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2

There's a minor variation of the very simple CAPM model that captures precisely the behavior you describe (high volatilities and correlations during a crisis). To be specific, let's say every security $S_A, S_B, \dots$ (or yield, if you want bonds in this) has a value linked by some constant beta $\beta_A, \beta_B, \dots$ between its return $r_A, r_B, ... 0 If one defines a crisis as a major unexpected event (ie. increases volatility) and also has a broad impact (ie affects more than one company or asset class), then I think the answer to your first question should be obvious... As for why the downside is usually more demonstrative of this affect, I think there are probably a number of factors involved ... 1 HF data have a lot of auto correlation so common models to deal with this problems are ARFIMA, FIGARCH, Fractional Integrated GARCH. Engle recently propose the multiplicative components GARCH for high frequency data, which can include a mean model like and ARMA. In this post they explain how to implement it in R with the rugarch package, it takes some time ... 1 If you are interested in evaluating forecasts accuracy, you could compare Value-at-risk forecasts. It has the advantage to take into account the forecast density (via quantile). Then you can compare easily their forecast accuracy via the Kupiec test for instance. Because if you just use points forecasts as it seems you are doing your results won't take the ... 0 I suggest taking a look at optimal hedge Monte Carlo and by extension the garam model from http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1428555. The basic issue is that a risk premium exists in option markets which is like Vol also unobserved. The better your guess of the future realized vol distribution, the better your guess of the risk premium ... 0 On the historical vs implied calibration: First, we model prices have to be arbitrate free. If you estimate the parameters based on history they do not necessarily have to agree with the current market prices. But then that's arbitrage. Or is it your historic estimation wrong? Instead of pondering whichever it is you can just fit your model to the market ... 3 It is not a "basic question", If I am correct : First you estimate your model on the return series and obtains parameters. You must estimate your model in such a way you obtain one-step ahead errors (that I will call computed errors in what follow) and associated time-serie of the predictive errors distributions parameters:$\hat{\mu_{t}}$and$ ...

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It is hard to give a definite answer to that question. Let me focus on the volatility for now (the answer for correlation is even harder). Schwert (1989) tries to determine the Economic Determinants of Stock Market Volatility. He finds that only lagged ex-post volatilities have strong forecasting power. Macro volatility, Industrial production and ...

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The smoothing factor is a way to specify the memory of your estimator. This view provides a simple and natural way to tune $a$. Say you want the $k$th term in the past to weight for 1% in your estimation. It gives you $$\frac{\alpha^k}{ A} = \frac{1}{100},$$ with $A$ your normalizing factor (see @Gordon's remark). Of course you can do better than that. For ...

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If $\log{(|R_t|)}$ is your first term, I'm not sure why this is a matrix. Modulus (determinant herein) applied to a matrix $R_t$ gives a scalar. If your implementation in python produces a matrix, that's likely because modulus is treated as an element-wise abs() function for each element of a matrix. It may be easier and faster to use rugarch (univariate ...

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In the proprietary/branded Risk Metrics implementation of EWMA, I believe a smoothing factor of .97 is used. Here is a paper that discusses different smoothing factors for EWMA http://www.tandfonline.com/doi/abs/10.1080/00036846.2014.982853?journalCode=raec20

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I actually discuss this question at length in chapter 1 of More Mathematical Finance. The essential point is that if you can write $$X=YZ$$ with $Y,Z$ independent $E(Z)=1$ and $Z>0$ then $X$ is more uncertain than $Y.$ It then follows from Jensen's inequality that the price of an option on $X$ that has a convex pay-off will be at least as high as the ...

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