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Dissertation paper

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. Goals.

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Dissertation paper

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  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 analysethe 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 • performance confirmed by Koopman and Hol (2002), Fleming (1998), Poon and Granger (2002)

  6. Stochastic volatility models yt= • mean eq.: • variance eq.: SV Model : - based on historical returns SVX Model: - implied volatility as explanatory variable SIV Model: - 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: • similar with likelihood function of an approx. Gaussian model (based on Kalman filter) multiplied by a correction term , • computational implementation:Ox, SsfPack 2.2 • given =>

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

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

  11. Descriptive statistics S&P 100 Autocorrelation and partial correlation of daily return Daily return series Histogram and distribution of return

  12. Descriptive statistics BET Autocorrelation and partial correlation of daily return Daily return series Histogram and distribution of return

  13. Empirical in-sample results BET index S&P 100 index

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

  15. Volatility forecast results S&P 100 index

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

  17. Volatility forecast results BET index

  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”

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