Model Group 1 (Models OU, OUJ, OUS and OUJS)

In this section, we will first compare the RMSEs² and the summary statistics for each parameter in each model. Then we will take a look at the regression statistics, and finally conduct further examinations.

6.1.1 Summary Statistics

For an overview of our results, summary statistics of the four models OU, OUJ, OUS and OUJS are shown in Table 6.1. For each model, for the RMSE and each parameter occurring therein, mean, median and standard deviation are given.

If we examine the RMSEs in Table 6.1, we see that introducing seasonality greatly improves the models. The RMSEs of models OUS and OUJS are less

1The models are presented in detail in Chap. 5.

2The RMSE is the root of the mean of all squared errors for a day. The squared errors for each day are the function that was minimised in our algorithm, see (2.28). So the RMSE shows how well the models fit the data.

Table 6.1.Summary Statistics models OU, OUJ, OUS and OUJS

For each trading day between 01 October 1999 and 30 September 2002, all parameters for the relevant models were independently estimated. Futures and forward contracts lead to implicit estimates minimising the RMSE (root mean squared error) for all contracts. In this summary statistics table, mean, median and standard deviation of each parameter as well as of the RMSE are reported. The models OU, OUJ, OUS and OUJS refer to an Ornstein–Uhlenbeck process, an Ornstein–Uhlenbeck process with jump, with season and with season and jump.

κ θ σ λ µJ σJ s0 s1 RMSE

Model OU

Mean 106,76 4,42 3,85 14,11

Median 6,10 4,62 2,32 13,74

Std. Dev. 397,15 1,07 7,61 3,05

Model OUJ

Mean 42,32 4,15 3,15 13,56 0,18 0,18 14,03

Median 5,65 4,39 2,29 4,36 0,12 0,12 13,85

Std. Dev. 149,58 1,07 4,51 56,29 0,36 0,19 2,98

Model OUS

Mean 135,54 4,80 4,29 26,57 0,64 5,31

Median 2,43 4,85 1,21 26,22 0,64 4,98

Std. Dev. 432,83 2,67 9,71 4,33 0,03 2,02

Model OUJS

Mean 145,74 4,33 2,52 27,36 0,07 0,26 25,47 0,64 5,71

Median 3,10 4,68 1,27 4,16 0,12 0,15 25,57 0,64 5,45

Std. Dev. 1501,16 1,02 5,47 113,33 0,35 0,47 5,20 0,03 2,07

than a half of the counterpart models OU and OUJ, while adding normal-distributed jumps do not improve models OU and OUS. Since models OU and OUS are em-bedded in models OUJ and OUJS for the limits λ→0 or µ_J →0 and σ_J →0, the RMSEs of the latter should be at least as good as those of OU and OUS. The poor values can be explained by the need for calculations of numerical integrals that are used in model OUJ as well as in model OUJS, see Sect. 5.4.1. The numerical prob-lems are also evident if we look at the histograms in Fig. 6.1, at least for OUJS:

while this model often has lower RMSEs, there are also a lot of outliers at very high values.

The differences between the models with and without season are also obvious in the histograms: the models without seasonal modelling are not able to explain observed prices up to a level that models OUS and OUJS easily can, i.e. in the models with season, the values of the RMSEs are more focused around 4.5 to 5.0, while in the models without seasonal modelling, most estimates seem to be around 14.

If we compare means and medians of the RMSEs in Table 6.1, they do not differ much for either model, and also the standard deviations are in an acceptable range.

0 5 10 15 20 25

The RMSEs for all estimations are plotted in a histogram for each model. The models OU, OUJ, OUS and OUJS refer to an Ornstein–Uhlenbeck process, an Ornstein–Uhlenbeck process with jump, with season and with season and jump.

The standard deviations diminish if we change from model OU over OUJ to OUS.

The slightly worse standard deviation for OUJS of 2.07 in comparison to 2.02 of model OUJS is again due to the numerical problems stated before. We already see here that modelling seasonal behaviour is very important. Jumps also could improve the models if numerical problems could be overcome.

