to the futures contract. It is therefore of the essence to choose a futures contract that is highly correlated to the risky commodity. Given their findings of a strong positive long-term correlation between the two, they assume that the prices are cointegrated.
This is also supported by empirical tests. Instead of imposing a cointegrated vector, they assume that the spread between the log prices is stationary. The long-term relationship is therefore an essential part of their analysis, which allows for inclusion of the basis-risk which varies in a stationary way over time. This is modeled as an OU-process. While their futures price is modeled as a GBM, we chose another path.
The second, and mainly contributing, article we focus on is one by Bertus, God-bey, and Hilliard (2009). The main motivation of this article is also how to best strategize a cross hedge. They derive minimum variance hedges by assuming that the commodity price moves according to the two-factor model (Gibson and Schwartz, 1990), and that the log spread moves as a mean reverting OU-process. It is therefore a jointly estimated 3-factor model. They specifically focus on the airline industry, and cross hedge jet fuel risk with crude oil futures given their high positive correla-tion.
Our concrete problem is a large energy corporation that wants to hedge away risk in German gasoil. There is no liquid German futures market for gasoil, hence the need to hedge with another commodity. For this purpose, we have chosen two potential hedges: ARA gasoil and Brent crude oil futures. We will for both of these look at different hedging strategies, and finally compare their effectiveness.
Figure 5.1: Spot Prices in EUR/mt
kg/litre in vacuum at 15 degrees Celsius1.
All the prices are quoted in euros per metric tonnes2, after the conversion we did.
First, we downloaded all the price data in Euros, and not in U.S. Dollars which is the standard currency. Futures prices of Brent crude oil was denominated in Euros per barrel. ARA gasoil was denominated in Euros per metric tonnes. We wanted all our data in Euros per metric tonnes, and to do that we used conversion factors from BP. 1 barrel equals 0.1364 tonnes. In addition, 1 hecto litre equals 0.0845 tonnes. To convert Brent crude from EUR/barrel to EUR/tonnes, we thus divided by 0.1364.
Since ARA gasoil was denominated in EUR/tonnes, but with a deliverable density quality of 0.0845 kg/hl, we needed to adjust the German gasoil prices to match this, and thus multiply these prices by 0.0845. As can be seen from Figure 5.1 all three prices series move quite similarly over time.
From Figure 5.1 it is not specifically clear whether the spot prices are mean reverting or not. They also do not seem to follow a GBM, as the variation in prices is limited to a certain interval. They do seem to revert to a long-term level of around 400 to 600 EUR/mt. The spot prices do not show any clear seasonal patterns. The reason for this might be that we are operating with monthly data. We are choosing to model the spot prices as mean reverting, because we want the model to fit any time period, and any frequency of prices. It is a well known phenomenon that
1Taken from Bloomberg when downloading the data
2Conversion rates can be found in Appendix A.2, Table A.1, and are taken from www.bp.com/conversionfactors.jsp
many energy prices are in fact mean reverting, and do not generally follow a GBM.
Summary statistics for the spot prices are reported in Table 5.1. As can be seen from Table 5.1, German gasoil spot prices are on average higher than that for Dated Brent and ARA gasoil, and they also have a higher volatility.
Statistic Dated Brent ARA gasoil German gasoil
N. of Obs. 95 95 95
Min. 219.434 306.472 339.035
Max. 680.858 788.902 867.675
Mean 445.520 531.056 585.356
SD 120.440 137.821 145.972
Table 5.1: Summary Statistics for Spot Prices
Figure 5.2 shows historical prices of both Brent and ARA gasoil futures. We have chosen to focus graphically and statistically on certain futures contracts for the sake of brevity and unnecessary repetition. Figure 5.2 thus shows front-month, 3-months, 6-3-months, 9-months and 12-months futures for each of the two commodities.
As becomes obvious by looking at Figure 5.2 is how similarly the different futures prices behave. Summary statistics for the different futures prices are given in Table 5.2.
