1. Predicting volatility: a comparative analysis between GARCH Models and Neural Network Models MCs Student: Miruna State Supervisor: Professor Moisa Altar - Bucharest, June 2002 -

2. Doctoral School of Finance and Banking2 Contents Introduction Models for return series  GARCH models  Mixture Density Networks Aplication and results Conclusion and further research Selective bibliography

3. Doctoral School of Finance and Banking3 1. Introduction Concepts of risk and volatility Objective: compare the GARCH volatility models with neural network based models for modeling conditional density

4. Doctoral School of Finance and Banking4 2. Models for time series returns 2.1 ARCH(p) models

5. Doctoral School of Finance and Banking5 2.2 GARCH (p,q) GARCH(1,1)

6. Doctoral School of Finance and Banking6 The unconditional variance from the GARCH (1,1) GARCH (1,1) it can be written as an infinite ARCH model :

7. Doctoral School of Finance and Banking7 2.3 Mixture Density Networks  Venkatamaran (1997), Zangari (1996) -used unconditional mixture densities for calculating VaR  Lockarek-Junge and Prinzler (1998) -used one neural network to model the density conditionally  Schittenkopf and Dorffner(1998, 1999) - concentrated on the performance of the of neural network based models to estimate volatility

8. Doctoral School of Finance and Banking8 Mixture Densities  the random variable is drawn from one out of many possible normal distributions  allows for heavy tails  preserves some convenient characteristics of a normal distribution

9. Doctoral School of Finance and Banking9 Neural Networks  have been used for medical diagnostics, system control, pattern recognition, nonlinear regression, and density estimation  relates a set of input variables x t t=1,…,k, to a set of one or more output variables, y t, t=1,…,k  it is composed of nodes

10. Doctoral School of Finance and Banking10  three common types of non-linearities used in ANNs

11. Doctoral School of Finance and Banking11 Multi-Layer Perceptron (MLP)  has one hidden layer The mapping performed by the MLP is given by

12. Doctoral School of Finance and Banking12 Mixture Density Network  combines a MLP and a mixture model  the conditional distribution of the data - expressed as a sum of normal distributions Estimation of MDN - by minimizing the negative logarithm of the likelihood function - by using backpropagation gradient descendent algorithm

13. Doctoral School of Finance and Banking13 RPROP algorithm  partial derivative of a weight changes its sign - the update value is decreased by a factor η -  If the derivative doesn’t change its sign - slightly increase the update value by the factor η +  0< η - <1< η +  η + =1.2  η - =0.5

14. Doctoral School of Finance and Banking14 3. Application and results Data used  daily closing values of the BET-C from 17.04.1998 to 10.05.2002  Returns calculated as follows: r t = ln(P t /P t-1 )  Two data sets: - a training one - a testing one  Softwere used: Eviews, Matlab Netlab

15. Doctoral School of Finance and Banking15 GARCH Estimation The daily BET-C returnsHistogram of the returns series

16. Doctoral School of Finance and Banking16 Mean equation Dependent Variable: RETURN_BETC Method: Least Squares Sample(adjusted): 2 1020 Included observations: 1019 after adjusting endpoints Convergence achieved after 2 iterations VariableCoefficientStd. Errort-StatisticProb. C-0.000193 0.000654-0.295154 0.7679 AR(1) 0.294192 0.029949 9.823033 0.0000 R-squared 0.086657 Mean dependent var-0.000189 Adjusted R-squared 0.085759 S.D. dependent var 0.015418 S.E. of regression 0.014742 Akaike info criterion-5.594207 Sum squared resid 0.221035 Schwarz criterion-5.584538 Log likelihood 2852.249 F-statistic 96.49198 Durbin-Watson stat 2.000042 Prob(F-statistic) 0.000000 Inverted AR Roots.29

