Predictive Trading Strategy for Physical Electricity Futures

: This article presents an original predictive strategy, based on a new mid-term forecasting model, to be used for trading physical electricity futures. The forecasting model is used to predict the average spot price, which is used to estimate the Risk Premium corresponding to electricity futures trade operations with a physical delivery. A feed-forward neural network trained with the extreme learning machine algorithm is used as the initial implementation of the forecasting model. The predictive strategy and the forecasting model only need information available from electricity derivatives and spot markets at the time of negotiation. In this paper, the predictive trading strategy has been applied successfully to the Iberian Electricity Market (MIBEL). The forecasting model was applied for the six types of maturities available for monthly futures in the MIBEL, from 1 to 6 months ahead. The forecasting model was trained with MIBEL price data corresponding to 44 months and the performances of the forecasting model and of the predictive strategy were tested with data corresponding to a further 12 months. Furthermore, a simpler forecasting model and three benchmark trading strategies are also presented and evaluated using the Risk Premium in the testing period, for comparative purposes. The results prove the advantages of the predictive strategy, even using the simpler forecasting model, which showed improvements over the conventional benchmark trading strategy, evincing an interesting hedging potential for electricity futures trading.


Context of This Research
New trading scenarios for electricity markets have arisen over the last three decades. Electricity can be traded as any other commodity: it can be bought or sold in deregulated electricity markets. The main difference with respect to other commodities is the limited storage capability that electricity has nowadays. Wholesale electricity markets are organized in a set of different niche markets, in which agents exchange energy and reserves from mid-term to very short-term periods. Mid-term periods, from days to some years ahead, are covered by forward markets, while short and very short-term periods are covered by day-ahead, intraday and real-time markets. In these markets, i.e., day-ahead, intraday and real-time markets, only agents who sell or buy electricity can trade, unlike forward markets where financial agents can also operate. In order to distinguish the electricity markets, the day-ahead market, organized with trade auctions, is called the spot market [1]. The spot market presents higher price fluctuations than the forward markets.
Forward electricity markets provide hedging opportunities against the price uncertainty of the spot market, and therefore, they contribute to reducing the operational risks. In Europe, the highest volume

Literature Review
The price at which agents sell electricity, through futures, is called the bid. The price at which agents buy electricity, through futures, is known as the ask. The difference between the maximum ask and the minimum bid is called the bid/ask spread. The agents participating in the electricity futures market indicate, by submitting asks or bids, their willingness to buy or sell electricity and their prices. They are, therefore, indirectly reflecting their perception about the average value of the prices that will be cleared in the spot market during the delivery period. In most commodity markets, it has been observed that when the end of the delivery period for a futures contract is approached, the futures price converges to the spot price of the underlying commodity [3]. An initial study [4] using electricity forward contracts price in the two US markets showed that the forward contracts price was a downwardly biased predictor of the future spot price under a low power demand and a moderate risk demand. On the other hand, several studies on European markets have shown the inefficiency of such markets, revealing that the futures prices are not unbiased predictors of the future spot prices [5][6][7]. Nevertheless, other studies, such as [8], have successfully used the futures price to predict the electricity price on the spot market. The differences in the predictive capacity of futures prices among diverse markets may be due to the different types of electricity supply in each market. For example, [9] shows that the storability of the fuels used to produce most of the energy in an electricity market leads the futures price to contain not only information about expected changes in the spot price but also risk premiums.
The risk premium (RP) can be obtained as the price of a futures contract minus the price of the underlying commodity [10]. The ex ante RP is calculated by using the expected price (price estimate) of the underlying commodity at the time of delivery, while the ex post RP is calculated by using its real price at this time. Electricity futures are slightly different from commodity futures, i.e., electricity is delivered over a period (day, weekend, week, etc.); it is not delivered at a specific moment. Since the calculation of the ex ante RP for electricity forward contracts requires a significant effort in modelling the dynamics of the electricity market in order to estimate (by forecasting) the average spot price during the delivery period, in economic studies, it is more common to analyse the ex post RP. Thus, the ex post RP corresponds to the sum of the ex ante RP and the average spot price forecasting error (i.e., the difference between the forecasted average spot price and the average real price during the delivery period) [11]. Several studies have analysed factors affecting the RP in different electricity markets: [12] found significant RPs in the electricity forward prices in an American market; [13] reported a term structure of the RP in the German electricity market due to the combination of two factors related to the agents; [14] reported a positive impact of the water reservoir level on the RP in the Nord Pool market; [15] found that the RP decreased over time as the delivery period profile, in order to maximize the profit, but less than a month in advance. For the MIBEL market Pinto et al. [35] present a decision support scheme, based on a spot price forecasting model, to optimize agents' profits. In this sense, several electricity market simulators have been developed in order to gain insight about the optimal trading strategy [36,37]. A recent work [38] has dealt with the direct forecast of the RP for year-ahead products in the German electricity market, although it leaves the user to choose the trading strategy to follow. Other approaches, designed for imbalance or intraday markets and based on statistical forecasting models, have explored risk-constrained strategies to increase profits and reduce risk by hedging against penalizing imbalance prices [39] or apply a density forecast provided by a stochastic latent moment model to predict imbalance volumes in order to optimize the positions on the intraday market [40].
In the different works published in the international literature, we can find diverse definitions for the term RP. The terms risk premium, forward risk premium, forward premium or market price of risk have been used interchangeably [14]. In this article, we used an ex post definition for the RP corresponding to the difference between the log of the average realized price in the delivery period and the log of the settlement futures price at the time of negotiation. This definition has been used in [19]. A similar definition, although without the log function, is used in [6,11,15,16]. The MIBEL Board of Regulators also defines the term risk premium on the basis of such difference [41]. On the other hand, some authors use the ex ante risk premium, which is defined as the difference between the futures price and the expected spot price in the delivery period [9,10,12,13]. With the definition used in this article, operations on the spot market correspond to a null value of RP. Additionally, in Section 2.2 of this article, we have established two different formulas for RP (with difference in sign) for buyers and sellers. Thus, with our convention for the RP signal used by the original predictive trading strategy presented in this article, a positive value of the signal represents always profits or savings for the agent, and a negative value represents losses, always with respect to the operations (purchases/sales) traded on the spot market.

