1. by Adjoa Numatsi and Erick Rengifo Economics Department, Fordham University, New York Exploratory analysis of GARCH Models versus Stochastic Volatility Models with Jumps in Returns and Volatility (Work in Progress) Conference on Quantitative Social Science Research Using R Fordham University – 18-19 June, 2009

2. OUTLINE Introduction Models and Data Preliminary Results Conclusion Further research

3. Introduction Multiple attempts to improve equity pricing models to reconcile theory with empirical features Important because misspecified models  mistakes in forecasting Example in option pricing: Black Scholes formula has shown biases (Rubinstein, 1985) due to 2 major assumptions on the underlying stock pricing process Stock prices: continuous path through time, their distribution is lognormal Variance of stock returns is constant (Black and Scholes, 1973) But asset returns are leptokurtic, and display volatility clustering (Chernov et al.(1999))

4. Both assumptions have been relaxed allowing for: discontinuities in the form of jump diffusion models (Merton (1976), Cox and Ross (1976)) stochastic volatility (Hull and White (1987), Scott (1987), Heston (1993)) Bates (1996) and Scott (1997) combined stochastic volatility models with jumps in returns However, the volatility process is still misspecified (Bakshi, Cao, and Chen (1997), Bates (2000), and Pan (2002)): Jumps in returns can generate large movement, but the impact is temporary Diffusive stochastic volatility is persistent but because its dynamics are driven by a Brownian motion  small normally distributed increments. Need for conditional volatility to move rapidly and also be persistent Duffie, Pan, and Singleton (2000) : models with jumps in both returns and volatility Introduction, cont…

5. Estimated by Eraker, Johannes, and Polson (2003). Results showed almost no misspecification But with all the features that these new models have, they are complex and it is time consuming to work with them. Our study will address this issue by comparing models with jumps and simple GARCH models GARCH models: Introduced by Bollerslev (1986) and Taylor (1986) Have time varying variance Discrete time models  empirically favored compared to continuous time models There are attempts to model GARCH with jumps (Duan, Ritchken, Sun (2006)) but we are interested in simple GARCH Introduction, cont…

6. Objectives Given the complexity of stochastic volatility models with jumps in both returns and volatility (SV2J thereafter), we want to provide a model that will allow to choose SV2J models only when they are relevant More specifically: We want to compare the performance of SV2J models and simple GARCH models in order to identify the market situation in which their respective performances are significantly different.

7. Clusters vs. Jumps Models should be able to capture specific behavior in the data Index returns display clusters and jumps Jumps do not always imply existence of autoregressive conditional heteroskedasticity (ARCH) process  ARCH type Models cannot capture dynamics However we do not always have jumps in mean and variance, but a smooth diffusion process where clusters can be found  ARCH type Models can do a good job then.

8. Clusters vs. Jumps, cont… 1 st case: a jump but no clusters. Ho = Homoscedasticity ARCH LM test ( p_value) = 0.321236. We do not reject the null  there is no ARCH effect. GARCH model is not good here 2nd case: clusters. ARCH LM test ( p_value) = 0.011259. We reject the null  There is ARCH process, which means GARCH is appropriate here.

9. Clusters vs. Jumps, cont… Alpha1 and beta1 are the GARCH and ARCH coefficients. They are significant R package used: fGarch

10. Clusters vs. Jumps, cont… We can also have both jumps and clusters (which is actually the case in our last example). Clusters are smooth movements increasing and decreasing. Jumps are not well defined in the literature, but they are characterized by one big move up or down. In our study we consider that differences in returns above 3 standard deviations from the mean are jumps What we are doing is to find situations in which SV models give results significantly different from GARCH, and therefore are worth the effort of estimating them.

11. Data FTSE100 daily returns from July 3, 1984 to Dec 29, 2006 (5686 observations) The volatility was generated by a rolling window approach Differences in returns and volatility above 3 standard deviations from the mean are considered as jumps

12. Data, cont… Y Mean0.031348 Standard Error0.013494 Median0.062642 Mode0 Standard Deviation1.017553 Sample Variance1.035415 Kurtosis8.268878 Skewness-0.5567 Range20.62556 Minimum-13.0286 Maximum7.596966 Sum178.2447 Count5686 V Mean1.042229 Standard Error0.015588 Median0.685022 Mode#N/A Standard Deviation1.17543 Sample Variance1.381636 Kurtosis18.20166 Skewness3.866057 Range9.187442 Minimum0.182197 Maximum9.36964 Sum5926.115 Count5686

13. Jumps in returns and volatility

14. Models: GARCH

15. Models: SV2J

16. Models: SV2J, cont…

17. Methodology Estimate a simple GARCH model and a SV2J model on a period where the market is stable, and on a period where there is instability in the market, and compare the performances. Market stability will be measured in standard deviations from long-term mean returns. Compare the results in terms of out-of-sampling forecast errors and the use of resources (basically time)

18. Estimation method There are packages in R dealing with GARCH (tseries, fGarch packages), and with Stochastic Differential Equations (sde package). But there are no packages yet for sde equation with double jumps  we have written a function in R to estimate our SV models, using MCMC method. We first derived the posterior distribution from the prior information, the distribution of the state variables and the likelihood. Then we derived the full conditional distributions from the posterior and programmed them in R. R packages used: Rlab, MCMCpack, msm

19. Estimation method: R code  R packages used: Rlab, MCMCpack, msm  Steps: 1.Read in the data 2.MCMC function to estimate the parameters and generate the state variables 3.Run the function and analyze the results

20. Preliminary Results We estimate: GARCH (1,1) Stochastic Volatility Model with jumps

21. Preliminary Results, cont… parameterssdmedian mu 0.9288264.49E-010.928147 mu_y -0.94437444.60E-01-0.97721 mu_v 141.314382.97E+0267.77618 theta 0.17800382.79E-020.171543 sigmasq_y 0.681036384.68E-020.694145 sigmasq_v 0.00973231.36E-020.007103 k 0.974602765.32E-020.995949 rho 0.234441021.63E-010.204819 rho_J 2.956022162.84E+002.562103 lamda_y 0.001300321.76E-030.000904 lamda_v 0.001300321.76E-030.000904 sigma_y 0.82524929 NaN sigma_v 0.09865241 NaN

22. Estimation challenges and future steps Convergence Choice of starting values, mostly for the state variables (especially jumps in returns and volatility) Explore R interface with WinBUGS (Bayesian Inference Using Gibbs Sampling)

23. Thank you!!!