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For a Long/Short Strategy, I have two stocks with different volatilities. How can I calculate the right position size for each leg?

*The pair trading is not coming from co-integration but more as a hedge (protection)

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up vote 2 down vote accepted

I would concur with Chrisaycock if the volatility between two assets is constant, but because it never is I find there are way better approaches. Generally correlations fly all over the map once things start to become interesting in the market, even for generally highly correlating assets. I would look to generate an algorithm that can determine a range of correlation coefficients within a given standard error at a minimum of xx% of all data points. For example, if your two assets do not correlate within a [0.7..0.8] correlation coefficient range at least 70% of the times (70% of all observations) then you probably deal with a very bad hedging instrument anyway. So, again, as I recommend just pretty much all the time: Before running analyses and tests, think intuitively what you try to achieve. You want to look for a very stable hedge instrument. Obviously all of what I said applies equally to intraday as well as longer frequency data points. No beta in the world will produce the desired/anticipated results if it is not stable, no matter how sophisticated you design your research approach.

Secondly, I would then look to cover the remaining data points and research why correlations broke down during those periods. If you can derive a mathematical relationship you could possibly formulate an approach in which you make adjustments to the hedge ratio through an adjustment in your beta/correlation coefficient.

Thirdly, you may want to "re-calibrate" at an optimal frequency in order to updated regime shifts in volatility/correlations. I could write pages about this approach because I have spent a lot of work on optimizing this approach as part of a bigger strategy.

Those are by far not exhaustive nor did I elaborate but I hope it serves as food for thought to actually first make you acknowledge that any assumption that even hints at constant relationships between assets in most markets is utterly flawed.

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Thanks Freddy, definitely food for thoughts. – Freewind Oct 20 '12 at 20:43

Here is a link to a paper with concrete details of calculating the hedge ratio for your position: http://quanttrader.info/public/betterHedgeRatios.pdf.

Certainly, you want to check that it is reasonable to hedge one position with the other, as Freddy warns. But assuming it is, the paper suggests that using total least squares is better than using ordinary least squares (OLS). It contains R code to implement the calculation.

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Assuming you have the beta of the hedging asset to the original asset, then

\begin{equation} \mbox{amount}_{hedge} = - \beta \times \mbox{amount}_{original} \end{equation}

As long as the relative volatility between the two assets is constant, then the beta should do the trick.

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