How Periodic Are the Oil Price Fluctuations?
Posted by Khebab on September 22, 2006 - 11:20am
Topic: Economics/Finance
Tags: oil prices, periodicity transform [list all tags]
The amplitude of the big slide in oil prices from $75 to $61 was a little bit a surprise for everybody. I'm trying to answer the following question: is this big drop significant or simply a consequence of a very volatile market?
There are some cycles in oil price fluctuations. For instance, the seasonal fluctuations in oil demand or even the change of oil contract at the end of each month. On top of that, there is the usual chaos of geopolitical events, Hurricanes, etc.. The objective is to see if we can apply the Periodicity Transform in order to capture eventual cycles and get an idea of future oil market volatility.
I consider the prices from 2002 to end of July 2006 (the data is from the EIA). We fit a straight line in the log domain (Fig. 1). The fit is quite nice with a correlation coefficient equals to 0.97, the slope is 0.2615/year which represents a 30% per year increase in prices.
Fig 1.- Linear fit in the log domain. Click to enlarge.
The residuals are Gaussian distributed with a standard deviation equals to 0.095:
Fig 2. - Residuals of the fit shown on Fig. 1 and corresponding pdf (the red line is the Gaussian model). Click to enlarge.
If we apply an exponential transform to go back to the normal price domain, we get a nice exponential trend (Fig. 3). The dotted lines are the 95% confidence interval derived from the above gaussian model for the residuals. Note that even if the residuals can be modeled as a gaussian additive noise, once the exponential transform is applied the noise becomes multiplicative and non gaussian (that's why the confidence interval becomes wider with time).
Fig 3. - Exponential trend and 95% confidence interval. The red points are the recent drop from Aug 1 to Sept 19. Click to enlarge.
The Periodicity Transform is quite a new tool proposed by William A. Sethares and Tom Staley in 1999:
Periodicity Transforms decompose a data sequence into a sum of simple periodic sequences by projecting onto a set of periodic subspaces, leaving residuals whose periodicities have been removed. As the name suggests, this decomposition is accomplished directly in terms of periodic sequences and not in terms of frequency or scale, as do the Fourier and Wavelet Transforms. In consequence, the representation is linear-in-period, rather than linear-in-frequency or linear-in-scale. Unlike most transforms, the set of basis vectors is not specified a priori, rather, the Periodicity Transform finds its own "best" set of basis elements. Technically, the collection of all periodic subspaces forms a frame, a more-than-complete spanning set. The Periodicity Transforms specify ways of sensibly handling the redundancy by exploiting some of the general properties of the periodic subspaces.
Applied on the residuals, we get 29 basis elements. I show the four most important periodic basis elements in terms of energy contribution:
Fig 4. The 4 top basis elements out of 29. Click to enlarge.
We can reuse the periodicity basis in order to estimate future fluctuations of the residuals:
Fig 5.- Modelisation of the residuals by the Periodcity Transform. Click to enlarge.
Back in the price domain, we get the following prediction:
Fig 6.- Periodicity Transform Extrapolation. Click to enlarge.
Fig 7.- Zoom in on Fig. 6.
A few comments:
- The recent drop in oil prices is still within the confidence interval of the exponential trend observed since 2002.
- There is about a 2% probability that prices could drop below $61. It seems that $60 is the actual lower support line for the oil prices right now.
- The periodicity transform gives us a rough idea of what future volatility could look like.
- The PT basis elements are hard to interpret in terms of known price cycles.
- The predictive power of the Periodicity Transform is probably limited because of the chaotic nature of price fluctuations. The PT seems to have predicted the recent big drop in prices and is predicting a big rise at the end of the year.
- Of course, this is a very simple approach that is assuming that the exponential trend observed since 2002 will stay a valid model for the coming years.
The twelve $10 moves are marked in red. These all occurred within 2 month periods. I've also counted 10 2-month periods where the price traded within a $10 range.
I've got one more I'll throw in here at the end of the line. But after that I'm done.
But I will tell you this. If you were to substitute Americans for those frogs you speak of. And if you substituted gasoline prices for the boiling water. You'd probably be right.
http://www.econbrowser.com/archives/2005/07/100_a_barrel_wh.html
This is a one standard deviation measure of how much prices are likely to change in a year. For shorter periods, we reduce it proportionately. For two months, 1/6 of a year, this is about 6%.
Now, that's one standard deviation. In general there is a 68% chance of staying within 1 SD, or plus or minus 6% price change. 68% of the time, prices will change by less than 6% in a 2 month period.
Two standard deviations would be twice as much, and correspond to 95%. 95% of the time, price will vary by no more than plus or minus 12%.
Three standard deviations would be plus or minus 18%, and corresponds to 99%. 99% of the time, prices will change by no more than 18% in a two month period.