Contributions and Structure of This Article
This article proposes an original trading strategy for both buyers and sellers in futures electricity markets (predictive strategy). The strategy is based on the use of a decision signal obtained with a new mid-term forecast of the average spot price for the delivery period. Depending on the forecast and on the value of the last settlement price for the futures, the decision signal indicates whether the agent should buy/sell the electricity through futures or to wait to buy/sell on the spot market.
In traditional trading strategy approaches, the "best guess" of the agents is the last known futures settlement price, reflecting the aggregated guess of all agents about an estimated average spot price in the corresponding delivery period. The predictive trading strategy proposed in this article achieves an advantage in relation to the traditional trading strategy, anticipating the good/wrong decision of buying/selling electricity through futures instead of waiting for the spot market. The predictive strategy succeeds when it is possible to forecast a mid-term average spot price with better accuracy than the traditional "best guess" of the agents, used as a benchmark in this article.
We also present a new mid-term average spot price forecasting model, implemented by a single hidden layer feed-forward neural network, trained with the extreme learning machine (ELM) algorithm, which exclusively uses available information from electricity derivatives and spot markets. The selection of the explanatory variables of the mid-term monthly average spot price forecasting model of this article is based on the experiences of market agents, and the set of explanatory variables is not exhaustive. The article is not focused on the exploration of improving forecasting performance; there is still room to improve the forecasts, in terms of both techniques and features. This kind of spot price forecast (several months ahead) is not as common as a short-term spot price forecast (1-7 days), but it is an issue of increasing interest for trading agents. For comparison purposes, a linear regression mid-term forecasting model is also tested. This alternative forecasting model is based on the ordinary least square (OLS) technique. Both the predictive trading strategy and the spot forecasting models have been successfully tested with historical data of the MIBEL, although they could be applied to other electricity markets.
Additionally, an original evaluation of the proposed predictive strategy is presented in this article. This evaluation is based on the comparison of the trading results achieved with the proposed predictive strategy with those obtained with two other reference strategies, defined as the "best" and the "worst" ones. These reference strategies assume knowing beforehand the actual values of the average spot price in the delivery period. They provide the inner limits of the scale where the RP of the predictive strategy performs.
The article is structured as follows: Section 2 presents the mid-term average spot price forecasting model and the trading strategy for agents in the futures market; Section 3 presents the results obtained for both buyers and sellers acting on the MIBEL and finally, Section 4 presents the conclusions.

