082 3 . 422 2 . 048 0 . 343 GARCH 2 . 837 1 . 348 5 . 339 3 . 174 0 . 405 GARCHX rv 3 . 134 1 . 228 4 . 603 2 . 738 0 . 421 GARCHX iv 2 . 872 1 . 297 5 . 079 2 . 720 0 . 419

Volatility forecasts : GARCH, SV, RV-UC, RV-ARFIMA 2003 15 2002 15 2002 2003 10 10 5 5 15 15 10 10 5 5

Volatility forecasts : GARCH, SV, RV-UC, RV-ARFIMA Sep 2002 Nov 2002 Sep 2002 Nov 2002 7.5 7.5 5.0 5.0 2.5 2.5 6 6 4 4 2 2