The Limits of Mathematics and NP Estimation in ... - Chichilnisky

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