Conditional Predictive Trading of Electricity Physical Futures
In this article, we present a novel approach for predictive trading and hedging of physical futures on electricity markets. It is applied to physical delivery futures, where the buyer or the seller will keep the product until the electricity delivery period (constant power, delivered in a predefined month). This commitment until the delivery is similar to that established in forward contracts, with the difference that for futures contracts, trading is carried out in a clearing house for derivatives exchange, without direct contact between buyer and seller. Instead, the forward contract also has a commitment to keep the product until delivery, but it is a direct contract between buyer and seller stipulated on the OTC market.
The objective is to decide, on each negotiation day (from a set of days before the delivery period), if the agent (buyer or seller) should buy/sell the electricity in advance through a futures contract, for a certain period ahead (days, weeks, months or years), or if he/she should wait for the spot market, on the eve of each day of the delivery period. At the time of negotiation, the price in the spot market is unknown; therefore, the decision about buying/selling the physical electricity through futures is based on an average of the futures prices at the time of negotiation and on the speculation about whether it will be higher or lower than the average spot price in the delivery period. The approach presented in this article uses a prediction that only utilizes electricity market price information at the time of negotiation in order to forecast the average spot price in the delivery period. Assuming that the forecast is a better indicator of the average spot price than the indicative price of the futures, we estimate a decision signal (buying/selling electricity through futures instead of waiting for spot market) based on the difference between the futures price and an average spot price forecast on the day of the decision, i.e., some time before the delivery. This section presents the proposed mid-term forecasting model and the predictive trading strategy illustrated with monthly futures (called "Futures" in the rest of the article), although it can be adapted to futures with different time frames (weekly, quarterly or yearly). Subsequently, the results obtained with the predictive strategy applied to a real electricity market (the Iberian Electricity Market, MIBEL) are presented in Section 3.

Proposed Mid-Term Forecasting Model of Average Spot Prices
The main component needed to set up a decision signal on a negotiation day n is provided by a mid-term forecasting model of the average spot price over the delivery period. Such forecasting model should only use electricity market price information at the time of negotiation, i.e., variables related to electricity prices on the days prior to the negotiation day. As mentioned above, we will describe the forecasting model for the trading of physical electricity monthly futures (Futures).
The forecasting model predicts the monthly average spot price for the delivery month. The model uses six input variables and is implemented with a single hidden layer feed-forward neural network trained with the ELM algorithm (referred to as the ELM neural network in this article), which randomly chooses the input weights and biases, thereby avoiding the tuning of other hidden layer parameters. We selected that model because it provided the best forecasting results with the cross-validation Energies 2020, 13, 3555 6 of 25 procedure within a set of models that included different types of feed forward neural networks with one hidden layer. The ELM algorithm was proposed by Huang et al. [42,43] as an alternative to gradient-based training strategies such as back-propagation and Levenberg-Marquardt algorithms. The ELM algorithm offers a very fast training process with minimal tuning requirements (number of neurons in the hidden layer). In recent years, the ELM algorithm has been successfully applied for forecasting purposes in the electricity field, with approaches for short-term load forecasting [44][45][46], generation forecasting in power plants based on renewable sources [47][48][49][50] and price forecasting in electricity spot markets [30,51,52].
The six input variables used by the forecasting model are the following: 1. DM: delivery month, with values 1 to 12. It represents the number of the month when the traded electricity will be delivered.

2.
LagD: period, in days, between the negotiation day (any day, i.e., n) and the last negotiation day before the start of the delivery period corresponding to DM.

3.
MF n,DM : Average monthly futures settlement price for delivery month DM in the last 7 days before negotiation day n. So, this input variable is calculated by (1): where MF p,DM represents the monthly futures settlement price for delivery month DM on the trading phase of day p, i.e., that established at the end of that phase on the previous negotiation day, and ND 7 corresponds to the number of futures negotiation days in the last 7 days. Observe that the MF p,DM values for nonmarket days (day p with no futures market) are not included in the summation.

4.
QF n,DM : Average quarterly futures settlement price for the quarter including the DM in the last 90 days before negotiation day n. This is calculated by (2), where the delivery quarter DQ must include the delivery month DM: where QF p,DM represents the quarterly futures settlement price for the quarter (including the DM) established at the end of the previous negotiation day and ND 90 corresponds to the number of futures negotiation days in the last 90 days. Observe that QF p,DM values for nonmarket days (day p with no futures market) are not included in the summation.

