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Actually, co-skewness is represented by a rank 3 tensor, rather than a matrix. I'm going to reproduce the formulation from Bhandari and Das, Options on portfolios with higher-order moments, but I'll add and omit some details. The co-skewness tensor is $$S_{ijk} = E \left[ r_i \times r_j \times r_k \right] = \frac{1}{T} \sum_{t=1}^T r_i(t) \times r_j(t) ... 7 Yes. Check out the Lower Partial Moments literature. In my view the best introduction to this is Narwrocki - A Brief History of Downside Risk Measures. Uryasev established equivalence between CVaR approach and low partial moments. If Markowitz had the tools at the time time, LPM utility functions would be the introductory optimization model as opposed to ... 5 There are many portfolio optimization paradigms that include a preference for skewness. These are generally alternatives meant to replace the modern portfolio management mean-variance framework developed by Markowitz. Skewness (or, more generally, higher moments) are only relevant in portfolio optimization if (a) assets are not normally distributed, and ... 4 That can be a somewhat difficult question to answer, given that the context may yield different distributions. Nevertheless, I think that you could try to fit the best distribution algorithmically. For instance, lately I found this package at Matlab file exchange: Finding the best distribution that fits the data Link (...) This is where Mike's ... 4 In my opinion you have two choices: You calculate annual returns from the daily returns that you have - I guess it is clear how. Subsequently you calculate your statistics on these 11 data points. When I look at your comment above, this could be what you want to achieve. Then you have the ex-post statistics on your data. The drawback is that 11 data ... 4 "Skewness" quantifies how asymetric a distribution is about the mean. "Kurtosis" quantifies how peaked or flat the distribution is. Skewness is defined as: E[ (X - mean)^3 ] = \frac{(\sum (x_i - x_{mean})^3 )}{N} and Kurtosis as: E[ (X - mean)^4 ] = \frac{(\sum (x_i - x_{mean})^4 )}{N} where X is your distro values (x_1, x_2, ... x_N), mean is the ... 4 There's a huge literature on this topic going back at least 30 years, and I am unfortunately not familiar enough with this literature to give you a great answer to your specific question. However, I will in this answer at least try to point you in some useful directions according to what I've found thus far. Kurtosis, by the way, seems like it is not ... 4 the question is very broad, Here is the brief summary of the role of all moments in portfolio optimization: expected value- the 1st moment represents the reward. All the even higher moments represent the likelihood of extreme values. Larger values for these moments indicate greater uncertainty. The odd moments represent measures of asymmetry. Skewness ... 3 Assuming you're talking about optimizing a portfolio that has options included in its investment universe. Skewness isn't directly modeled in the optimization, although many formulations involve using implied vol as the currency numeraire. (i.e. modeling the components of skewness, instead of skewness itself) The main impact on the optimization though is ... 2 Have a look at PortfolioAnalytics in R. > library(PerformanceAnalytics) > data(managers) > CoSkewness(managers, managers) 2 Assuming you have return time series$$ r_1(1), r_1(2), \ldots, r_1(T) \qquad \text{and} \qquad r_2(1), r_2(2), \ldots, r_2(T) $$for the 2 assets and asset weights w_1 and w_2, we can follow the calculation of the N-asset portfolio skewness laid out in another answer for a similar question. To extend it to include portfolio kurtosis, we need the ... 1 What is the data basis that you start from? If you just have the covariance matrix, then you can only calculate portfolio variance or volatility by$$ w^T \Sigma w where $w$ are the portfolio weights and $\Sigma$ is the covariance matrix. If you have the individual asset continuously compounded returns $r^j_t$ where $j$ indexes assets, $j=1,\ldots,N$, and ...

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I think the best answer is to test different distributions with your specific data set and see which fits the data the best... skewness and kurtosis are just a small piece of information , there is still a good deal of information we don't know and won't have with out having the data set in front of us

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Selling 2 ATM calls against 100 underlying shares result in Delta neutral. "Given the extra downside protection, and potential need for a stop order if the asset price rises too high, is the added risk of the naked leg justified?" The risk is if the move is more than extrinsic premium collected. One thing to watch out this type of trade is skewness. ...

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