Risk Latte - Going for the “big step” correlation in assets

Going for the "big step" correlation in assets

Rahul Bhattacharya
January 16, 2007

Of late correlation amongst assets has become an important dimension of risk. Correlation impacts pricing of exotic derivatives (the basket and multi-asset derivatives) as well as impacts their risk profile. It is the "shape" risk (as opposed to volatility which is the "volume" risk) and is presently causing traders, asset managers and risk managers some trauma. Correlation can and do wreak havoc on entire portfolio of traders (though most of the time it can best be analyzed after the damage is done, something like cancer).

Long before the advent of Excel worksheets, when the computing world inside the dealing rooms and trading floors was still dominated by Lotus 1-2-3 and HP scientific calculators, and long before the profession of risk management was marginalized and commoditized by exams like FRM and PRM and housewives started talking about VaR in kitty parties, futures traders have been using the concept of "minimum variance hedging" (the FRM fanatics can look up John Hull's book or any one of the other twenty million books that talk about futures and options). It is a simple technique where a certain hedge ratio is found to create a portfolio of the original asset (or the instrument) and its hedge whereby the total variance is minimized. Two lines of elementary differential calculus or years of experience (take your pick) can get you there. Since differential calculus is trivial these days - to the newly minted FRMized risk managers - and years of experience on the trading floor mean nothing to them, we will not talk about that technique. Suffice it to say that "minimum variance hedging" has stood the test of time and has proven effective to generations of futures and options trader (well, in another era). In fact, a large number of pit and futures traders still use this technique quite extensively.

A key input of that model is the correlation between the asset (the traded instrument) and the instrument with which the asset will be hedged. Say, you are trading in Russian crude oil but have no relevant futures market to go for hedge; but you have the Brent oil futures market. You can then use the Brent oil futures as a hedge for the Russian crude oil trades. And the hedge is put in such a way that the overall portfolio comprising both the Russian crude oil contracts and the Brent crude oil futures should have the lowest possible variance (squared volatility). This means that there is an exact number of Brent crude futures that you must sell (or buy as the case may be) against the original trade in Russian crude oil which will minimize the variance of the overall portfolio. And that number - the hedge ratio - is a linear function of the correlation between the two, i.e. the Russian crude oil and the Brent crude oil futures. And this correlation number can make all the difference to your hedge (ignore volatility risk for the moment). Slight errors in estimation of the correlation and the risk can increase (or decrease significantly).

How do we calculate this correlation? Naturally, implied correlation, a favourite of the option traders, is out because not only is it difficult and cumbersome to estimate, one of the assets, the Russian crude, has no listed futures or options price. We have to take the historical correlation between the two. And most risk managers would say a EWMA (exponentially weighted moving average) type correlation would be best suited. Some others may use the correlation using simple un-weighted time series of the two assets. But if you ask a pit trader on CBOE he will most like say use "big step" correlation between the two.

What is "big step" correlation?

Take a hundred day price series of the original asset (you could choose to take a longer series as well, but that requires judgment). Divide this series into two smaller series, one for the small price movements and the other for large price movements. You will have to use some sort of a filter or a simple algorithm or heuristic to do that. Then take the price series of the second asset (or instrument) which you want to use as your hedge and subdivide that into two as well - the series of small price movements and the series of large price movements - by ensuring that all the movements between the two series on the given dates are matched. Then calculate the corresponding correlations between the two series of small price movements of the two assets and the two series of large price movement of the two assets.

And then simply ignore the correlation between the small price movements and take correlation between the two series of large price movement of the assets. This is the "big step" correlation. This correlation then goes into the calculation of the hedge ratio for putting in the hedge.

Rationale for "big step" correlation

Between most assets the correlation between two series of small price movements is quite low. Sometimes these small co-movements are caused by pure noise and most of the time they are produced by interaction of many micro-factors which have an effect of canceling one other out. However, large price movements are caused by large external events in the economy (political, economic, catastrophic, etc.) which affect both the series either in the same way or in the different way with different magnitude. Hence the shape of the correlation plot - remember correlation is a "shape risk" - which the radius of the ellipse (of the correlation scatter plot) is determined primarily by the large price movements in both the series.

Risk management is all about managing the "global" (large) movements in asset prices.

Note: Richard Flavell's Swaps & Derivatives talks about big step correlation and minimum variance hedging as well and is a good piece of work

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