Question 1: (a) Estimate the simple GARCH(1,1) model on S&P 500 daily log returns using the maximum likelihood estimation technique. First estimate σ 2 t = ω + αR2 t + βσ2 t−1 , with Rt = σt−1zt , and zt ∼ NID(0, 1). (In the estimation process, set σ 2 0 = V ar(Rt) and use the sample variance as estimate of this initial value. Use α = 0.1, β = 0.85 and ω = 0.000005 as starting values for the optimization process.) (b)

Question 1: (a) Estimate the simple GARCH(1,1) model on S&P 500 daily log returns using the maximum likelihood estimation technique. First estimate σ 2 t = ω + αR2 t + βσ2 t−1 , with Rt = σt−1zt , and zt ∼ NID(0, 1). (In the estimation process, set σ 2 0 = V ar(Rt) and use the sample variance as estimate of this initial value. Use α = 0.1, β = 0.85 and ω = 0.000005 as starting values for the optimization process.) (b)

Include a leverage effect in the variance equation. Estimate: σ 2 t = ω + α(Rt − θσt−1) 2 + βσ2 t−1 , with Rt = σt−1zt , and zt ∼ NID(0, 1). (Use starting point: α = 0.1, β = 0.85, ω = 0.000005, and θ = 0.5.) What is the sign of the leverage parameter? Explain how is captured in this model by deriving Covt(σ 2 t+1, Rt+1).

(c) Include the option implied volatility VIX series from the Chicago Board Options Exchange as an explanatory variable in the GARCH equation. Use MLE to estimate : σ 2 t = ω + α(Rt − θσt−1) 2 + βσ2 t−1 + γVIX2 t /252, with Rt = σt−1zt , and zt ∼ NID(0, 1). (Set starting values to α = 0.04, β = 0.5, ω = 0.000005, θ = 2, and γ = 0.07

(d) Use the Mincer-Zarnowitz regression to assess each of the three volatility models. These consists in testing H0; β0 = 0, β1 = 1 in the regression: R 2 t = β0 + β1σˆ 2 t−1 + ut . (Use the heteroskedasticity and autocorrelation robust estimator of the asymptotic variance in the testing procedure.) Which model is the best? Explain.