Machine Learning for Multi-step Ahead Forecasting of Volatility Proxies
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1 Machine Learning for Multi-step Ahead Forecasting of Volatility Proxies Jacopo De Stefani, Ir. - jdestefa@ulb.ac.be Prof. Gianluca Bontempi - gbonte@ulb.ac.be Olivier Caelen, PhD - olivier.caelen@worldline.com Dalila Hattab, PhD - dalila.hattab@equensworldline.com MIDAS ECML-PKDD Hotel Aleksandar Palace, Skopje, FYROM Monday 18 th September, 2017
2 Problem overview First series CAC40 [ / ] Last Volume (100,000s): 345, Moving Average Convergence Divergence (12,26,9): MACD: Signal: Jan Mar May Jul Sep Nov Jan Mar May Jul Sep Nov /32
3 What is volatility? Definition Volatility is a statistical measure of the dispersion of returns for a given security or market index. 1 High volatility Low volatility 0.5 rt t [days] 3/32
4 A closer look on data - Volatility proxies Calendar Day 0 Calendar Day P h 0 P h 1 Pt 10 P c 0 Pre-opening P c P o 0 P l 0 1 f f 1 f P o 1 P l 1 P o t t [days] P h t P l t P c t Volatility proxy σ P t 4/32
5 Models for volatility Volatility models Past volatility Averagebased HA MA ES EWMA STES Simple Regression SR-AR SR-TAR SR-ARMA Random Walk ARCH Symmetric ARCH (q) Asymmetric E GJR- Q ST- RS- Extended Component- R Machine Learning Univariate NN k-nn SVR Multivariate 5/32
6 Models for volatility Volatility models Past volatility Averagebased HA MA ES EWMA STES Simple Regression SR-AR SR-TAR SR-ARMA Random Walk ARCH Symmetric ARCH (q) Asymmetric E GJR- Q ST- RS- Extended Component- R Machine Learning Univariate NN k-nn SVR Multivariate 5/32
7 Models for volatility Volatility models Past volatility Averagebased HA MA ES EWMA STES Simple Regression SR-AR SR-TAR SR-ARMA Random Walk ARCH Symmetric ARCH (q) Asymmetric E GJR- Q ST- RS- Extended Component- R Machine Learning Univariate NN k-nn SVR Multivariate Established Research 5/32
8 Models for volatility SVR k-nn NN Univariate Machine Learning Multivariate ES MA Future Research Established Research STES EWMA HA R Component- Averagebased Volatility models Extended ST- RS- SR-TAR SR-AR Simple Regression Past volatility ARCH Asymmetric Q GJR- SR-ARMA E Random Walk Symmetric ARCH (q) 5/32
9 Multistep ahead TS forecasting - Taieb [2014] Definition Given a univariate time series {y 1,, y T } comprising T observations, forecast the next H observations {y T +1,, y T +H } where H is the forecast horizon. Hypotheses: Autoregressive model y t = m(y t 1,, y t d ) + ε t with lag order (embedding) d ε is a stochastic iid model with µ ε = 0 and σ 2 ε = σ 2 6/32
10 Multistep ahead forecasting for volatility State-of-the-art NAR [σp t d σ P t 1 ] m(σ P ) [ˆσ P t ˆσ P t+h ] 1 Input 1 Output 7/32
11 Multistep ahead forecasting for volatility State-of-the-art NAR [σp t d σ P t 1 ] Proposed model NARX [σt d P [σt d X σt 1 P ] σt 1 X ] m(σ P ) m(σ P, σ X ) [ˆσ P t ˆσ P t+h ] [ˆσ P t ˆσ P t+h ] 1 Input 1 Output 2 inputs 1 output 7/32
12 Multistep ahead forecasting for volatility State-of-the-art NAR [σp t d σ P t 1 ] Proposed model NARX [σt d P [σt d X σt 1 P ] σt 1 X ] Future work [σt d P [ [σ X M t d σt 1 P ] ] σ X M t 1 ] m(σ P ) m(σ P, σ X ) m(σ P,, σ X M) [ˆσ P t ˆσ P t+h ] 1 Input 1 Output [ˆσ P t ˆσ P t+h ] 2 inputs 1 output [ˆσ P t [ [ˆσ X M t ˆσ P t+h ] ] ˆσ X M t+h ] M + 1 inputs M + 1 outputs 7/32