5.
S n : Average spot price in the last 7 days before day n; this is calculated by (3): where S p,h represents the spot price for day p hour h. Observe that the spot prices for day n were established on the previous day.
where MF n,DM represents the average monthly settlement futures price for delivery month DM in the last 7 days before negotiation day n and MF n−7,DM corresponds to its value 7 days before. The output variable,Ŝ n,DM , corresponds to the forecast of the monthly average spot price for the delivery month DM forecasted on day n.
As indicated above, the mid-term forecasting model is implemented with a single hidden layer feed-forward neural network trained with the ELM algorithm. In our work, the neural network has 6 nodes (neurons) in the input layer and only one node in the output layer (one output). The training dataset is composed by N samples, T R 6 is the input vector for the sample j; and o j is the desired output value for such sample (target value). The output of a neural network with L nodes in the hidden layer and linear activation function in the output node can be represented by (5): where a i = [a i1 , a i2 , . . . , a i6 ] T is weight vector that stores the weights between the input layer nodes and hidden layer node i; b i is the bias of hidden layer node i; β i is the weight between the hidden layer node i and the output layer; g is the activation function of the nodes in the hidden layer and a i ·x j represents the inner product of a i and x j . The Equation (5) can be represented, in compacted form, by (6): where H is the hidden layer output matrix, β is the output weights vector and O is the target vector. So, the matrix H and the vectors β and O can be represented by (7): In the training process of the single hidden layer feedforward neural network with the ELM algorithm, the bias of each neuron (node) in the hidden layer and the weights connecting input and hidden layers are randomly chosen (i.e., the a i and b i values). From these random values, the matrix H is determined according the input samples, and the training process is converted into solving, by least squares, the linear equations Hβ = O, to obtain β, what can be represented by (8): The least squares solution of (8) is estimated by (9): where [53] of the hidden layer output matrix.
Since the training process of the neural network does not need to adjust the values of the weights between the input layer and the hidden layer and the biases of the hidden layer nodes, the ELM algorithm generates an optimal solution in a fast training process while maintaining a great capacity of generalization [54]. Only the number of nodes or neurons in the hidden layer needs to be optimized in order to achieve the best forecasting performance.

