stochastic process is a family of random variables X(h, ) of two variables h H, on a common probability space ( ,F,P), which assumes real values and is P-measurable as a function of for a fixed h. The parameter h is interpreted as time, with H being a time interval and X(h,·) represents a random variable on the above probability space , while X(·, ) is called a sample path or trajectory of the stochastic process.
3.6.1 Stochastic process
A stochastic process (Wh) 0 is defined as Brownian motion (BM) if has the following characteristics:
W0=0, that is, BM starts at zero.
(Wh) 0 is a process with homogeneous and independent increments, i.e., distribution of futurechanges does not depend on past realizations.
Any increment Wh-Wt is normally distributed with a mean zero and the variance h-t, t<h, i.e., the variance increases linearly with the length of time interval.
The paths of (Wh) 0 are continuous but nowhere differentiable.
3.6.2 Ornstein-Uhlenbeck process
The mean-reversion (MR) process is one of the most applied stochastic processes to simulate electricity prices (Gibson and Schwartz, 1990; Hirsch, 2009; Möst and Keles , 2010). Therefore, it can be considered an alternative to the Box-Jenkins time series models. The MR process called Ornstein-Uhlenbeck (OU) (Uhlenbeck and Ornstein, 1930) can be formulated for the price changes with the following stochastic differential equation (SDE):
( )
h h h
dX k µ X dh dW (3.26)
3.6 Stochastic differential equations – Ornstein-Uhlenbeck process 59 The first term k(µ-Xh) of Eq. (3.26) describes the drift component. The parameter k determines the reversion rate of the stochastic process to its long-term mean µ. The essence of the mean-reversion concept for the case of a price time series is the assumption that any stochastic price fluctuations are temporary and the price will tend to move to the mean price over time. As mentioned above, in the electricity markets, the price fluctuations and the mean reversion are generally explained by entering expensive generators as a result of an extreme meteorological situation, power plant outages and transmission congestions.
The second term dWh corresponds to the standard Brownian motion. The stochastic driver is the Wiener process movement dWh= hdh1/2, where h is a standard normally distributed random variable.
3.6.3 Calibration of SDE
The SDE is solved by Euler discretization (Lari-Lavassani et al., 2001), applying Ito’s Lemma with the following exact solution (Karatzas and Shreve, 2000):
2k
The parameters a,b, are determined by ML or LSQ. The resubstitution of the parameters a,b, results in the original parameters of the exact solution k,µ, . With the help of the estimated parameters, the exact solution of the SDE is applied to generate the price path.
3.6.4 OU process to simulate electricity prices
In the first step, prices are logarithmized and the price logs are passed to the simulation tool instead of the prices themselves. The logarithm is used as it limits the volatility and leads to a variance stabilization. Since the electricity prices display typical patterns, the models developed to describe the behavior of electricity prices should capture the deterministic components (trend, daily, weekly, and annual cycles) of electricity prices.
The deterministic patterns (daily, weekly, annual seasonality) are removed from the log-price series. The remaining stochastic component is then used to estimate the parameters of the corresponding stochastic process. Finally, the deterministic components are added to the simulated stochastic component, and then, the simulated price logs are retransformed receiving a simulated price path.
Model parameter estimates are calculated for the stochastic component extracted from the logarithmized Finnish day-ahead electricity prices of the years 2007–2009. At a closer inspection of Figure 3.13 it becomes evident that the simulated price path partly follows the actual series. Rather, this is a consequence of the excessive "jumpiness" of an optimal mean-reverting model. The residuals emerging from this optimal mean reverting model are normally distributed. Since an Ornstein-Uhlenbeck model is always normally distributed by definition, this property is transferred to the model residuals when there are frequent spikes in the simulated series that do not coincide with the spikes in the actual series (Naeem, 2009).
Figure 3.13. Ornstein-Uhlenbeck simulation (left) and normalized histogram of the model residuals with normal distribution (right).
Relevant statistics of the original and simulated prices are collected in Table 3.6. To achieve a more robust result, an expected value for the measurements is determined based on 50 simulations for the OU process.
It should be concluded that the conventional mean-reverting Ornstein-Uhlenbeck model, even when calibrated optimally with the actual electricity market prices, is not able to capture the statistical characteristics of the actual series.
3.6 Stochastic differential equations – Ornstein-Uhlenbeck process 61 Table 3.6. Basic statistics for original Finnish day-ahead electricity prices and price paths
simulated by the Ornstein-Uhlenbeck process.
Original prices, [euro/MWh] Simulated prices, [euro/MWh]
Mean 39.67 40.01
3.6.5 OU process with colored noise
The mean-reversion process driven by an exponential colored noise can be formulated for the price changes with the following SDE (Mtunya, 2010):
( )
h h h
dX k µ X dh dh (3.28)
The terms of Eq. (3.28) have the same meanings as in Eq. (3.26), h is an exponentially colored noise process generated to mimic the behavior of both the spikes and the usual volatility of the prices. The colored noise process h produces a sequence of correlated random variables (h1), (h2),… with the same standard deviation in each. Colored noise is a Gaussian process, and it is well known that this process can be completely described by their mean and covariance functions (Arnold, 1974).
The Ornstein-Uhlenbeck process is extended and repeatedly integrated to obtain the colored noise of the first and second orders forcing along the series:
1 1 1 Wiener process with dWh~N(0,dh).
The system of Eqs. (3.29)–(3.30), with (0)=0 (i.e., starting with no noise) and t<h, has the following solutions:
( )
All the relevant process parameters are estimated by the ML methodology. The system of Eqs. (3.31)–(3.32) generates a stationary, zero-mean, correlated Gaussian process
2(h). The generated colored noise process 2(h) is applied to Eq. (3.28) to model the price. Therefore, the mean-reverting log-price equation is as
( ) 2
h h
dX k µ X dh dh (3.33)
With the use of colored noise forces, the correlation of the noise terms that influence the price time series is modeled more accurately, and it becomes possible to take into account the spiking characteristics and volatility clustering of the prices.
3.6.6 OU process with colored noise to simulate electricity prices
Prices are logarithmized and deterministic patters are removed. The corresponding stochastic component of the price logs are passed to the simulation tool, deterministic patterns are added, and the simulated price logs are retransformed receiving a simulated price path. Simulation of the Finnish day-ahead electricity prices of the years 2007–
2009 with the use of the MR process driven by an exponential colored noise is presented in Figure 3.14.
The relevant statistics of the original and simulated prices based on 50 simulations are collected in Table 3.7.
Table 3.7. Basic statistics for the original Finnish day-ahead electricity prices and price paths simulated by the Ornstein-Uhlenbeck process with colored noise.
Original prices, [euro/MWh] Simulated prices, [euro/MWh]
Mean 39.67 41.97