13 Multistep ahead forecasting for volatility Direct method A single model f h for each horizon h. Forecast at h step is made using h th model. Dataset examples (d = 3, h = 3): Direct NAR Direct NARX x y x y σ P 3 σ P 2 σ P 1 σ P 5 σ4 P σ3 P σ2 P σ6 P σ P T 5 σ P T 6 σ P T 7 σ P T 2 σ P 3 σ P 2 σ P 1 σ X 3 σ X 2 σ X 1 σ P 5 σ4 P σ3 P σ2 P σ4 X σ3 X σ2 X σ6 P σ P T 5 σ P T 6 σ P T 7 σ X T 5 σ X T 6 σ X T 7 σ P T 2 8/32
14 Experimental setup [σt d P [σt d X m(σ P, σ X ) σt 1 P ] σt 1 X ] Data: Volatility proxies σ X, σ P Price based from CAC40: σ i family - Garman and Klass [1980] Return based (1,1) model - Hansen and Lunde [2005] Sample standard deviation [ˆσ P t ˆσ P t+h ] 2 TS Input 1 TS Output Models: Feedforward Neural Networks (NAR,NARX) k-nearest Neighbours (NAR,NARX) Support Vector Regression (NAR,NARX) Naive (w/o σ X ) (1,1) (w/o σ X ) Average (w/o σ X ) 9/32
15 ????????????? Correlation meta-analysis (cf. Field [2001]) Volume σ 1 σ 6 σ 4 σ 5 σ 2 σ 3 r t σ σ SD 100 σ SD 50 σ SD σ G Volume σ 1 σ 6 σ 4 σ 5 σ 2 σ 3 r t σ σ SD 100 σ SD 50 σ SD σ G time series (CAC40) Time range: to OHLC samples per TS Hierarchical clustering using Ward Jr [1963] All correlations are statistically significant 10/32
16 NARX forecaster - Results ANN 11/32
17 NARX forecaster - Results ANN 12/32
18 NARX forecaster - Results KNN 13/32
19 NARX forecaster - Results KNN 14/32
20 NARX forecaster - Results SVR 15/32
21 NARX forecaster - Results SVR 16/32
22 Conclusions Correlation clustering among proxies belonging to the same family, i.e. σ i t and σ SD,n t. All ML methods outperform the reference method, both in the single input and the multiple input configuration. Only the addition of an external regressor, and for h > 8 bring a statistically significant improvement (paired t-test, pv=0.05). No model appear to clearly outperform all the others on every horizons, but generally SVR performs better than ANN and k-nn. 17/32
23 Thank you for your attention! Any questions/comments? Find the paper at: 18/32
24 Bibliography I References Tim Bollerslev. Generalized autoregressive conditional heteroskedasticity. Journal of econometrics, 31(3): , Andy P Field. Meta-analysis of correlation coefficients: a monte carlo comparison of fixed-and random-effects methods. Psychological methods, 6(2):161, Mark B Garman and Michael J Klass. On the estimation of security price volatilities from historical data. Journal of business, pages 67 78, /32
25 Bibliography II Peter R Hansen and Asger Lunde. A forecast comparison of volatility models: does anything beat a garch (1, 1)? Journal of applied econometrics, 20(7): , Rob J Hyndman and Anne B Koehler. Another look at measures of forecast accuracy. International journal of forecasting, 22(4): , Souhaib Ben Taieb. Machine learning strategies for multi-step-ahead time series forecasting. PhD thesis, Ph. D. Thesis, Joe H Ward Jr. Hierarchical grouping to optimize an objective function. Journal of the American statistical association, 58 (301): , /32
26 Appendix 21/32
27 System overview Raw OHLC data Missing values imputation User choice Data preprocessing Imputed OHLC data Model choice {ANN, KNN} Proxy generation σt, i σt SD, σt G Model identification Correlation analysis Evaluation choice User choice {RO, RW} m, θ Forecaster 22/32