Predictive Trading Strategy of Physical Futures
In the case of trading physical Futures, the general objective is to hedge against the volatility of spot market prices. Thus, the objective, at the time of negotiation, is to choose between buying/selling Energies 2020, 13, 3555 8 of 25 electricity through Futures several days, weeks or months in advance or waiting and accepting the spot market price settled for every hour of every day in the delivery period. Buyers and sellers have opposite strategies for this physical product (e.g., if the seller succeeds in selling electricity through Futures, then the buyer succeeds in waiting for the spot market).
Assuming that, at the negotiation moment, the monthly average spot price forecast is closer to the actual value (the ex post monthly average spot price) than the Futures price for the delivery period, we can define a signal indicating whether it is more favourable to trade in the futures market. This is the basis for deciding whether to buy/sell electricity through Futures or wait for the spot market. As will be presented in the analysis of the forecasting results, this assumption is not always true. However, if the error of the monthly average spot price forecast is lower than the error obtained by using the Futures price as this forecast then the assumption is more true than false, and there is a chance of success.
Based on this assumption, the predictive trading strategy for buyers and for sellers is defined by (10) and (11), respectively.
for buyers ifŜ n, DM < MF n,DM then buy on spot market ifŜ n,DM ≥ MF n,DM then buy on futures market (10) for sellers ifŜ n, DM ≤ MF n,DM then sell on futures market ifŜ n,DM > MF n,DM then sell on spot market (11) Obviously, in order to participate in the futures market, the seller should submit a bid at a price equal to the Futures settlement price established at the end of the previous trading day. On the other hand, the buyer should submit an ask with the same price value if he/she wants to participate in the futures market following the proposed strategy. The submission of a bid or an ask does not guarantee the realization of the transaction, because an opponent with an ask or bid is needed to close the contract. From a seller's perspective, if a Futures contract is closed, the final price will be at least equal to that submitted in the bid; from a buyer's perspective, if the Futures contract is closed, the final price will be at most equal to that submitted in the ask.
The success of the trading approach could be evaluated by calculating the ex post Risk Premium (RP) under the worst-case scenario, i.e., when the Futures contract is closed with the price submitted in the bid (for a seller) or in the ask (for a buyer). Therefore, the RP assessment test used in this article is the most pessimistic assessment, considering the changes in market uncertainty that may occur on the trading day. In any case, the RP evaluates the premium associated with the option of buying/selling electricity through Futures instead of waiting for the spot market.
Therefore, at each negotiation moment, the RP resulting from the proposed predictive trading strategy can be formulated as a conditional buying/selling only on some negotiation days n, and only for some delivery months, for which a favourable signal is obtained. Thus, RPs for buyers and sellers using the predictive trading strategy (RP buyers,predictive n,DM and RP sellers,predictive n,DM ) are calculated by (12) and (13), respectively: for buyers for sellers where S DM represents the average actual (ex post) value of the spot price in the month DM. Note that in this article, we calculate the RP as a logarithmic difference, as in [14,55]. Additionally, as we mentioned in Section 1.2, we have defined the RP for buyers with the opposite sign to the RP for sellers. With such sign convention for the RP formulation, a positive value represents always profits or savings for the agent and a negative value represents losses. If the agent does not buy/sell through futures and does it on the spot market, then the associated RP is null.
The computer results will be evaluated by comparing the RP values resulting from the predictive strategy, with the conventional ex post RP values obtained from Equations (14) and (15), i.e., simply buying/selling electricity through Futures on any negotiation day. The main difference between the predictive and the conventional strategies is that the predictive strategy uses a conditional decision, about whether or not to buy/sell, instead the conventional strategy assumes always buying/selling electricity through Futures instead of trading in the spot market. Another possible trading strategy for the agents is to trade exclusively on the spot market, which corresponds to a null RP.
for buyers RP buyers,conventional n,DM for sellers RP sellers,conventional Thus, the benchmark measure of ex post RP that will be labelled in the charts (shown later in Section 3.2) as RP Futures, consists in always buying/selling electricity through Futures, on every negotiation day (n), for all holding periods or maturities (LagM from 1 to 6) and at the last settlement price, i.e., MF n,DM .
In order to evaluate the scale for the ex post RP using the predictive trading strategy, we analysed the values achieved with the best and the worst possible trading strategies, defined as those resulting from knowing beforehand the actual values of the monthly average spot price in the delivery month. Thus, the ex post RP for the best possible strategy (for buyers and sellers) is calculated using (16) and (17), respectively, which always leads to a nonnegative RP value, i.e., profits for the agents.
for buyers for sellers We can also evaluate the worst possible strategy, which always leads to a negative RP value (losses for the agents), that is symmetrical to the one obtained with the best possible strategy. The worst strategy and the associated ex post RP (for buyers and sellers) are calculated using (18) and (19), respectively. for buyers for sellers In the "testing" period, the RP is computed for each negotiation day and each LagM (LagM represents the maturity or holding period in months with values from 1 to 6). In order to summarize the results for the entire testing period, an average value of the RP indicator, RP, is used. The average ex post RP for both perspectives (buyers and sellers) is calculated for an overall evaluation of the performance of the above-defined strategies using (20) for a maturity and using (21) for all maturities. These RP Indexes are applied for the four possible strategies (predictive, conventional, best and worst strategies) and for both types of agents, i.e. buyers and sellers.
where RP agents,strategy n,DM represents the RP for negotiation day n corresponding to the delivery month DM following the trading strategy (predictive, conventional, best and worst) and for the agents (buyers and sellers); DM i is the first delivery month and DM f the last delivery month in the testing period; ni LagM,DM is the first negotiation day and n f LagM,DM is the last negotiation day for the delivery month DM with a maturity LagM; and ND LagM,DM is the total number of negotiation days for delivery month DM with a maturity LagM. Observe that only the RP agents, strategy n,DM values corresponding to negotiation days are included in the summation.
The overall balance of the ex post RP can be computed by summing the profits/losses for the buyers/sellers. Thus, the following overall RPs are defined using (22)

Results of Conditional Predictive Trading of Physical Futures
The new conditional predictive trading strategy of electricity physical futures presented in this article has been successfully tested with historical data of an electricity market, the Iberian Electricity Market (MIBEL). The available data from the MIBEL corresponded to 56 months, from January 2015 to August 2019. These data include the prices of monthly futures for base-load in the Spanish zone (called SPEL base monthly futures) and the prices established in the daily spot market in the same period. The data were divided into two sets: The training dataset with data corresponding to the period from January 2015 to August 2018, with a total of 5060 records, and the testing dataset with data from September 2018 to August 2019, with a total of 1004 records. Section 3.1 gives the results corresponding to the proposed forecasting model in the prediction of the monthly average spot prices (described in Section 2.1). Afterwards, Section 3.2 provides the results corresponding to the predictive trading strategy of physical Futures in the MIBEL (described in Section 2.2).

