The model presented in the previous Section 3.2.2 has been tested on two dierent commodity markets: Silver and New Zealand electricity; both of them discussed in Section 1.3.1.
Some applications of the proposed model are presented below. The title of each simu-lation encompasses the information related to the parameter values used for simulating the corresponding commodity prices. Tables with statistical features such as mean, standard deviation, skewness and kurtosis are available in Appendix B.
In order to apply the stochastic interacting traders model to commodity markets, two dierent local interaction kernels have been used and compared: a Gaussian kernel and an interacting kernel K. In the following gures the red line represents the evolution of commodity prices obtained using a Gaussian kernel as the local interaction kernel, while the green line represents the stock prices simulated through an interacting kernel K, which reects the idea that each trader can perceive only the inuence of its neighbours within a range of 5% of the maximum distance.
3 THE DYNAMICS OF TRADERS: A NEW PROPOSAL 77 3.3.1 Silver commodity market
Silver: 30days 100traders 2000speed 0,05iScale 2volatility
Figure 16: Silver, real and simulated prices
(a) Root mean square error for Gaussian
Ker-nel (b) Root mean square error for Kernel K
Figure 17: Root mean square error
3 THE DYNAMICS OF TRADERS: A NEW PROPOSAL 78 We started to simulate the Silver price, Fig. 16, using the price of the previous 30 days to obtain the prices of the next month, and the parameters are chosen as in Figures 7 and 9.
The choice of the interacting scale parameter β = 0.05 is for being coherent with the idea of 5% of interaction, already included in the interacting kernel K.
Only the speed of the mean reversion level has been increased and as a consequence the simulated prices, both with Gaussian and with K kernel, seem to have more spikes than the real price.
However, from Fig. 17 we can deduce that the forecasted price follows in the mean the real price.
Silver: 30days 100traders 1500speed 0,25iScale 2volatility
Figure 18: Silver, real and simulated prices
3 THE DYNAMICS OF TRADERS: A NEW PROPOSAL 79
(a) Root mean square error for Gaussian
Ker-nel (b) Root mean square error for Kernel k
Figure 19: Root mean square error
Decreasing the speed of the mean reversion γ and increasing the interaction scale parameter β we have evidence that the model is not able to change the slope, yet.
Silver: 90days 100traders 1500speed 0,25iScale 2volatility
Figure 20: Silver, real and simulated prices
3 THE DYNAMICS OF TRADERS: A NEW PROPOSAL 80
(a) Root mean square error for Gaussian
Ker-nel (b) Root mean square error for Kernel K
Figure 21: Root mean square error
The inability of the model to change the slope is amplied when we use the previous 90 days to predict the next 3 months. Nevertheless, the mean values of the forecasted prices are still close to the real prices, as shown in Fig. 21.
Silver: 30days 100traders 1500speed 0,25iScale 5volatility
Figure 22: Silver, real and simulated prices
3 THE DYNAMICS OF TRADERS: A NEW PROPOSAL 81
(a) Root mean square error for Gaussian
Ker-nel (b) Root mean square error for kernel K
Figure 23: Root mean square error
An improvement is gained when, in order to escape from the inability of the model to change the gradient, we increase the volatilityσ
Silver: 30days 150traders 1500speed 0,25iScale 5volatility
Figure 24: Silver, real and simulated prices
3 THE DYNAMICS OF TRADERS: A NEW PROPOSAL 82
(a) Root mean square error for Gaussian
Ker-nel (b) Root mean square error for Kernel K
Figure 25: Root mean square error
Finally, we increased the number of traders but there were no evident improvements.
3 THE DYNAMICS OF TRADERS: A NEW PROPOSAL 83 3.3.2 New Zealand electricity market
New Zealand: 30days 100traders 500speed 0,05iScale 120volatility
Figure 26: New Zealand, real and simulated prices
(a) Root mean square error for Gaussian
Ker-nel (b) Root mean square error for Kernel K
Figure 27: Root mean square error
3 THE DYNAMICS OF TRADERS: A NEW PROPOSAL 84 The electricity market is one of the most challenging markets to simulate due to its high volatility. Nevertheless, the stochastic interacting traders model seems to simulate the New Zealand electricity price, keeping the mean behaviour of the real data. As shown in Fig. 27, the model is not capable to uctuate as much as the real price but it is able to reproduce big spikes, and this is a promising feature.
New Zealand: 30days 100traders 1800speed 0,1iScale 120volatility
Figure 28: New Zealand, real and simulated prices
3 THE DYNAMICS OF TRADERS: A NEW PROPOSAL 85
(a) Root mean square error for Gaussian
Ker-nel (b) Root mean square error for Kernel K
Figure 29: Root mean square error
Indeed increasing the mean reversion speed and the interaction scale we got a better result. The negative spike is a consequence of the high strength of the mean reversion and of the permission of the mathematical model to assume negative values.
New Zealand: 90days 100traders 1800speed 0,1iScale 120volatility
Figure 30: New Zealand, real and simulated prices
3 THE DYNAMICS OF TRADERS: A NEW PROPOSAL 86
(a) Root mean square error for Gaussian
Ker-nel (b) Root mean square error for Kernel K
Figure 31: Root mean square error
When the number of days to forecast is increased to 90 days, we lost the accuracy of the model, as expected.
New Zealand: 30days 150traders 1800speed 0,15iScale 120volatility
Figure 32: New Zealand, real and simulated prices
3 THE DYNAMICS OF TRADERS: A NEW PROPOSAL 87
(a) Root mean square error for Gaussian
Ker-nel (b) Root mean square error for Kernel K
Figure 33: Root mean square error
Increasing the number of market participants, the eect of the mean reversion and the global interaction, decreases. Therefore the model reduces its ability to reproduce peaks.
New Zealand: 30days 100traders 250speed 0,1iScale 180volatility
Figure 34: New Zealand, real and simulated prices
3 THE DYNAMICS OF TRADERS: A NEW PROPOSAL 88
(a) Root mean square error for Gaussian
Ker-nel (b) Root mean square error for Kernel K
Figure 35: Root mean square error
Better results are achieved with lower mean reversion speed and higher volatility. In these situations, we can note that the model reproduces the spikes without going down in the negative zone.