If we examine the parameters in Table 6.1, we see that mean and median greatly differ for κ, λ, and, to a lesser extent, σ, µ_J and σ_J. The estimations do not seem to be robust. Comparing the standard deviations supports these findings.

One of the parameters that seems to be relative stable is θ, whose means and medians lie between 4.15 and 4.85 for all models. θ is the parameter of the long-term mean of the time series, i.e. the log spot price. The mean and median of the log spot price are 4.91 and 4.93, refer to Fig. 4.4. Thus, θ lies within a range that seems reasonable. The difference can emerge from the different time horizons of historical and implicit data, but also from risk premia. For histograms of the

estimations of θ, see Fig. 6.2. The most robust values of θ seems to have model

The parameter values ofθfor all estimations are plotted in a histogram for each model. The models OU, OUJ, OUS and OUJS refer to an Ornstein–Uhlenbeck process, an Ornstein–Uhlenbeck process with jump, with season and with season and jump.

The parameterκshows the velocity of the data to return to the long-term mean θ. The means of κ seem quite high, but the medians that are between 2.43 for model OUS and 6.10 for model OU are in a more encouraging range. Histograms of the estimates can be viewed in Fig. 6.3. The histograms show thatκ in models OUS and OUJS is more biased against zero than in OU and OUJ.

The values for σ, the standard deviation of the process, are most of the times below the standard deviation of the log spot price of 6.69 (compare to the annualized standard deviation on p. 38). But they are quite similar for all models, higher for OU and OUJ than for OUS and OUJS, at least in the medians. See Fig.

6.4 for histograms of the estimates. The histograms show that in OU and OUJ the estimates are more dispersed than in the models OUS and OUJS. But some estimates are near 6 and 7 for all models.

0 5 10 15 20 25 30

The parameter values ofκfor all estimations are plotted in a histogram for each model. The models OU, OUJ, OUS and OUJS refer to an Ornstein–Uhlenbeck process, an Ornstein–Uhlenbeck process with jump, with season and with season and jump.

0 5 10 15 20

The parameter values ofσ for all estimations are plotted in a histogram for each model. The models OU, OUJ, OUS and OUJS refer to an Ornstein–Uhlenbeck process, an Ornstein–Uhlenbeck process with jump, with season and with season and jump.

λ can be interpreted as the expected number of jumps per year, and more than four jumps (≈ the medians of models OUJ and OUJS), but less than 27 (≈ the mean of model OUJS) seem realistic. The histograms of λ in Fig. 6.5 show that

0 5 10 15 20 25 30

The parameter values ofλJ,µJ andσJ for all estimations are plotted in a histogram for each parameter and for each model containing a jump. The models OUJ and OUJS refer to an Ornstein–Uhlenbeck process with jump and to an Ornstein Uhlenbeck process with season and jump.

most estimates are very close to zero, with a second mode around 3.0 in model OUJ.

The mean jump size µ_J of model OUJ of 0.18 in the mean is much higher than the mean of µJ in model OUJS with 0.07, but the medians in both models are the same with 0.12. Also the standard deviations of µ_J are nearly the same. For OUJS, the mean of theσJ, that is the standard deviation of the jump size, of 0.26 is above that of OUJ with 0.18,³ but the medians again do not differ very much.

The estimations of σJ in model OUJS with a standard deviation of 0.47 are more volatile than in model OUJ with a standard deviation for σ_J of 0.19.

The histograms for µJ and σJ in Fig. 6.5 support the findings that the estimations for OUJ and OUJS are quite similar.

The seasonality parameters s0 and s1 are quite similar in the two models. In model OUS, s₀ is 26.57 and 26.22, and this is both in mean and median slightly higher than in model OUJS with 25.47 and 25.57. If we plot the spot price, and the

3The rounded means and medians forµJ andσJ are really the same, this is no typing error!

spot price at t0 = 10/01/99 plus the deterministic functions, calculated from the mean ofs₀ ands₁ of the two models, like in Fig. 6.6, the deterministic functions of models OUS and OUJS cannot be distinguished on the graph.