Simulation and model assessment

In this section we do a simulation study and assess the performance of the model in terms of reproducibility of both the EEX data under investigation and the stylized features of energy futures markets. Firstly, we compare simulated paths of some exemplary futures contracts to the corresponding observed trajectories, in order to assess the qualitative behavior of model simulations. Then, we compute fundamental statistics of the model by averaging the results of a set of simulations and compare them to our data.

In order to simulate the trajectory of a certain contract, we use the dynamics in Equation (3.14). We define it for given input parameters in MATLAB via the sdemrd class, specifically designed for mean-reverting equations. Then, the function simByEuler discretizes the SDE and computes the price for each day by following an Euler scheme for stochastic differential equations, with the parameters which have been estimated in Section 5. For all futures we have the same averaged mean-reversion speed ˆλ, while each contract (indexed by i) has its own parameter Ψi in the long-term volatility coefficient and ˆΦi for the drift component. Since we are interested in the simulation of a single contract, we compute the long-term means from the single-forward time series calibration described in Appendix A.1. Finally, the diffusion parameters are computed with the non-parametric procedure, being this the case that performed better in terms of residuals (cf. Section 5).

For a given contract we plot the historical daily prices observed during its trading time interval and run a single simulation for the sake of qualitative comparison. We take as examples two contracts for each delivery period of the EEX data considered: calendar, quarter, month. Specifically, we plot Cal-18 and Cal-19 in Figures 3.7 and 3.8, which are, respectively, the longest traded one and the last one in our observation period. In Figures 3.9 and 3.10, Q2/16 and Q1/18 are compared, being the first and last observed quarter. Finally, we plot in Figures 3.11 and 3.12 the prices of the two monthly futures Jan/17 (cold month) and Jul/17 (warm month). Together with price trajectories, we indicate the long-term mean of the model

by a dotted line.

From a visual perspective, the simulated paths mimic the observed movements reasonably well, being capable to follow different trends for different periods as well. This is particularly encouraging, if we consider that the overall observation window is long about one year and a half and, moreover, we observe contracts with heterogeneous delivery periods and trading lives, some of which are traded for less than 40 days, such as Jul/17 (see Figure 3.12). The mean-reversion effect is generally evident and well reproduced by our model, especially if we focus on the mean-trend described by the dotted line. However, if we move from this line, in some cases the simulations seem not to take into account certain idiosyncratic movements. This can be noticed in Figure 3.7 (Cal-18) and Figure 3.10 (Q1/18). One explanation could be related to the mean-reversion rate of decay, which determines the excursions from the long term equilibrium. Recall that, by arbitrage, the parameter λ must be equal for all traded futures, but if we calibrate it for each contract separately we observe different values in practice. This assumption of course reflects on the accuracy in reproducing the single contract prices pathwise, but it is required by our no-arbitrage arguments.

For a more rigorous discussion of the fitting quality, we investigate the statistical features of the model and make a comparison with the historical data. A standard way to do it is by computing moments. In Table 3.3 we report the values of the first four moments, the minimum and the maximum of both empirical and simulated returns, i.e. the daily price increments, of all contracts. For the simulated returns, we run 1000 simulations and then average the results over all samples. The values are classified among different delivery periods in order to distinguish different behaviors (if present) among them.

The general performance is quite satisfactory. The mean is positive but very close to zero, both in the model as for the observed prices, with the notable exception of the calendars, where the model reproduces the data’s positive trend very well. The standard deviation is adequately captured for all delivery periods with a small error for the monthly contracts, which prove to be the most difficult to model also looking at the other statistics: let us remark that, in these markets, monthly contracts are available for trading for relatively short periods. Regarding the skewness, the results are very good, since both the model and the observed returns are statistically not skewed. Let us recall that the standard error of the skewness estimator is approximatelyp6/n, where n is the number of daily observations per contract. In our case, n approximately ranges from 40 to 65 for monthly contracts, from 60 to 250 for the quarters and from 40 to 270 for the calendars. This means that, for example, in the case of some quarter and calendar contracts, where we have the highest number of observations (n = 270), the standard error is about 0.15. Then, roughly speaking, any value between −0.3 and 0.3 is not statistically different from zero with a 5% significance level, as it is observed from the market even for the monthly contracts (which have much less trading days). With the same procedure, a similar computation can be done for the kurtosis, which gives results still within the confidence interval.

To conclude, our framework seems capable to describe fairly well the main stylized features of power futures contracts, which are of different nature, in a comprehensive way. In other words, this assessment analysis suggests that the model adequately reproduces the main trajectorial and statistical properties of the Phelix Base futures prices. Specifically, in the statistic study presented in Table 3.3, we have not found evidence of non-normality in futures’ daily price increments. Most importantly, this study has been carried out by calibrating the model directly on the observed prices of the traded contracts, i.e. without introducing artificial forward curves or ex post estimated risk premium, besides working under the no-arbitrage assumption. 50 100 150 200 250 300 350 400 Days 22 24 26 28 30 32 Eur/MWh CAL18 50 100 150 200 250 300 350 400 Days 22 24 26 28 30 32 Eur/MWh

Figure 3.7: Historical (red) and simulated (blue) path of the contract Cal-18. The dotted line represents in both plots the estimated long-term mean of the contract.

3.7 Concluding remarks 81