Results of the Forecasting Model in the Prediction of the Monthly Average Spot Price
The ELM neural network, applied as the implementation of the mid-term forecasting model of the monthly average spot price, used the six input variables described in Section 2.1 and the sigmoidal activation function. Initially, the number of nodes (neurons) in the hidden layer was selected by a fivefold cross-validation procedure with the data from the training dataset. An extensive set of ELM neural networks with number of neurons in the hidden layer ranging from 6 to 100 was analysed. The ELM neural network with the lowest average mean square error with the 5 folds had 25 neurons in the hidden layer, and afterwards, it was trained with the complete training dataset.
After the training, the proposed forecasting model was applied for the forecast of the monthly average spot price in the testing period, from September 2018 to August 2019. This was a particularly difficult period with spot and Futures prices exceptionally high in September 2018 compared with the values belonging to the training dataset. Figure 1 plots the monthly average Futures price (for different holding lapses, i.e., LagM values) and the daily average spot prices for the training and testing periods (the first months of the training period are not represented in the figure). We can see that shorter LagM values correspond to faster reaction to the variation of the spot price. Thus, the shorter maturities are more reactive and more difficult to forecast. The influence of the average spot price input variable is more important for these shorter maturities. The statistics for the training and testing periods are shown in Table 1 and Table 2. The testing period shows higher maximum, minimum and mean values for the Futures price. The standard deviation (SD) is slightly higher for the testing period. The decrease in the SD for longer maturities (large LagM) indicates high volatility for shorter maturities. The forecast was carried out for each day of the testing period, providing six forecasts each day for the corresponding six types of holding periods, LagM, with a forecasting horizon from 1 to 6 months. Figure 2 and Figure  The statistics for the training and testing periods are shown in Tables 1 and 2. The testing period shows higher maximum, minimum and mean values for the Futures price. The standard deviation (SD) is slightly higher for the testing period. The decrease in the SD for longer maturities (large LagM) indicates high volatility for shorter maturities. The forecast was carried out for each day of the testing period, providing six forecasts each day for the corresponding six types of holding periods, LagM, with a forecasting horizon from 1 to 6 months. Figures 2 and 3 plot the ex post monthly average spot price, S DM (average actual spot price in the delivery month DM), labelled as "Ex post monthly average spot"; the forecast of the monthly average spot price,Ŝ n,DM , labelled as "Spot forecast"; and the monthly futures settlement price for negotiation day n, labelled as "Futures," for each LagM (1 to 6). For delivery months in the second half of 2018, the forecast generally leads to a negative error. In the following first half of 2019, the Futures and spot prices were more regular in relation to their historical behaviours and, consequently, both were relatively close. The forecast error is lower for smaller holding periods or maturities (LagM) but it increases significantly with larger values. Note that for the negotiation days in the first half of each month for the maturity LagM = 6 (Figure 3c), there is no forecast. This is because the values of the input variable . MF n,DM are not available for those days, i.e., the negotiation of Futures with a 6-month maturity (LagM = 6) begins on the first trading day of each month and 14 days have to elapse to obtain the value of such input variable because Futures trading for the corresponding delivery month has not taken place before. The root mean square error (RMSE), in relation to the ex post (actual) monthly average spot price (S DM ) was computed for the forecast of the monthly average spot price (Ŝ n,DM ) according to the expression (26). Furthermore, for comparative purposes, the RMSE was computed by (27) using the Futures settlement price (MF p,DM ) as "forecast": where RMSE Spot price forecast LagM represents the root mean square error of the spot forecast for each maturity LagM; RMSE Futures price LagM represents the root mean square error of the average Futures settlement price for each maturity LagM; S DM is the average actual spot price in the delivery month DM; DM i is the first delivery month and DM f the last delivery month in the testing period; ni LagM,DM is the first negotiation day and n f LagM,DM is the last negotiation day for the delivery month DM with a maturity LagM and ND LagM,DM is the total number of negotiation days for delivery month DM with a maturity LagM. Note that only negotiation days n are included in the summations.
The RMSE results, plotted in Figure 4, show a general better proximity (lower RMSE) of the monthly average spot price forecast than of the Futures settlement price. By comparing the RMSE values ("All" in Figure 4), we conclude that the monthly average spot price forecast improves the prediction achieved using the Futures settlement price from 7.95 to 6.12 €/MWh (RMSE values). As expected, the error of the Futures settlement price is lower for shorter LagM (holding period or forecasting horizon). The monthly average spot price forecast improves the RMSE for almost all the six types of maturities, with the exception of LagM = 1, where the Futures settlement prices are very close to the ex post monthly average spot prices. These RMSE values summarize the observations shown in Figures 2 and 3 of the three price features (ex post monthly average spot price, forecast of the monthly average spot price and Futures settlement price) throughout the testing period.

