1. Dissertation paper Modelling and Forecasting Volatility Index based on the stochastic volatility models MSc Student: LAVINIA-ROXANA DAVID Supervisor: Professor MOISA ALTAR, PhD. Bucharest, 2006

2. Goals to model and to forecast an index based on the stochastic volatility models to analyse the predictive ability of these models one of them is based on implied volatility calculated from option prices to perform an empirical evidence for 2 indices: S&P 100 index analysis is performed based on volatility index VXO BET index

3. Outline Importance of volatility Stochastic volatility models model specification and estimation volatility forecasting methodology Data description Model estimates and volatility forecast results Conclusions

4. Importance of volatility volatility risk is considered as one of the prime and hidden risk factors on capital markets volatility forecasting plays an important role in financial decision making predicting volatility is quite difficult to be accurately performed historical (realised) volatility and implied volatility GARCH and Stochastic Volatility (SV) models

5. Stochastic volatility models allow a stochastic element in the time series evolution of the conditional variance process time-varying volatility breaches the constant volatility assumption underlying the Black-Scholes formula  by incorporating stochastic and implied volatility from options prices new level of precision is reached  new level of precision is reached o performance confirmed by Koopman and Hol (2002), Fleming (1998), Poon and Granger (2002)

6. Stochastic volatility models mean eq.: yt =yt = variance eq.: SV Model : SVX Model: SIV Model: - based on historical returns - implied volatility as explanatory variable - obtained for

7. Model estimation parameters are estimated by simulated maximum likelihood, using Monte Carlo importance sampling SV/ SVX models - special cases of non-linear state space models: where: based on state vector:

8. Model estimation likelihood function of a SVX model is calculated via Monte Carlo technique of importance sampling: o similar with likelihood function of an approx. Gaussian model (based on Kalman filter) multiplied by a correction term o computational implementation: Ox, SsfPack 2.2 given =>,

9. Volatility forecasting methodology rolling window principle oone –step ahead volatility forecast: o N-step ahead volatility forecast: o measuring predictive forecasting ability: o regression model: o error statistics: RMSE, MAE, Theil, Variance Prop, Covariance Prop H0: a=0 si b=1

10. Data description S&P 100 indexVXO index 09/19/1997 - 12/30/2005 BET index 2060 daily returns 2084 daily returns2084 daily obs.

11. Descriptive statistics S&P 100 Daily return series Histogram and distribution of return Autocorrelation and partial correlation of daily return Period No. of Obs. T 09/19/97 to 12/30/2005 2084 09/05/2000 – 12/30/2005 1338 SeriesS&P 100 (OEX) VXOS&P 100 (OEX) VXO RtRt R t 2 RtRt Mean0.0104781.60080.68134-0.0275181.49370.54602 Standard deviation 1.26523.51930.687651.22193.18320.78182 Skewness-0.0463896.5425-0.325030.199684.95320.00098878 Excess Kurtosis2.835364.875-0.398522.562131.849-1.0014

12. Descriptive statistics BET Daily return series Histogram and distribution of return Autocorrelation and partial correlation of daily return Period No. of Obs. T 09/19/1997 – 12/30/2005 2060 09/05/2000 – 12/30/2005 1314 SeriesBET RtRt R t 2 RtRt Mean0.000915920.000316090.00200510.00020871 Standard deviation0.0177550.000885830.0143070.00057780 Skewness-0.155287.3312-0.3350012.804 Excess Kurtosis5.917371.5196.0773267.98

13. Empirical in-sample results Period09/19/97 - 12/30/200509/05/2000 - 12/30/2005 T20841338 Model SVX SIV SVX SIV 0.5139010.520230.546870.55351 -0.540020-0.772380 γ1.11091.11151.10621.1146 0.0734830.0829110.0103301.0440e-006 Period 09/05/1997 – 12/30/2005 09/05/2000 – 12/30/2005 T20601314 ModelSV 1.85571.3967 0.932090.94676 γ0.0901440.076754 0.117980.077643 S&P 100 index BET index

14. Volatility forecast results S&P 100 index o based on the period: 09/05/2000 - 05/08/2005 (1172 obs.) o evaluation period: 05/09/2005 – 12/30/2005 ( 166 obs.) o based on the parameters of this initial sample o roll it forward by one trading day, keeping the sample size constant at 1172 obs. o horizon forecast of 1, 5 and 10 days