Statistic Brent 1M Brent 6M Brent 12M GO 1M GO 6M GO 12M
N. of Obs. 95 95 95 95 95 95
Min. 239.546 255.525 246.421 299.935 294.726 290.702 Max. 675.728 664.977 665.862 802.301 825.972 819.141 Mean 447.997 454.984 457.061 530.455 539.177 545.662 SD 117.793 107.744 101.313 136.589 127.024 118.997
Table 5.2: Summary Statistics for Brent and ARA gasoil (GO) Futures Prices
When looking at optimal hedge ratios further on, correlation coefficients between German gasoil prices and the different futures prices will become important. We have mentioned that correlation is the most important criteria for finding a suitable cross hedge, and this will be discussed in more detail in the following chapter. Table 5.3 shows the correlation coefficients between German gasoil and the different futures
(a) Brent Futures Prices
(b) ARA gasoil Futures Prices
Figure 5.2: Brent and ARA gasoil Futures Prices in EUR/mt
prices. The correlation coefficients range from 0.935 to 0.975, indicating that the correlation is high even for the lowest values. It is generally higher between German gasoil and ARA gasoil futures than it is for German gasoil and Brent futures.
What is interesting further, is how the log spread evolves between the illiquid German gasoil and the more liquid futures prices mentioned above. Figure 5.3 shows the log spreads between some chosen futures contracts for each of the two commodities3. Not surprisingly, the spread between German gasoil and ARA gasoil futures is a bit more stable than that between German gasoil and Brent crude oil
3Graphs of all log spreads can be found in Appendix A.1, Figures A.1 and A.2
Correlation Coefficients between GGO and Futures Prices
GGO GGO GGO GGO
BR 1M 0.963 BR 7M 0.951 GO 1M 0.975 GO 7M 0.959 BR 2M 0.961 BR 8M 0.948 GO 2M 0.973 GO 8M 0.956 BR 3M 0.960 BR 9M 0.945 GO 3M 0.970 GO 9M 0.954 BR 4M 0.958 BR 10M 0.942 GO 4M 0.968 GO 10M 0.951 BR 5M 0.955 BR 11M 0.938 GO 5M 0.965 GO 11M 0.948 BR 6M 0.953 BR 12M 0.935 GO 6M 0.961 GO 12M 0.944
Table 5.3: Correlation Coefficients between German gasoil and Brent (BR) and ARA gasoil (GO) Futures Prices
futures. For both Brent and ARA gasoil, the stationarity also seems to weaken as time to maturity increases. This makes perfect sense as front-month futures contracts are often very similar in its characteristics as the spot price underlying the futures. In this particular case, since the correlation is so high between the different prices, see Table 5.3, the same logic appears to be true.
Statistic GGO and Brent 1M GGO and Brent 6M GGO and Brent 12M
PP -7.207*** -6.038*** -4.411***
(0.010) (0.010) (0.010)
ADF -3.824*** -2.815** -2.337
(0.021) (0.239) (0.437)
Statistic GGO and GO 1M GGO and GO 6M GGO and GO 12M
PP -9.998*** -8.025*** -5.475***
(0.010) (0.010) (0.010)
ADF -4.590*** -3.416** -2.600*
(<0.010) (0.057) (0.333)
Table 5.4: Stationarity Tests of the Log Spreads. P-Values are reported in parenthesis.
***/**/* means rejection of unit root at a 1 %/5%/10% significance level, respectively.
In order to test for stationarity, ADF-tests have been performed for all possible spreads. In Table 5.4, the results from the ADF-tests of some chosen spreads are reported. We have performed two different tests for stationarity, the well-known Augmented Dickey-Fuller test and the Phillips-Perron test. Both of them checks
(a) GGO and BR1M (b) GGO and BR6M (c) GGO and BR12M
(d) GGO and GO1M (e) GGO and GO6M (f ) GGO and GO12M
Figure 5.3: Log Spreads between German gasoil and Brent and ARA gasoil Futures.
These six graphs show the log spreads between the illiquid German gasoil and front-month, 6-months and 12-months Brent crude oil and ARA gasoil futures contracts
whether there exists a unit root, and the existence of a unit root is rejected if the ADF or PP-statistic is more negative than -3.5 (1%), -2.9 (5%) and -2.6 (10%), where the numbers in parenthesis show the statistical significance. When a unit root is rejected, the time series is stationary. The Phillips-Perron (PP) unit root test is different from the ADF test, primarily because of how it deals with serial correlation and heteroskedasticity in the errors. Two advantages of the PP test are that it is robust to general forms of heteroskedasticity in the error term, and that one does not have to specify a lag length for the test regression.
As can be seen from Table 5.4, German gasoil and ARA gasoil futures are more stationary in their log spreads than German gasoil and Brent are. There are also some differing evidence with respect to the two tests, but both of them do show some evidence of stationarity, at least in the closer time to maturity futures contracts.
This is also what is shown graphically in Figure 5.3.