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The formula works for total variance, not "strike specific" variance that you need to construct basket vol surface from components, because single historical correlation (or correlation matrix) just does not provide enough information to uniquely reconstruct expected distribution of basket returns (unless for a trivial case where all components are gaussian, ...

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In their paper on their S&P 500 Implied Correlation Index the CBOE has defined a measure for the market-capitalization weighted average correlation of the S&P 500 index which could be applied to portfolios in general. The equation $$\rho_{av} = \frac{\sigma^2 - \sum_{i=1}^N w_i^2\sigma_i^2}{2 \sum_{i=1}^N \sum_{j>i}^N w_i w_j \sigma_i ... 2 I think you might be looking for the portfolio return variance:$$\sigma_p^2 = \sum_i \sum_j w_i w_j \sigma_i \sigma_j \rho_{ij},$$where \rho_{ij} is the Pearson product-moment correlation coefficient between the returns on assets i and j and \rho_{ij} = 1 for i=j. In your case you could either weigh the assets equally or according to the real ... 1 This sounds like quadratic hedging. If you have the return of the assets r_X and r_Y with negative correlation \rho between the two (we could think of bonds and stocks) and more variance in one of them then the problem of weighting the two by w is (assume zero expected returns for ease of presentation)$$ \text{risk} = E[(w r_X + (1-w) r_Y)^2] ...

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One way to do this is to weight the assets in the portfolio to make it beta neutral against some benchmark. This would minimize risk in terms of the benchmark. $W_i = \left | \frac { \beta_i } { \sum_{}{} \left | \beta \right | } \right | = \left | \frac{ Cov(R_i, R_m) / Var(R_m) } { \sum_{i}^{n} \left | Cov(R_i, R_m) / Var(R_m) \right | } \right |$

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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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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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