In this subsection, we briefly describe different approaches to model fuel prices. Thereby, we focus on two popular approaches, the GBM and the mean-reverting process. We conclude this subsection with some critical remarks about the quality of fuel price forecasts and the implications for the power generation expansion problem.
1.4.2.1 Literature Review
Early works in the field of fuel price modeling are often based on GBMs. For example, Gibson and Schwartz [GS90] describe the spot price of oil as a GBM. The choice of the GBM can be explained with the popularity of the famous work of Black and Scholes [BS73], who developed a model to price options based on the assumption that the price follows a GBM.
However, in the nineteen nineties, it was questioned whether the GBM assump- tion is pertinent. Bessembinder et al. [Bes+95] studied for several commodities whether investors anticipate mean-reversion in spot asset prices. They found mean-reversion in all examined markets. For crude oil, they reported that 44% of a typical price shock is expected to be reversed over the subsequent eight months. Schwartz [Sch97] formulates a commodity price model in which the logarithm of the spot price is assumed to follow a mean-reverting process of the Ornstein– Uhlenbeck type. Besides this one-factor model, Schwartz also formulates a two- factor and a three-factor model. The advantage of these multi-factor models is their ability to consider additional parameters influencing commodity prices. A main difficulty of such multi-factor models is that the factors that are used in these models are often not directly observable.
When fuel prices are used as an input parameter for fundamental electricity mar- ket models, the approaches the most commonly used to simulate fuel prices are GBMs (e.g., Aronne et al. [Aro+08], Botterud [Bot03]) or mean-reverting processes (e.g., Hundt and Sun [HS09]). For this reason, we focus in the following on a comparison of these two processes.
1.4.2.2 Comparison of GBM and Mean-Reverting Models
Pindyck [Pin99] studied the long-run evolution of oil, coal and natural gas prices using data of up to 127 years. Based on this data, Pindyck addressed the question whether these fuel prices should be modeled as a GBM or as a mean-reverting process. Unit root tests and variance ratio tests suggest that the logarithms of the prices are mean-reverting. However, the rate of mean-reversion is slow. Further- more, the trends to which the prices are reverting fluctuate over time.
Even though the results indicate that energy prices follow rather a mean-reversion process than a GBM, Pindyck examined the effects of investment decisions if a GBM process is used instead. He mentions that if the rate of mean-reversion is slow, the dependence of the optimal decision on the equilibrium price is low.
1.4. MODELING ENERGY PRICES 27 0 50 100 150 200 250 300 350 400 2000 2005 2010 2015 2020 2025 Index (1997 = 100) Year Data 1930 - 1996
Data 1974 - 1996 Observed prices
Figure 1.7: Expected gas price development and 95% confidence interval for gas prices modeled as a GBM. Drift and standard deviation estimations based on dif- ferent periods. Data taken from [Pin99]. Observed gas prices taken from the US Energy Information Administration.
However, this holds only if the implied volatility of energy prices is relatively constant, as assumed by the GBM. As both conditions are fulfilled, Pindyck con- cludes that the GBM assumption is unlikely to lead to large errors in the optimal investment rule.
Considering the data Pindyck used for his analysis, another difficulty of predict- ing fuel prices becomes obvious. The annual average price changes vary signif- icantly depending on the considered time period. While the average annual gas price increased by 1.88% in the years between 1930 and 1996, considering just the years of 1970–1996, average annual increases of 4.9% are observed. The standard deviation also increased from 0.1079 to 0.1459.
Figure 1.7 shows the expected gas prices and a 95% confidence interval for GBMs with drift and standard deviation estimated based on the two before mentioned periods. In addition, we added the observed gas prices until 2008.14
It can be seen that for several years, the observed prices are outside the 95% confi- dence interval for the GBM based on the mean and standard deviation estimations of the years between 1930 and 1996. For the GBM based on the data between 1970 and 1996, observed prices remained in the 95% confidence interval. However, this confidence interval gets quite wide.
14We took the price of gas used by electricity generators from the US Energy Information Admin-
istration (available at http://tonto.eia.doe.gov/dnav/ng/ng_pri_sum_dcu_nus_a.htm, last time accessed on December 02, 2009) and deflated the prices using the Producer Price Index for all commodities (available at http://www.bls.gov/ppi/, last time accessed on December 02, 2009) as Pindyck did it.
While the majority of researchers assume fuel prices to be mean-reverting, this is not undisputed. Geman [Gem07] examined the development of crude oil and natural gas prices during 15 years. She identified a mean-reversion pattern for crude oil prices in the years between 1994 and 2000. In 2000, the pattern changed into a random walk. For natural gas prices, a mean-reversion pattern is found until 1999. It also changed into a random walk in 2000.
1.4.2.3 Some Critical Remarks on Fuel Price Forecasts
Even though a wide variety of fuel price forecasting models exist, the quality of forecasts derived from the models is often questioned. Manera et al. [Man+07] present a survey of different models used to forecast oil prices. Manera et al. note that there is no consensus about the appropriate forecasting model. Findings vary across models, time periods and data frequencies. The authors conclude that the best performing econometric model for oil forecasts is still to appear in literature. A similar conclusion is made by Fattouh [Fat07]. The author reviewed the three main approaches used for analyzing oil prices: non-structural models, the infor- mal approach and the demand–supply framework. Fattouh concludes that each of these approaches suffers from major limitations, especially when used to make predictions.
While those two papers surveyed oil price models, the results are probably also applicable to gas price models. At least in Germany this is the case, as the gas price is directly linked to the oil price. Summarizing the results of the literature on fuel price forecasting, two things should be retained. First, a perfect forecast model does not exist. Second, even if the appropriate stochastic process for a fuel price model could be identified, the forecasts might still differ depending on the historical data used to estimate the parameters of the process. In our opinion, these findings emphasize the importance to consider a wide variety of different fuel price scenarios in a power plant investment model.