In this cases, prices fell from about 78 to 61. This is a drop of 22%, or more than a three standard deviation change, in fact almost four standard deviations. It should happen something like one in ten thousand times. Either oil price volatility has increased markedly (and I don't know that we see much evidence for that prior to the recent change) or else this was an extremely rare occurance.
Well, I suppose that's no surprise, we already knew it was rare, prices haven't fallen this much this fast in many years. This just shows how rare it is.
Now, one caveat - these statistics are based on a normal distribution. However financial prices are slightly "leptokurtic", a fancy word that just means "fat tailed". (If you like to live dangerously, try telling your girlfriend she's leptokurtic and say it's latin for beautiful.) What it means is that extreme moves are more likely than would be the case for a classical normal curve. Market psychology is more likely to lead to extremes than the kinds of physical processes that are often described by normal curves.
The bottom line is that this 22% drop in two months is in fact rare, but maybe not quite as rare as the calculations above would suggest. Four SD drops do happen in the market but they are certainly noteworthy.
And let me point out that Khebab's prediction graph shows us climbing back up to around 85 in November. An increase from 61 to 85 in two months would be 6.5 standard deviations! That would be even that much more remarkable than the drop we've already seen.
If I look at the lognormal distribution properties:
the variable mu is a linear dependent function of time in my case (log(Price)= 0.2615*(t-2002)+3.1). We have the following relationship for the mean and the variance of lognormal distributed variable X:
so the standard deviation is proportional to the mean. The coefficient in my case would be sqrt(exp(0.091^2)-1)= 0.0912= 9.1% wich is not far from the 6% you are using. So volatility would be given by the following function:
0.0912 * exp(0.2615*(t-2002)+3.1 + 0.091^2/2)
using this formula for mid-September 2006, we get a volatility between $6.89 and $7.03. The 95% confidence interval would be [$61.2, $89.2]. The recent drop of $22 is 22/75.2= 22.6% which is 22.6/9.3= 2.43 SD. The move toward $85 would be 3.44 SD with my value.
Can you point me towards some references on this issue? I of course see the same thing in the MSM that you do, but have not seen the evidence that contradicts that story.
Thanks.
Anyone have any current figures for forward coverage in oil and/or gas?
There are more graphs here.
Matt Simmon's link (http://www.simmonsco-intl.com/files/060997.pdf ) is old -- from 1997 -- but he's the only guy I can find who addresses this important topic -- what are the "minimum" level of oil inventories? Does anyone have any newer information than this? I suspect that as our usage has grown our "minimum" level of oil inventories for proper function of our petroleum industry is higher than 280M to 300M barrels, but I am not certain.
It is easy to get this idea of a contradiction -- the latest EIA weekly oil report released 9/20/06 begins with the title "How Low Can it Go?" and relies on both "technical" chart data (it is stated that the decline represents "the second-largest uninterrupted decline in the history of the survey (dating back to August 1990") and then also uses inventories as a reason why oil prices are dropping. A chart is very conspicuous that shows higher than average crude oil inventories (of course no mention of world oil inventories and the fact that we've withdrawn from the SPR). No other reasons are given for the price decline. see: http://tonto.eia.doe.gov/oog/info/twip/twip
This bigger picture being a shift in the way the world community thinks of Oil. Here in Australia, petrol consumption has decreased by 5% over the past year, even though domestic GDP growth remains strong (we're benefiting from the commodity boom due to the rapid expansion of demand from China and India).
In market-speak one could say the fundamentals have changed once Oil passed through $70 per Barrel and that consumers around the world have decided to act differently.
So to me the question is: Will $70 per barrel be sufficient to continue this drive away from consuming Oil, or will further price increases (like the 30% per year trend line) be required to balance supply and demand?
However, gasoline consumption is still up 2% from last year in the US despite higher prices:
Not much sign of the population falling here. Although it probably is aging - compensated for by migration from East Europe.
Whatever the reasons, I am an optimist in the aspect of that policy makers around the world are starting to embrace the fact that the oil age is approaching to an end. I expect it to become a race very, very soon. The ones that manage to implement alternatives will prosper in the long term, the others will lag behind.
Additionally, dollar amounts should be adjusted for inflation. Not sure you did the latter.
Nonetheless, very interesting work.
You can't read history backwards.
The drop from $78 didn't happen when the conflict ended, nor did the run-up begin when the conflict started. That's being projected on the price graph. Im not going to say that it was coincidence, or that it was unrelated, but I will say that there is no provable correlation. And I said that from Day 1. Nobody ever thought the Levant had anything to do with oil.
Oil is not seasonal.
I too believe that $60/barrel is the floor, the "bounce" price -- right at the low end of your 95% confidence interval.
OPEC issued a public warning recently that prices below $60 would lead to production cuts. In light of your modelling, I would say that somebody else has done a similar analysis. The fundamentals justify the rising 95% interval now and for some time to come. OPEC and The Oil Drum -- finally on the same page!
-- Dave
BTW, can I email you? I've got some topics I don't have the time for. You might. And you're one of the few I trust.