In the electric power industry, modeling of commodity prices focuses on the price path simulation of prices of fuels, such as coal, gas and certainly oil. Some mod-els (e.g. Muche (2007)) consider also CO2-certificate prices, since CO2-certificate trading is established in electricity markets. However, one of the main uncertainty for electric power producers is fuel prices. Different stochastic models have been explored in the last few years, to handle uncertain fuel prices. Again, they use mean-reversion processes and ARMA processes to describe the stochastic devel-opment of the commodity prices. Some of the commodity price models consider trend and seasonality of the price development similar to electricity price models (see Heydari and Afzal (2008)). And some financial models include a second fac-tor, the convenience yield3, which also follows also an MR Process (see Schwartz (1997)). These models contain the correlation of the convenience yield and the commodity prices. But more interesting is the correlation between different fuel prices. Thereby the dependency of other fuel prices on the oil price plays a key role. Analogue to electricity price simulations, the logarithms of the primary en-ergy prices are generally modelled instead of the prices themselves (Xf = lnpf).
Thereby f represents the index of fuel (primary energy carrier PEC) types. More precise approaches (e.g. Weber (2005)) model the derivatives of the price logs with the help of a mean-reversion process.
d(dXf) = κf(μdXf− dXf) + σdXfdWf [3.3]
whereas dWf = εf
√dt is a Wiener Process. Thereby the error term εf of the Wiener process dWf is standard normal distributed. For estimation purposes, the continuous model is again changed into a discrete one based on discrete time periods for the fuel price simulation; i.e. the marginal time interval is replaced by a discrete time periodΔt= 1. Based on the discrete approach, the oil price is modelled firstly, as oil is still the most important world energy carrier. The oil
3The convenience yield can be defined as the surplus of holding the commodity itself instead of a future contract. It plays a major role in times of scarcity of resources.
3.1. Stochastic processes for modeling uncertainties in electric power generation
price simulation is followed by the simulation of the other fuel prices considering correlations based on the oil price.
Δ(ΔXoil) = κoil(μΔXoil− ΔXoil) + σΔXoilεΔXoil∼ N(0,1) [3.4]
For other energy carriers the mean-reversion model is extended by a term for the difference to the long-term equilibrium oil price (XOil− θfXf) and oil price changes (ΔΔXOil) as further explanatory variables, considering correlations and dependencies on the oil price:
Δ(ΔXf) = κf(μΔXf− ΔXf) + βf(XOil− θfXf) + γfΔΔXOil+ σΔXfεΔXf [3.5]
The new parameters represent the tendency to the oil priceβf, the price ratioθf
between the specific fuel price Xfand the oil price and at last a factorγfdescribing the dependency on the oil price change. After estimating these parameters from historical data via the least-squares method, the extended mean-reversion process can be applied for different fuel prices.
However, these mean-reversion models for fuel prices do not take deterministic components as trend and seasonality into account. But as mentioned above, there are models considering these components similar to electricity price modeling.
The trend function again generally contains a constant growth rate, whereas the seasonality is described by again a trigonometric function (see 3.2). However, as coal prices are noted quarterly, there are no significant seasonal effects noticeable, so the models for coal do not include seasonality functions. In contrast to coal prices, gas prices possess strong seasonal effects, which can be described by a trigonometric functions.
After removing the deterministic components trend (and seasonality), the re-ceived stochastic residues of fuel prices are modelled via ARMA processes. But the ARMA model can only be applied to a stochastic process, if the process is at least a weak stationary one and the error term (see 3.6) follows a White noise process (see Hackl (2008)). The residual series of the detrended coal prices form a strong stationary process, so the an ARMA process can be applied to the stochastic series. If the a stochastic series is not stationary, it can be transformed into a sta-tionary one using a filter (see Box et al. (2008)). This filter can be the differences of two sequent residues forming a new residue series.
Sometimes a filter has to be applied d-times to receive a stationary process.
The composition of filtering the stochastic price series d-times and the proper ARMA(p,q) process is also called autoregressive integrated moving average pro-cess (ARIMA(p,d,q) propro-cess). For example, coal prices are modelled by Schmoeller (2005) with the help of a ARIMA(1,1,0) process, while the gas prices are de-scribed by an ARIMA(2,0,1) process.
These approaches describe independent models for coal and gas prices. But in fact there is a correlation between both price processes. Therefore the ARIMA (1,1,0) process for coal is extended, taking into account the correlation of the coal price in t with the average gas price in t− ρ:
Xcoal,t= αXcoal,t−1+ γ ¯Xgas,t−ρ+ εt [3.6]
As mentioned above, coal prices are noted quarterly, so the coal price logs are not modelled on the basis of daily price changes. However, if future expected prices are required, e.g. for real option models (see 3.3), the AR(1) process for electricity prices can be formulated also for coal prices (see Eq. 3.19), based on the expected value from the perspective of today’s price logarithm X0. As the risk-neutral process is required for real options, the AR(1) process is extended by a termλ · σ/κ, representing the market price of risk (see Hull (2005)).
E(XRN,t) = e−κtX0+
α −λ · σ κ
(1 − e−κt) ; Var0(XRN,t) =σ2
2κ(1 − e−2κt) [3.7]
Due to the log-normal distribution assumption of the prices, the expected prices are calculated from their expected logs and variance as follows (see Jaillet et al.
(2004)):
E(pf u,RN,t) = eE(XRN,t)+12Var0(XRN,t) [3.8]
This model for the coal price is similar to the one factor model developed by Schwartz for the simulation of commodity prices. The one factor model is ex-tended in the two factor approach by Gibson and Schwartz, which is based on two
3.1. Stochastic processes for modeling uncertainties in electric power generation
mean-reversion processes, one for the commodity spot prices and a second one for the convenience yield, regarding correlation between both parameters.
CO2-certificate prices are also simulated with the aid of ARMA processes or mean-reversion processes, whereas risk-neutral processes are considered (see Wag-ner (2007)) if the simulated prices are used again in a real option model. As CO2 -certificates and coal are storable products (in contrast to electricity), no larger price jumps are expected in their price process. Therefore it is sufficient to apply a standard mean-reversion process without any jump component for CO2-certificate prices. At last, it is worth mentioning that the correlation of electricity prices and CO2-certificate prices is also considered in the CO2-certificate price model by Muche (2007). Therefore the error term of the risk-neutral ARMA process of the CO2-certificate prices is extended by the product of the correlation coefficientρec
and the error term of the electricity prices:
εCO2,t= εe,tρec+ εCO 2,t
1− ρec2 [3.9]
TherebyεCO 2,trepresents the original error random variable of the CO2-certificate price process,εe,t the error random variable of the electricity prices. However, only a few models simulate CO2-certificate prices. The behaviour of this highly volatile market parameter should be further addressed in future research.