Results of the Predictive Trading Strategy of Physical Futures with the ELM Neural Network Model
In the testing period (last 4 months of 2018 and first 8 months of 2019), the Futures settlement price was generally higher than the ex post monthly average spot price. For this reason, the sellers

Results of the Predictive Trading Strategy of Physical Futures with the ELM Neural Network Model
In the testing period (last 4 months of 2018 and first 8 months of 2019), the Futures settlement price was generally higher than the ex post monthly average spot price. For this reason, the sellers

Results of the Predictive Trading Strategy of Physical Futures with the ELM Neural Network Model
In the testing period (last 4 months of 2018 and first 8 months of 2019), the Futures settlement price was generally higher than the ex post monthly average spot price. For this reason, the sellers obtained large economic profits (positive ex post RP) and the buyers suffered economic losses (negative ex post RP). Obviously, the amount of profits was equal to the amount of losses.
The profits or losses for buyers and sellers are perfectly symmetrical for the benchmark measure of ex post RP (labelled as RP Futures in the charts and the dotted lines presented later in Figures 5  and 6). This observation is not valid for the proposed predictive strategy based on forecast information (labelled as RP Forecast in the charts and the thin lines presented later in both figures). The RP equal to zero means a decision of waiting for the spot market, when the expectation is to lose (economic losses); otherwise, if the expectation is to earn (economic profits), then the RP can take positive or negative values for buying electricity through Futures (this defined behaviour can be observed in the charts of RP evolution throughout the testing period). In Figure 5a, for LagM = 1, we can see that most of the time the sellers have a positive RP value and the buyers have a negative RP value for both trading strategies, predictive and conventional ones. The predictive strategy for sellers (represented by the black thin line) succeeded most of the time (RP ≥ 0). However, on some negotiation days in October and November 2018, the predictive strategy failed because it wrongly suggested selling in the spot market instead of selling electricity through Futures. In February and March 2019, the predictive strategy for sellers did not predict that selling electricity through Futures would become a wrong option for some weeks. However, in June 2019, the predictive strategy succeeded when it suggested selling on the spot market instead of selling electricity through Futures. For buyers, the predictive strategy (represented by the red thin line) failed  Situations of successes or failures of the predictive strategy can be observed in the charts for other maturities (LagM = 2 to 6 in Figure 5 and LagM = 4 to 6 in Figure 6 Figure 6), the predictive strategy pointed to waiting for the spot market for buyers and to selling through Futures for sellers, for most of the period.
The average ex post RP from the perspectives of buyers (Figure 7) or sellers (Figure 8), and for each maturity LagM, was calculated in order to evaluate the overall performance of the trading strategies. In Figure 7, the average ex post RP was always negative from the perspective of buyers following the conventional strategy (labelled as "RP Futures (Buyers)"), and it remained negative even with the conditional predictive strategy (labelled as "RP Forecast (Buyers)"). However, with such predictive strategy, the losses decreased very significantly (RP value from −11.2% to −0.3% for all maturities, "All" in Figure 7), with a very significant improvement in each maturity. Observe that red and blue lines in the chart (Figure 7) represent the best and worst possible RP values (best and worst strategies). Then, for the maturities, Figure 7 clearly allows to observe the significant improvements of the losses, provided by the predictive strategy, in the inner scale limits delimited  Figures 5 and 6 present the four RP indicators throughout the testing period, in six separated subfigures, one for each LagM maturity. The dotted lines represent the RP of the benchmark or "conventional" strategy, red is the RP for buyers (labelled "RP Futures (Buyers)") and black is the RP for sellers (labelled "RP Futures (Sellers)"). The thin lines represent the RP achieved following the conditional predictive strategy, red is the RP for buyers (labelled "RP Forecast (Buyers)") and black is the RP for sellers (labelled "RP Forecast (Sellers)"). When the thin line follows the dotted line, it means that the predictive strategy is aimed at buying/selling electricity through Futures, whereas when the thin line is zero, it means that the predictive strategy is aimed at buying/selling on the spot market. The agent, by following the predictive strategy, succeeds if the RP thin line value is positive and loses if the RP thin line value is negative. The black and red thin lines never overlap, i.e., if one of the them presents a zero value, then the other shows a positive or negative value.
