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Recent Developments in SeasonalVolatility Models 11Recent Developments in Seasonal Volatility ModelsThe Ψ-weights are Ψ j = φ j , j ≥ 1. Therefore, ∑ ∞ j=0 Ψ2 j= 1 + φ 2 + φ 4 + ...= 1 . Then, the1 − φ2 kurtosis of y t is:K (y) = 6(σ2 b + β2 )[1 − (β 2 + σb 2)][1 − (β 2 + σ 2 {b1 − (3σb 4 + β4 + 6β 2 σb 2) + 3)]2σ 2 }1 − (3σb 4 + β4 + 6β 2 σb 2) exp v1 − φ 2 .Example 5.2. RCA with pure seasonal autoregressive [AR(1) s ]SVprocessy t =(β + b t )y t−1 + ɛ tɛ t = Z t e 1 2 h t(1 − ΦB s )h t = ω + v tThe Ψ-weights are Ψ j = Φ j , j ≥ 1. Therefore, ∑ ∞ j=0 Ψ2 j= 1 + Φ 2 + Φ 4 1+ ... =1 − Φ 2 . Then,the kurtosis of y t is:K (y) = 6(σ2 b + β2 )[1 − (β 2 + σb 2)][1 − (β 2 + σ 2 {1 − (3σb 4 + β4 + 6β 2 σb 2) + 3b )]2σ 2 }1 − (3σb 4 + β4 + 6β 2 σb 2) exp v1 − Φ 2 .Example 5.3. RCA with multiplicative seasonal autoregressive [AR(1)x(1) s ]SVprocessy t =(β + b t )y t−1 + ɛ tɛ t = Z t e 1 2 h t(1 − φB)(1 − ΦB s )h t = ω + v tThe Ψ-weights are Ψ 1 = φ + Φ, andΨ j =(φ + Φ)Ψ j−1 + φΦΨ j−2 , j ≥ 2. Then, the kurtosisof y t is:K (y) = 6(σ2 b + β2 )[1 − (β 2 + σb 2)][1 − (β 2 + σb 21 − (3σb 4 + β4 + 6β 2 σb 2) + 3)]21 − (3σb 4 + β4 + 6β 2 σb 2) eσ2 h twhere σh 2 (1 + φ s )σv2t=(1 − φ 2 )(1 − Φ 2 )(1 − Φφ s ) .Recently, Gong & Thavaneswaran (2009) discussed the filtering of SV models. The predictionof discrete SV models can be obtained by using the recursive method proposed inGong & Thavaneswaran (2009).6. Option pricing with seasonal volatilityOption pricing based on the Black-Scholes model is widely used in the financial community.The Black-Scholes formula is used for the pricing of European-style options. The modelhas traditionally assumed that the volatility of returns is constant. However, severalstudies have shown that asset returns exhibit variances that change over time. Duan (1995)proposes an option pricing model for an asset with returns following a GARCH process.Badescu & Kulpeger (2008); Elliot et al. (2006); Heston & Nandi (2000) and others derivedclosed form option pricing formulas for different models which are assumed to follow aGARCH volatility process. Most recently, Gong et al. (2010) derive an expression for the callprice as an expectation with respect to random GARCH volatility. The model is then evaluated41
42 Advances in Econometrics - Theory and Applications12 Will-be-set-by-IN-TECHin terms of the moments of the volatility process. Their results indicate that the suggestedmodel outperforms the classic Black-Scholes formula. Here we extend Gong et al. (2010) andpropose an option pricing model with seasonal GARCH volatility as follows:dS t = rS t dt + σ t S t dW t (27)( ) [ ( )]StSty t = log − E log = σ t Z tS t−1 S t−1(28)θ(B)Θ(L)σ 2 t = ω + α(B)y 2 t (29)where S t is the price of the stock, r is the risk-free interest rate, {W t } is a standard Brownianmotion, σ t is the time-varying seasonal volatility process, {Z t } is a sequence of i.i.d. randomvariables with zero mean and unit variance and α(B), θ(B) and Θ(L) have been defined in(4).The price of a call option can be calculated using the option pricing formula given inGong et al. (2010). The call price is derived as a first conditional moment of a truncatedlognormal distribution under the martingale measure, and it is based on estimates of themoments of the GARCH process. The call price based on the Black-Scholes model withseasonal GARCH volatility is given by:C(S, T) =S(f [E(σ 2 t )] + 1 2 f ′′ [E(σ 2 t )]( 13 κ(y) − 1))E 2 (σt 2 )− Ke(g[E(σ −rT t 2 )] + 1 ( ) )2 g′′ [E(σt 2 1)]3 κ(y) − 1 E 2 (σt 2 ) , (30)where f and g are twice differentiable functions, S is the initial value of S t , K is the strike price,T is the expiry date, σ t is a stationary process with finite fourth moment, and κ (y) = E(y4 t ) .[E(y 2 t )]2Also, f [E(σt 2)], g[E(σ2 t )], f ′′ [E(σt 2)],andg′′ [E(σt 2 )] are given by:⎛⎞f [E(σt 2 )] = N(d) =N ⎝ log(S/K)+rT + 1 2 E(σ2 t√) ⎠ ,E(σt 2)[ ⎛f ′′ [E(σt 2)] = √ 1 − 2π()g[E(σt 2 )] = N d −√E(σ t 2) ⎝ E(σ2 t ) − 2(log(S/K)+rT)4E(σ 2 t ) √E(σ 2 t )⎛⎞= N ⎝ log(S/K)+rT − 1 2 E(σ2 t√) ⎠ ,E(σt 2)⎞⎠( )[E(σ 2t )] 2 − 4(log(S/K)+rT) 28[E(σt 2)]2 ⎛⎞+ ⎝ 6(log(S/K)+rT) − E(σ2 t )] {}√ ⎠ × exp − (2(log(S/K)+rT)+E(σ2 t ))28[E(σt 2)]2 E(σt 2)8E(σt 2) ,
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