15. Volatility forecast results S&P 100 index Forecasting Model Forecasting Horizon 1510 SVX ModelR2R2 0.6285850.1426460.197848 RMSE0.3196051.5222210.608028 MAE0.2611041.2948260.470655 Theil0.4580940.3194130.265894 Variance Prop0.5866340.4517100.384273 Covariance Prop0.4133660.5482900.615727 SIV ModelR2R2 0.4233360.2356410.016977 RMSE1.7197100.3192990.894773 MAE1.3055280.3059760.767675 Theil0.1570080.1910820.202357 Variance Prop0.2116490.3464120.769449 Covariance Prop0.7883510.6535880.230551

16. Volatility forecast results BET index o based on the period 09/05/2000 - 05/04/2005 (1132 obs.) o evaluation period: 05/05/2005 – 12/21/2005 ( 158 obs.) o based on the parameters of this initial sample o roll it forward by one trading day, keeping the sample size constant at 1132 obs. o horizon forecast of 1, 5 and 10 days

17. Volatility forecast results BET index Forecasting Model Forecasting Horizon 1510 SV Model R2R2 0.9727520.8473830.523865 RMSE0.16462110.396858.760473 MAE0.1466856.9518047.561559 Theil0.0562710.2581090.159677 Variance Prop0.0069060.1602370.041377 Covariance Prop0.9930940.8397630.958623

18. Conclusions Analysis of results SVX and SIV models are appropriate for prediction forecasting horizon N=1 provide a better prediction than N=5 and N=10 (more relevant) Forecast accuracy depends on: selecting periods as in-sample and out-of-sample selecting forecasting horizon, forecast evaluation measure volatilities ranges for in-sample unexpected change in future volatility is difficult to be predicted availability of data about implied volatility (volatility index)

19. Conclusions Proposal and further research direction better identification and selection of periods intra-day data (high frequency data) instead of daily data applying the model for a stock (instead of index) for which there are options on that stock (for calculating implied volatility)

20. Bibliography (selection) Alexander, C. (2001), “Market Models: Chapter 5: Forecasting Volatility and Correlation” Andersen T. G., T. Bollerslev, P. F. Christoffersen and F. Diebold (2006), “Volatility and correlation forecasting”, Handbook of Economic Forecasting, Volume 1 Blair, B., S.-H. Poon and S. J. Taylor (2000), “Forecasting S&P 100 volatility: The Incremental Information Content of Implied. Volatilities and High Frequency Index Returns” Fleming, J. (1998), “The quality of market volatility forecast implied by S&P100 index option prices” Ghysels, E., A. Harvey and E.Renault (1995), “Stochastic Volatility” Jungbacker, B. and S. J. Koopman, “Monte Carlo likelihood estimation for three mult Jungbacker, B. and Koopman S. J., “On Importance Sampling for State Space Models” Bos, C. (2006), “The method of Maximum Likelihood” Koopman, S. J. and E. Hol (2002), “Forecasting the Variability of Stock Index Returns with Stochastic Volatility Models and Implied Volatility” Koopman, S. J. and N. Shephard (2004), “Estimating the likelihood of the stochastic volatility model: testing the assumptions behind importance sampling” Koopman, S. J. (2005), “Introduction to State Space Methods”

21. Bibliography (selection) Jungbacker B. and S. J.Koopman, “On Importance Sampling for State Space Models” Holger, C., Mittnik S. (2002), “Forecasting Stock Market Volatility and the Informational Efficiency of the DAX index Options Market” Hull, J., “Options, futures and other derivatives” Koopman, S. J., “Modelling Volatility in Financial Time Series: Daily and Intra-daily Data” Koopman, S. J., N. Shephard and J. Doornik (1998), “Statistical algorithms for models in state space using SsfPack 2.2” Koopman, S. J., M. Ooms and M.-J. Boes (2006), “Case Econometrics and Quantitative Finance: Modeling Volatility for Forecasting and Option Pricing, MSc. Econometrics” Koopman, S. J., B. Jungbacker and E. Hol (2004), “Measuring price and volatility from high-frequency stock prices” Koopman, S. J. and K. M. Lee, “Simulated Maximum Likeliood in Stochastic Volatility Modelling” Lopez, J. A. (1999), “Evaluating the Predictive Accuracy of Volatility Models” Notger C. (2004), “Volatility and its Measurements: The Design of a Volatility Index and the Execution of its Historical Time Series at the DEUTSCHE BORSE AG”, im Fach Bank-, Finanz- und Investitionswirtschaft in WS 2004/2005 Poon, S.-H. and, C. W. J Granger (2003), “Forecasting Volatility in Financial Markets: A Review”