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6

I personally use the simple Garch(1,1) for volatility filtering in the risk management area. In fact in most cases I don't even estimate the parameters, I stick 0.94 for mean reversion, 0.04 for the squared error and I get the constant by matching the series variance. My experience is that there is no point pretending to finetune parameters when vol is ...


3

Not sure your question is about having a process for covariance or to have multivariate GARCH. The standard viewpoint on a stochastic volatility for covariance is to use a Whishart process. See for instance Philipov, A. and M. E. Glickman (2006, July) Multivariate stochastic volatility via wishart processes. Journal of Business & Economic Statistics 24 ...


3

I think you're looking for multivariate GARCH models of which this is an overview paper. Multivariate GARCH models have one big drawback: they are pretty hard to estimate due to the number of correlations. This paper by Caporin and McAleer might be of interest in that regard.


3

The formula is $$ \mu = \lambda CX $$ in your notation. You find it in many places, e.g. here. The assumption is that you know $\lambda$ which is a strong assumption. Furthermore it only holds if investors are unconstrained (long/short not long only). It is intuitive as it says that given the weighting the return expectation increases with risk aversion ...


2

The clearest and most intuitive article I have seen so far is Kritzman et al., Regime Shifts: Implications for Dynamic Strategies in FAJ (May / June 2012) It not only shows how you can use HMM for financial modelling but it also goes through the actual estimation algorithm (Baum-Welch) step-by-step and even gives full Matlab-code. From the abstract: ...


2

One common "model" is to assume the correlation to be constant, such as in a CCC-MVGARCH model. If you want a review of different multivariate GARCH models, you could look at: Silvennoinen and Täräsvirta 2009, Multivariate Garch models, in Handbook of financial time series.


2

Interesting question, as All the answers (including mine) could not be generalized unfortunately. As far as I am concerned, I use a univariate EGARCH for risk modelling purposes (Filtered Historical Simulation (FHS), etc.). 1 - EGARCH, merely because GARCH models do not take into account so-called leverage effects, which is crucial to me for skewed and ...


1

You most probably don't want to estimate the covariance of prices but rather the covariance of returns. Thus for equities you can take the return of the traded price. For bonds: if the maturity is long enough (say bigger than 2 years), then you can take the returns of traded prices. The pull to par should not be too relevant here. if the maturity is short ...


1

The first objective is to minimize the variance by choosing a proper control variate. First note that an expectation value is just a constant, so the covariance between an expectation value and a random variable is zero: $$\text{Cov}\left(\mathbb{E}[Y], X\right) = 0$$ Similarly for the variance of an expectation value, $\text{Var}(\mathbb{E}[Y])=0$. The ...


1

I agree on Richard. the simpler you choose, the better it is so as to get reliable estimates. What's your data frequency? purpose? For model construction as far as I am concerned, daily data from 2010 is enough. Otherwise, you could use a proxy asset for asset D depending on its nature. To clarify, if D is an ETF let's say CAC 40 ETF, concatenate its return ...


1

You can consider old prices for Stocks B, C and D to be "missing data" and apply techniques used by Statisticians to deal with such missing data. One approach, the EM algorithm, suggests you estimate the covariance for the common period, use that covariance matrix and the available data to generate pseudo data for the back period for the third stock and ...


1

The short answer: Take all time series starting from 2010 (at most). The covarianc-matrix tells you something about the assets for a certain amount of time. E.g. if I estiamte the covaraince matrix of those 4 assets taking into account data from the last year (!) then I can expect that this matrix remains valid for the coming 1-3 months - if the markets ...


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Are your 407 stocks all different? No A and B listings contained that are strongly if not perfectly correlated? The observation that the daily covariance matrix is singular makes me wonder. You can try the package corpcor for another shrinkage estimator.


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For the terminal distributions, I don't have the closed-form solution to hand, but it's computable, since we can price power options (with payoffs like $(S_T^n-K)^+$). You need to find $$ E[S_T C_{K,T}] = \int_K^\infty x(x-K) \cdot p_{BS}(x) dx \\=-Ke^{(r-q)T} C_{K,T} + \int_K^\infty x^2 \cdot p_{BS}(x) dx $$ The latter formula is just a power-option ...


1

This is the challenge for below-mean semivariance in optimization. Since the mean becomes a moving target, the observations that impact the min function change. Estrada proposed a heuristic method for optimization and Beach(2011) discusses the decomposition and semi covariances. Below target semivariance assumes investors do not change their target ...


1

Yes. Correlations max out at 1. However if the correlation is near 1 and the volatility of the spot is significantly larger than the volatility of the future the hedge ratio will be greater than 1. The intuition is if that vol of the future is much smaller than the vol of the spot you might need a lot more futures to minimize the high spot variance.


1

Say that you did the calculations in the classic regression way. If you stick the returns of your 4 asset returns in a $(T\times 4)$ matrix $Y$, and your 3 factor returns in a $(T\times 3)$ matrix $X$, then your betas would solve the multiple regressions, collected in a $(3\times 4)$ matrix $$Y = X\cdot \beta + \epsilon$$ You could also add a column of ones ...


1

One of the more prominent proponents has a current paper on the reasons - and why this anomaly "will persist": Why the Low Volatility Anomaly Will Persistby Eric G. Falkenstein, March 2015 Abstract Common explanations of the low volatility anomaly involve biases or frictions that cause investors to overpay for high volatility assets, giving them a ...



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