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In practice, for heavily traded assets (above 60% quantile of average daily dollar volume), individual asset return is pretty scalable across different time frame by a factor of $\sqrt{T}$. However, for covariance among different assets, moving between different time frame is not linearly scalable (although it should be in math). This is known as "Epps ...

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This paper states that heteroskedasticity is a stylized fact in daily as well as intra-day returns: https://statistik.econ.kit.edu/download/doc_secure1/HandbookITandFinan.pdf

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A variance swap can be replicated with vanilla European options. If you take derivative with respect to variance, you need to do the same thing on both sides. That is, you need also take derivative with respect to variance on those vanilla options. However, the resulting derivative is not the vega in the usual sense, which is the derivative with respect to ...

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This is why Markowitz says that the diversification of the portfolio is always preferable. You have a lot of certain past data and some fallible speculations to evaluate the variance and expected return of a title. Inherently the best possible evaluation method, and there are several main ones, is not a foolproof inference. But, if you diversify your ...

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In literature you'll find many approaches to compute the variance. As mentioned already, the standard ideas are to use MLE, Shrinkage on the Covariance Matrix (Ledoit, Wolf), Shrinkage on the inverse of the Covariance Matrix (Kourtis,Dotsis) which makes sense as in fact the inverse of the Covariance Matrix determines the shape of the efficient frontier. ...

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