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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 ...

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I am not sure if I understood your question correctly but I will try to answer it anyway. If you have a standard normal random vector $z \sim N(\mathbb{0},I_n)$ (where $z,0 \in \mathbb{R}^{n\times1}$ and $I_n \in \mathbb{R}^{n\times n}$ is the identity matrix) and you want to transform it into a multivariate normal $x \sim N(\mu,\Sigma)$ you do it the ...

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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: ...

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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.

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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 ...

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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 ...

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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.

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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.

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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 ...

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With this solution you have to split your covariance matrix somewhat, but it should give you a vector with betas based on you conditional covariances. Example with two indexes, $x1$ and $x2$, and one asset $y$. $$[\sigma_{y,x1}, \sigma_{y, x2}]\begin{bmatrix} \sigma_{x1}^2 & \sigma_{x1,x2} \\ \sigma_{x1,x2} & \sigma_{x2}^2 \end{bmatrix}^{-1}$$

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