Crude Oil and Natty Gas: Implied Volatility and Futures

Continuing in our series on exploring fundamental relationships in futures and options via regression analysis (see https://commodityvol.com/aux/doc/WEB/WEBSCROLL/WEB/VIXSP/ for our VIX exposition). We turn our attention to the energy markets. A well known stylized fact in the energy markets is the existence of the “Heat Equation.” Roughly speaking the theory posits that energy trading is driven by the ratio of energies present between two commodities. Referring to https://www.investopedia.com/terms/n/natural-gas-equivalent.asp, we see that there is 6000 times more energy in a barrel of oil than in 1000 cubic feet of natural gas. In a perfect world, abstracting from processing costs, transportation cost, alternative (non-energy) uses, we would expect that changes in one product's price would correspond to changes in the other subject to a division or multiplication by that constant. This is also called the “Natural Gas Equivalent”. The argument here is related, though slightly different, to the same type of ratio which exists between different precious metals.

On the surface things look almost good for our model. On average, since approximately 2016, there is a positive relationship between the price of oil and the price of natural gas. The first suggestion that all is not well occurs when one observes that in the most recent period, the relationship is negative. Then there is the value of 0.0071 estimated as the regression slope coefficient between natty gas and oil. If we take 1/6000 we might expect 0.00166 should be the number. Moreover, the Rsq of the overall regression as well as the rolling 20 regressions are all perilously close to 0. In levels, it doesn’t seem like the heat equation has survived.

In time series analysis there are all sorts of problems that might arise if one or both series have unstable dynamics (“see unit root” problems). One of the easiest ways to deal with this feature of the data is to transform the data. An easy to understand and relatively trivial transformation is the log difference, looking at the series in terms of percent changes. We see the following.

There is no positive relationship between percent changes in the price of oil and natural gas. In fact, we are on the cusp of a statistically significant and negative relationship between the two sets of front month futures.

Perhaps the heat equation doesn’t work, but there is some systematic relationship between changes in the 50 Delta implied volatility of the future and the % change in prices. Let’s start with Natty vol against Natty futures (all front month).

The only durable effects seem to be a 1 day reversal and a 10 day follow through on rising prices. The effect is not terribly durable. What about effects from including crude futures?

The one day changes in Natty gas prices seem to have the strongest effect overall with the one day percent change in crude futures coming second. In the most recent 20 day period, we see much more complex 5 day reversals as well as longer term effects from the crude market. Overall, here as well, the results are very weak. Very little of the variance is being explained.

What about relationships in vol? Is there something to understand in the lagged adjustment process in this market? Here there is something truly interesting and systematic. We do find weak evidence that there is a connection between natural gas futures and crude.

On the overall results, the reversal pattern still exhibits itself with respect to lagged vol. However, there is a statistically significant effect that longer lags of crude vol exerts on natural gas vol. Moreover, the R squared, though not screaming, hits 10%. If we look at the most recent result, we see that the reversal pattern is still there (though not statistically significant), but the effect of crude vol on natural gas vol increases.

What have we discovered in this note? First the heat equation in levels or returns does not seem to be the manner in which futures trade. We try to assess predictability of the front month natural gas futures on its lagged returns, we find weak results. There is definitely a reversion effect at work, but overall the Rsq is very low. Finally we concluded with a regression analysis of changes in the front month 50 delta vol of natty gas versus its lagged realization and the lagged changes in vol for the crude contract. Our Rsq jumps to about 10% and we see, in addition to reversion, a sizable positive connection between implied vol in crude and that in natural gas.

We provide these as a reference for what's happened over the past year.

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