28 System overview Raw OHLC data Missing values imputation User choice Data preprocessing Imputed OHLC data Model choice {ANN, KNN} Proxy generation σt, i σt SD, σt G Model identification Correlation analysis Evaluation choice User choice {RO, RW} m, θ Forecaster 22/32
29 System overview Raw OHLC data Missing values imputation User choice Data preprocessing Imputed OHLC data Model choice {ANN, KNN} Proxy generation σt, i σt SD, σt G Model identification Correlation analysis Evaluation choice User choice {RO, RW} m, θ Forecaster 22/32
30 Correlation analysis - Methodology [ σ i(1), σ SD(1), σ G(1)] corr( ) corr(σ (1) ) [ σ i(j), σ SD(j), σ G(j)] corr( ) corr(σ (j) ) Metaanalysis toolkit corr(σ AGG ) [ σ i(n), σ SD(N), σ G(N)] corr( ) corr(σ (N) ) 40 Time series (CAC40) Time range: to OHLC samples per TS 23/32
31 NARX forecaster - Methodology Disturbances σ J p Original DGP σ J f d e σ X p Model m (θ, σ J p, σ X p ) ˆσ J f m (, σ J p, σ X p ) θ {ANN,KNN} {RO, RW} Structural Parametric identification identification Model identification 24/32
32 Volatility proxies (1) - Garman and Klass [1980] Closing prices ˆσ 0(t) = Opening/Closing prices [ ( )] 2 ˆσ 1(t) = 1 2f P (o) t+1 ln + P (c) t }{{} Nightly volatility [ ln ( )] 2 P (c) t+1 = r P (c) t 2 (1) t [ ( )] (c) 2 1 P 2(1 f) t ln P (o) t }{{} Intraday volatility (2) OHLC prices ˆσ 3(t) = a f [ ˆσ 2(t) = 1 2 ln 4 ln [ ln ( P (o) t+1 P (c) t )] 2 } {{ } Nightly volatility ( )] (h) 2 P t (3) P (l) t + 1 a 1 f ˆσ2(t) }{{} Intraday volatility (4) 25/32
33 Volatility proxies (2) - Garman and Klass [1980] OHLC prices u = ln ( ) (h) P t P (o) t d = ln ( ) (l) P t P (o) t c = ln ( ) (c) P t P (o) t (5) ˆσ 4(t) = 0.511(u d) [c(u + d) 2ud] 0.383c 2 (6) ˆσ 5(t) = 0.511(u d) 2 (2 ln 2 1)c 2 (7) ˆσ 6(t) = a f log ( P (o) t+1 P (c) t ) 2 } {{ } Nightly volatility + 1 a 1 f ˆσ4(t) }{{} Intraday volatility (8) 26/32
34 Volatility proxies (3) (1,1) model - Hansen and Lunde [2005] p q σt G = ω + β j(σt j G )2 + α iε 2 t i j=1 where ε t i N (0, 1), with the coefficients ω, α i, β j fitted according to i=1 Bollerslev [1986]. Sample standard deviation σ SD,n t = 1 n 1 (r t i r) n 1 2 i=0 where r t = ln ( ) P (c) t P (c) t 1 r n = 1 n t j=t n r j 27/32
35 Hyndman and Koehler [2006] - Error measures RelX MdRAE Percent- Better Relative Measures MRAE Relative Errors GMRAE MASE smdape Error measures MAPE MdAE MdAPE Scale independant smape Scale dependant MAE RMdSPE RMSPE MSE RMSE 28/32
36 Hyndman and Koehler [2006] - Scale dependant Scale dependant MdAE e t = y t ŷ t MSE : 1 nt=0 n (y t ŷ t ) 2 RMSE : 1 n nt=0 (y t ŷ t ) 2 MAE MAE : 1 n nt=0 y t ŷ t MSE RMSE MdAE : Md t {1 n} ( y t ŷ t ) 29/32
37 Hyndman and Koehler [2006] - Scale independant MAPE : 1 n nt=0 100 yt ŷt y t MAPE smdape MdAPE : Md t {1 n} ( 100 yt ŷt y t ) Scale independant smape RMSPE : 1 nt=0 n (100 yt ŷt y t ) 2 MdAPE RMSPE RMdSPE RMdSPE : Md t {1 n} ((100 yt ŷt y t ) 2 ) smape : 1 n nt=0 200 yt ŷt y t+ŷ t smdape : Md t {1 n} (200 yt ŷt y t+ŷ t ) 30/32
38 Hyndman and Koehler [2006] - Relative errors MdRAE r t = e t e t MRAE MRAE : 1 n nt=0 r t GMRAE MdRAE : Md t {1 n} ( r t ) Relative Errors GMRAE : n t = 0 n r t 1 n MASE MASE :( Tt=1 1 T e t 1 T T 1 i=2 Y i Y i 1 ) 31/32
39 Hyndman and Koehler [2006] - Relative measures Percent- Better RelX Relative Measures RelX : X X bench Percent Better : P B(X) = n forecasts I(X < X b) where X: Error measure of the analyzed method X b : Error measure of the benchmark 32/32
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