a shortcoming of traditional simulation-based methods. Nonetheless, a sig- nificant drawback of this valuation approach is that it is computationally intensive, and particularly so for problems with multiple concurrent options (Hahn and James, 2008).
Monte Carlo methods are straightforward to apply for European options. However, they can be difficult to apply to many complex real problems, such as a compound option. Different types of tree-building procedures, such as trinomial trees or time-dependent drift and volatility specifications, commonly result in trees that are computationally complex and/or path- dependent (Hahn and James, 2008).
7.2
Modelling the price processes
We have chosen to model the electricity price as an Ornstein-Uhlenbeck (OU) process, which is a so-called mean-reverting process. Compared to the common practice of using a geometric Brownian motion (GBM) for modelling the power price, a mean-reverting price process captures the clear tendency of electricity prices to revert towards a long-term level.
Even though modelling the power price as a mean-reverting process is considered more realistic compared to a price that follows a GBM, there are complications associated with this model. For instance, we need to estimate more parameters in order to describe the OU process. While we need only to estimate the volatility for a GBM (the mean is directly observable from the sample), we must estimate a long-term price level as well as a rate of mean reversion in order to capture the characteristics of an OU process.
In addition, a mean-reverting process does not capture all properties of electricity prices. For example, it is not successful in capturing the jumps that are often seen in electricity prices (this is evident in Fig. 1), and Blanco and Soronow (2001) point out that the rate of mean-reversion is not constant, but changes according to how large a particular jump is, in which direction it was and why it occurred. In modelling these processes the choice between accuracy and applicability will work in opposite directions. In order to create a binomial tree for the option in a relatively straightforward fashion, the price must follow a simple process. This is a goal in itself in this thesis, as we aim to make the models useful for practitioners who are not experts in the field. It would, however, be very interesting to try and model the rate of mean-reversion as parameter that can vary with time.
We model the price of tradable green certificates (TGC’s) as a GBM. The reasoning was that there is no marginal cost of production for TGC’s, so it does not tend to revert back to a mean like the price of power (this can be seen in Fig. 2). Additionally, a demand for renewable power is
7.2 Modelling the price processes
created artificially, and the demand is based on a quota defined by the government rather than by the individual consumer’s desire for electricity from renewable sources. Given this, the price might become very high or very low at a certain point in time, e.g. when we approach the end of some period for which there exists a set goal for new renewable production. It is then very likely that the government will step in and prevent this price from becoming way too high or low. In that case, it can be hard to justify using a GBM, since in practice there would be a roof and a floor for the price.
Independently of high or low prices, political actions that will affect the subsidy scheme of TGC’s are likely to occur from time to time. This possibility of political action aimed at building more renewable electricity production likely affects the volatility of the TGC price. The sensitivity analysis of volatility for the Monte Carlo valuation was shown in Table 9, and this analysis showed that the value of the real option is more dependent on changes in the volatility of the TGC price than that of the power price. Thus, if we assume that volatility of the TGC price is positively correlated with political intervention, then we can state that the option value will be higher in times where political action is expected or has just occurred and the market is still absorbing the implications of the amendments to the scheme.
It is important to be aware that the historical TGC price data used in this thesis does not fully represent the common market between Norway and Sweden that exists now. The common market was introduced in 2012, and the data from earlier years is from the Swedish TGC market that was in operation from 2003 throughout 2011. One can only speculate about how the market will evolve in the future compared to the past. Nonetheless, in this thesis we have assumed that the historical data will be representative of the future.
When we are already describing the underlying stochastic processes, it is natural also to discuss the correlation between them. As mentioned in Section 6.4.3, the correlation estimation done here is not one that can be trusted, as it was investigated only superficially. This is justified as this report aims to demonstrate the differences between valuation models, not to determine parameters as accurately as possible (except the parameters for the price processes). It is worth noting that TrønderEnergi is believed to have a broad understanding for the relationship between TGC and power prices, so we do not think that this is where we can make our biggest contribution.