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Recent Developments in SeasonalVolatility Models 13Recent Developments in Seasonal Volatility Models43g ′′ [E(σt 2)] = √ 1 [ ⎛ ⎞⎝ E(σ2 t )+2(log(S/K)+rT)√⎠2π 4E(σt 2) E(σt 2)⎛⎞+ ⎝ 6(log(S/K)+rT)+E(σ2 √ t ) ⎠8[E(σt 2)]2 E(σt 2)( )[E(σ 2t )] 2 − 4(log(S/K)+rT) 2[E(σt 2)]2 ] {exp − (2(log(S/K)+rT) − }E(σ2 t ))28E(σt 2) ,where N denotes the standard normal CDF, and under the option pricing model with seasonalGARCH volatility,E(σ 2 t )=κ (y) =ω() ,pP1 − ∑ φ i)(1 − ∑ Φ ii=1i=13∞ .3 − 2 ∑ Ψ 2 jj=17. Concluding remarksIn this chapter we propose various classes of seasonal volatility models. We consider timeseries processes such as AR and RCA with multiplicative seasonal GARCH errors and SVerrors. The multiplicative seasonal volatility models are suitable for time series whereautocorrelation exists at seasonal and at adjacent non-seasonal lags. The models introducedhere extend and complement the existing volatility models in the literature to seasonalvolatility models by introducing more general structures.It is well-known that financial time series exhibit excess kurtosis. In this chapter we derivethe kurtosis for different seasonal volatility models in terms of model parameters. We alsoderive the closed-from expression for the variance of the l-steps ahead forecast error of i)y n+l in terms of ψ-weights and model parameters, and of ii) squared series Y n+l in termsof Ψ-weights, model parameters and the kurtosis of ɛ t . The results are a generalization ofexisting results for non-seasonal volatility processes. We provide examples for all the differentclasses of models considered and discussed them in some detail (i.e. AR(1)-GARCH(p, q) ×(P, Q) s , RCA(1)-GARCH(p, q) × (P, Q) s and RCA(1)-seasonal SV).The results are primarily oriented to financial time series applications. Financial time seriesoften meet the large dataset demands of the volatility models studied here. Also, financial datadynamics in higher order moments are of interest to many market participants. Specifically,we consider the Black-Scholes model with seasonal GARCH volatility and show that themoments of the seasonal volatility process can be used to evaluate the call price for Europeanoptions.8. ReferencesBadescu, A. & Kulperger, R. (2008). GARCH option pricing: A semiparametric approach.Insurance, Mathematics & Economics, Vol. 43, No. 1, 69 – 84, ISSN 0167-6687
44 Advances in Econometrics - Theory and Applications14 Will-be-set-by-IN-TECHBaillie, R. & Bollerslev, T. (1990). Intra-day and inter-market volatility in foreign exchangerates. Review of Economic Studies, Vol. 58, No. 3, 565 – 585, ISSN 0034-6527Bollerslev, T. (1986). Generalized autoregressive conditional heteroscedasticity. Journal ofEconometrics, Vol. 31, 307 – 327, ISSN 0304-4076Bollerslev, T. & Ghysels, E. (1996). Periodic autoregressive conditional heteroscedasticity.Journal of Business and Economic Statistics, Vol. 14, No. 2, 139 – 151, ISSN: 0735-0015Doshi, A.; Frank, J.; Thavaneswaran, A. (2011). Seasonal volatility models. Journal of StatisticalTheory and Applications, Vol. 10, No. 1, 1–10, ISSN 1538-7887Duan, J. (1995). The GARCH option ricing model. Mathematical Finance, Vol. 5, 13–32, ISSN0960-1627Elliot, R.; Siu, T.; Chan, L. (2006). Option pricing for GARCH models with Markov switching.International Journal of Theoretical and Applied Finance, Vol. 9, No. 6, 825–841, ISSN0219-0249Engle, R. (1982). Autoregressive conditional heteroskedasticity with estimates of the varianceof U.K. inflation. Econometrica, Vol. 50, 987–1008, ISSN 0012-9682Frank, J. & Garcia, P. (2009). Time-varying risk premium: further evidence in agriculturalfutures markets. Applied Economics, Vol. 41, No. 6, 715 – 725, ISSN 0003-6846Franses, P. & Paap, R. (2000). Modelling day-of-the-week seasonality in the S&P 500 index.Applied Financial Economics, Vol. 10, No. 5, 483 – 488, ISSN 0960-3107Ghahramani, M. & Thavaneswaran, A. (2007). Identification of ARMA models with GARCHerrors. The Mathematical Scientist, Vol. 32, No. 1, 60 – 69, ISSN 0312-3685Ghysels, E. & Osborn, D. (2001). The econometric analysis of seasonal time series, CambridgeUniversity Press, ISBN 0-521-56588-X, United Kingdom.Gong, H. & Thavaneswaran, A. (2009). Recursive estimation for continuous time stochasticvolatility models. Applied Mathematics Letters, Vol. 22, No. 11, 1770 – 1774, ISSN0893-9659Gong, H.; Thavaneswaran, A.; Singh, J. (2010). A Black-Scholes model with GARCH volatility.The Mathematical Scientist, Vol. 35, No. 1, 37–42, ISSN 0312-3685Heston, S. & Nandi, S. (2000). A closed-form GARCH option valuation model. The Review ofFinancial Studies, Vol. 13, No. 3, 585 – 625, ISSN 0893-9454Nicholls, D. & Quinn, B. (1982). Random coefficient autoregressive models: An introduction.Lecture Notes in Statistics, Vol. 11, Springer, ISSN 0930-0325, New York.Paseka, A.; Appadoo, S.; Thavaneswaran, A. (2010). Random coefficient autoregressive (RCA)models with nonlinear stochastic volatility innovations. Journal of Applied StatisticalScience, Vol. 17, No. 3, 331–349, ISSN 1067-5817Taylor, S. (2005). Asset price dynamics, volatility, and prediction, Princeton University Press, ISBN0-691-11537-0, New Jersey.Thavaneswaran, A. & Ghahramani, M. (2008). Volatility forecasts with GARCH errors andapplications. Journal of Statistical Theory and Applications, Vol. 1, No. 1, 69 – 80, ISSN1538-7887Thavaneswaran, A.; Appadoo, S.; Ghahramani, M. (2009). RCA models with GARCHinnovations. Applied Mathematics Letters, Vol. 22, 110–114, ISSN 0893-9659Thavaneswaran, A.; Appadoo, S.; Peiris, S. (2005a). Forecasting volatility. Statistics andProbability Letters, Vol. 75, No. 1, 1–10, ISSN: 0167-7152Thavaneswaran, A.; Appadoo, S.; Samanta, M. (2005b). Random coefficient GARCH Models.Mathematical and Computer Modelling, Vol. 41, No. 1–7, 723–733, ISSN 0895-7177
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