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Yes, that's what we wish to see from the correctly-specified model. Now, let me try to answer your 2nd and 3rd questions together as they are based on the same confusion. There are two different concepts: model-implied volatility and model-implied BSIV (Black-Scholes Implied Volatility). I think you are confused because of mixing them up. So yes, people ...

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As indicated by @AlexC and @amdopt, the formula is exact for log returns and approximate for discrete returns. Define the factor by which a price changes as $k$ so that price tomorrow $P_{t+1}$ is the price today times $k$ : $P_{t}*k$.Then the change in the price over a business year is $$\prod_{i \in [1, 252]}{k}$$ The log of the change is by properties of ...

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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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In econometrics, if you have access to high-frequency (HF) data, then the realised variance approach works better than simply computing the standard deviation. The reason is that you use much more data and thus can utilise the additional information HF data carries, thus RV typically performs better than, say, GARCH models and a plain standard deviation. ...

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In general for stock data, classical GARCH models are designed to model daily volatilities, but not the intraday ones, because, for instance, they do not capture diurnal patterns. So, I would say that models you are estimating are not valid for the 1-minutes returns. And of course, the presence of the microstructure noise makes them even less valid. As a ...

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