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When you model log-returns $(Y_t)$ by $Y_t=\varepsilon_t$ where $\varepsilon_t|\mathcal{F}_{t-1}\sim N(0,\sigma^2_t)$ and a standard GARCH($p,q$) model with $$\sigma_t^2=\omega+\sum_{i=1}^p \alpha_{i}\varepsilon^2_{t-i}+\sum_{i=1}^q \beta_i \sigma^2_{t-i},$$ where $\omega>0, \alpha_i,\beta_i\geq0$. This model assumes indeed a constant mean of zero for the ...

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Your function returning (minus) the log-likelihood seems weird to me, I would go with function y = findGARCH_LLy(params,S,rf) % Finds log-likelihood for the GARCH option pricing model. alpha0 = params(1); alpha1 = params(2); beta1 = params(3); lambda = params(4); N = length(S); % Define the returns (pad first return with zero) r = [0, diff(log(S))]; % ...

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It is a good idea indeed to use GARCH for intraday volatility because it is as clustered as daily volatility. Moreover, if you want to account for autocorrelations, you should consider using other variables like the bid-ask spread, the traded volume and the volume of the book at first limits. It is done in Endogeneous Dynamics of Intraday Liquidity by ...

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