Modelling Volatility in Financial Time Series: Daily and Intra-daily Data Siem Jan Koopman s.j.koopman@feweb.vu.nl Vrije Universiteit Amsterdam Tinbergen Institute Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 1
Data Daily return R n 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 R n,o = 100(ln P n,o − ln P n − 1 ,D ) . Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 2
Data Realised volatility is computed as D � σ 2 n = R 2 R 2 ˜ n, 0 + n,d , n = 1 , . . . , N, 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 R 2 co ˜ n,d , σ 2 ˆ oc d =1 where � N 10 , 000 σ 2 n =1 (log P n,D − log P n, 0 ) 2 , ˆ = oc N 10 , 000 � N σ 2 n =1 (log P n, 0 − log P n − 1 ,D ) 2 . ˆ = co N Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 3
Data 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 introduces bias into the implied volatility measure but magnitude of the bias is small for near-the-money and close-to-maturity options. Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 4
S&P 100 Volatility 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 0 . 020 1 . 889 0 . 920 − 0 . 612 26 . 46 3 . 253 Mean Stand.Dev. 1 . 374 4 . 058 1 . 359 0 . 981 5 . 998 0 . 208 − 0 . 122 Skewness 7 . 918 5 . 109 0 . 245 1 . 266 0 . 744 Exc.Kurt. 5 . 621 110 . 8 39 . 80 0 . 524 1 . 482 0 . 135 − 8 . 994 0 0 . 004 − 5 . 484 16 . 84 2 . 834 Minimum Maximum 5 . 702 80 . 89 15 . 38 2 . 733 50 . 48 3 . 922 Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 5
S&P 100 Volatility 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 Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 6
Stochastic Volatility in Continuous Time 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 σ 2 ( s ) d s. 0 Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 7
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 statistical 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 ) , j =1 where z j ( t ) is independent Lévy process (with non-negative increments, known as a subordinator) and λ j is unknown. Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 8
OU Type Models for SV Bandorff-Nielsen and Shephard (2001, 2002) : The SDE defining τ j ( t ) implies its acf to be = e − λ j | s | . τ j ( t ) , τ j ( t + 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 , is n ≡ (1 − e − λ j h ) 2 n , τ j 2( e − λ j h − 1 + λ j h ) e − λ j h ( m − 1) , corr ( τ j n + m ) = m = 1 , 2 . . . , where h is the length of the day interval. Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 9
OU Type Models for SV 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 � 1 − 4 ϑ 2 1 − n , τ j corr ( τ j n +1 ) − φ j j θ j = , with ϑ j = . j ) − 2 φ j corr ( τ j n , τ j 2 ϑ j (1 + φ 2 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. Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 10
SV Models for Daily Returns The discrete time SV model is based on the continuous process for returns. By discretisation the return process at daily intervals and by assuming an AR for log-volatility, we obtain ε n ∼ NID (0 , 1) , R n = µ + σ n ε n , 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. Note that this is a non-linear state space model. Taking log R 2 n as the dependent variable, the model becomes a linear non-Gaussian state space model. Efficient estimates can be obtain by using Importance Sampling, see Sandmann and Koopman (1998). Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 11
Measuring Realised Volatility High-frequency price data (tick by tick) is subject to irregularities in recording and market micro-structure. Current practice of computing realised volatility is to construct five minute returns and compute rv from these. However, such data can be messy and a regular series of daily series of 5 minute returns is not always available. Also bid-ask spreads in data can be huge (ways to capture these require a lot of extra data and modelling). Some approaches of obtaining a regular set of 5-minute return to linearly interpolate between ask-bid bounces as in Andersen, Bollerslev, Diebold and Ebens (2001). More flexible spline interpolations are used by Hansen and Lunde (2003) and Fourier methods are used by Malliavin and Mancino (2002) and Barucci and Reno (2002). First we adopt a model-based version of these interpolations. Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 12
Spline Model in State Space Representation Consider the smoothing problem where the log price p ( t ) is a continuous function of t > 0 . To smooth p ( t ) by function µ ( t ) , we observe tick prices (bid and asks) p ( t i ) for i = 1 , . . . , n where 0 < t 1 < . . . < t n < T ( t i is a tick). We can choose µ ( t ) to be a twice-differentiable function on ( 0 , T ) which minimises � T � 2 n � ∂ 2 µ ( t ) [ p ( t i ) − µ ( t i )] 2 + λ � dt, ∂t 2 0 i =1 This problem can be represented as a state space model ε ( t i ) ∼ N [0 , σ 2 ( t i )] , p ( t ) = µ ( t ) + ε ( t ) , t = t 1 . . . , t n , with state equation � � � � � � � � µ ( t ) 0 1 µ ( t ) 0 d = dt + σ ζ . ν ( t ) 0 0 ν ( t ) d W ( t ) Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 13
Spline Model in State Space Representation In discrete time, we obtain the model p i = µ i + ε i with � � � � � � � � µ i +1 1 δ i µ i ξ i = + , i = 1 , . . . , n, ν i +1 0 1 ν i ζ i where the disturbances are Gaussian and correlated with each other. The distance δ i is for the distance in seconds between tick prices (can be zero !). The variance of ζ i , as a ratio of the variance of ε i , equals q = 1 /λ . This model can be used to smooth out the micro-structure in tick prices and to obtain a regular set of 5 minutes quotes from which rv can be computed, for example. In standard smoothing q (or λ ) is fixed. Here, we estimate q for each day by standard maximum likelihood methods using the Kalman filter (see www.ssfpack.com ). It turns out that the q estimates are very close to rv, up to a constant. Modelling Volatility in Financial Time Series:Daily and Intra-daily Data – p. 14
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