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Very interesting question. I am not an expert on the subject, however, I was able to find a collection of papers on the subject that should get you started. Here is a good and very informative paper that walks you through several tick by tick volatility estimators that seek to reduce the volatility imposed by market micro-structure: Efficient estimation of ...


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For any continuous distribution we can define $$VaR_{\alpha}=-F^{-1}(1-\alpha)$$ where $F^{-1}$ is the inverse of the CDF. Now suppose that you have a distribution which comes from a location-scale family with location parameter $\mu$ and scale parameter $\sigma$ then $$F^{-1}(1-\alpha)=\mu+\sigma \phi^{-1}(1-\alpha)$$ where $\phi^{-1}$ is the inverse CDF ...


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Both approaches can be useful. For stocks, sorting into quantiles is popular because it's easy to understand and explain it's a simple matter to build factor portfolios and track or backtest their performance, while the translation from expected returns to a portfolio is a bit more involved more robust than a single-stock regression, because it is less ...


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The chart you linked to offers data for the "instantaneous forward rate" which are the rates you are looking for (f(tj,tj+τk)). Regarding the construction of the zero-coupon yield curves (cited from the ECB website): "The ECB estimates zero-coupon yield curves for the euro area and derives forward and par yield curves. A zero coupon bond is a bond that ...



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