17. Doctoral School of Finance and Banking17 ARCH LM test for serial correlation in the residuals from the mean equation ARCH Test: F-statistic 33.88049 Probability 0.000000 Obs*R-squared 120.0804 Probability 0.000000 Test Equation: Dependent Variable: RESID^2 Method: Least Squares Sample(adjusted): 6 1020 Included observations: 1015 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. C 0.000126 1.95E-05 6.461881 0.0000 RESID^2(-1) 0.292038 0.031465 9.281234 0.0000 RESID^2(-2) 0.092484 0.032770 2.822200 0.0049 RESID^2(-3) 0.027229 0.032769 0.830916 0.4062 RESID^2(-4) 0.008512 0.031505 0.270192 0.7871 R-squared 0.118306 Mean dependent var 0.000217 Adjusted R-squared 0.114814 S.D. dependent var 0.000573 S.E. of regression 0.000539 Akaike info criterion-12.20960 Sum squared resid 0.000293 Schwarz criterion-12.18535 Log likelihood 6201.370 F-statistic 33.88049 Durbin-Watson stat 1.999460 Prob(F-statistic) 0.000000

18. Doctoral School of Finance and Banking18 Estimation of GARCH (1,1) Dependent Variable: RETURN_BETC Method: ML - ARCH Sample(adjusted): 2 1020 Included observations: 1019 after adjusting endpoints Convergence achieved after 23 iterations Bollerslev-Wooldrige robust standard errors & covariance CoefficientStd. Errorz-StatisticProb. RETURN_BETC(-1) 0.342440 0.035452 9.659369 0.0000 Variance Equation C 4.42E-05 1.34E-05 3.303162 0.0010 ARCH(1) 0.345598 0.073219 4.720056 0.0000 GARCH(1) 0.483342 0.111485 4.335486 0.0000 R-squared 0.084258 Mean dependent var-0.000189 Adjusted R-squared 0.081551 S.D. dependent var 0.015418 S.E. of regression 0.014776 Akaike info criterion-5.785426 Sum squared resid 0.221616 Schwarz criterion-5.766087 Log likelihood 2951.675 F-statistic 31.13013 Durbin-Watson stat 2.094529 Prob(F-statistic) 0.000000

19. Doctoral School of Finance and Banking19 MDN Estimation  feed forward single-hidden layer neural network  4 hidden units  3 Gaussians  m-dimensional input x t-1,…,x t-m  3n dimensional output : weights, conditional mean, and conditional variance

20. Doctoral School of Finance and Banking20 Evaluation of the models  Normalized mean absolute error  Normalized mean squared error

21. Doctoral School of Finance and Banking21  Hit rate  Weighted hit rate

22. Doctoral School of Finance and Banking22 Results ModelNMAEHRLoss functionWHRNMSE NNLearning sample 0.7505840.5927322.9092790.5602680.888448 Testing sample 0.8311390.578704 2.820094 0.5878780.784555 Garch(1,1)Learning sample 0.594350.6854642.9326130.5694740.76637 Test sample 0.621550.7129632.7443260.5758860.983746 GARCH(1,1) 0.9054860.6368992.8966390.575640.806182

23. Doctoral School of Finance and Banking23 4. Conclusion and further research Recurrent neural networks The structure of the network used Trading or hedging strategies Methodoligies for measuring market risk

24. Doctoral School of Finance and Banking24 5. Selective bibliography Bartlmae, K. and R.A. Rauscher (2000) – Measuring DAX Market Risk: A Neural Network Volatility Mixture Approach, www.gloriamundi.org/var/pub/bartlmae_rauscher.pdf. www.gloriamundi.org/var/pub/bartlmae_rauscher.pdf Bishop, W. (1994) - Mixture Density Network, Technical Report NCRG/94/004,Neural Computing Research Group, Aston University, Birmingham, February. Jordan, M. and C. Bishop (1996)– Neural Networks, in CDR Handbook of Computer Science, Tucker, A. (ed.), CRC Press, Boca Raton. Locarek-Junge, H. and R. Prinzler (1998) - Estimating Value-at-Risk Using Neural Networks, Application of Machine Learning and Data Mining in Finance, ECML’98 Workshop Notes, Chemnitz. Schittenkopf, C. and G. Dockner (1999) – Forecasting Time-dependent Conditional Densities: A Neural Network Approach, Vienna University of Economic Studies and Business Administration, Report Series no.36. (1998) – Volatility Prediction with Mixture Density Networks, Vienna University of Economic Studies and Business Administration, Report Series no.15. Venkatamaran, S. (1997) – Value at risk for a mixture of normal distributions: The use of quasi-Bayesian estimation techniques, Economic Perspectives (Federal Bank of Chicago), pp. 3-13. Zangari, P. (1996)- An improved methodology for measuring VaR, in RiskMetrics Monitor 2.