In Figure 5a, for LagM = 1, we can see that most of the time the sellers have a positive RP value and the buyers have a negative RP value for both trading strategies, predictive and conventional ones. The predictive strategy for sellers (represented by the black thin line) succeeded most of the time (RP ≥ 0). However, on some negotiation days in October and November 2018, the predictive strategy failed because it wrongly suggested selling in the spot market instead of selling electricity through Futures. In February and March 2019, the predictive strategy for sellers did not predict that selling electricity through Futures would become a wrong option for some weeks. However, in June 2019, the predictive strategy succeeded when it suggested selling on the spot market instead of selling electricity through Futures. For buyers, the predictive strategy (represented by the red thin line) failed in October and November 2018 because it indicated to buy electricity through Futures, and failed again in February and March 2019 suggesting to wait for the spot market. In the other 8 months, the predictive strategy for buyers rightly pointed to buy electricity on the spot market.
Situations of successes or failures of the predictive strategy can be observed in the charts for other maturities (LagM = 2 to 6 in Figure 5 and LagM = 4 to 6 in Figure 6 Figure 6), the predictive strategy pointed to waiting for the spot market for buyers and to selling through Futures for sellers, for most of the period.
The average ex post RP from the perspectives of buyers (Figure 7) or sellers (Figure 8), and for each maturity LagM, was calculated in order to evaluate the overall performance of the trading strategies. In Figure 7, the average ex post RP was always negative from the perspective of buyers following the conventional strategy (labelled as "RP Futures (Buyers)"), and it remained negative even with the conditional predictive strategy (labelled as "RP Forecast (Buyers)"). However, with such predictive strategy, the losses decreased very significantly (RP value from −11.2% to −0.3% for all maturities, "All" in Figure 7), with a very significant improvement in each maturity. Observe that red and blue lines in the chart (Figure 7) represent the best and worst possible RP values (best and worst strategies). Then, for the maturities, Figure 7 clearly allows to observe the significant improvements of the losses, provided by the predictive strategy, in the inner scale limits delimited by the best and worst possible strategies, (−11.7% and 0.5%). Furthermore, for buyers, the predictive strategy led to results very close to the ones of the best strategy.
Energies 2020, 13, x FOR PEER REVIEW 18 of 25 by the best and worst possible strategies, (−11.7% and 0.5%). Furthermore, for buyers, the predictive strategy led to results very close to the ones of the best strategy. From the perspective of sellers (Figure 8), the predictive strategy ("RP Forecast (Sellers)") was extremely favourable with positive RP value in each maturity (LagM), very slightly lower than the ones of the conventional strategy ("RP Futures (Sellers)") and of the best strategy ("RP Best (Sellers)"). Observe that the RP of sellers for all maturities was 10.9% ("All" in Figure 8), which indicates significant overall profit for sellers under the predictive strategy. Thus, for sellers, the predictive strategy led to profits (in the maturities) very close to the ones of the best strategy. Lastly, observe that the decreases of sellers' profits are much less significant than the improvement in the decreases of buyers' losses.
In general, the proposed predictive trading strategy, using the monthly average spot price forecast, achieved gains or losses depending on several volatility conditions. We concluded that the predictive strategy has interesting potential as a hedging strategy. In the testing period (a representative year with special volatility perturbations in spot and Futures prices), the conditional predictive strategy provides excellent hedging against losses for buyers and it slightly reduces the profits for sellers. In Figure 7 and Figure 8, in an overall balance (RP for all maturities), the conditional predictive strategy provides a positive balance of 10.6% (overall benchmark strategy of buying From the perspective of sellers (Figure 8), the predictive strategy ("RP Forecast (Sellers)") was extremely favourable with positive RP value in each maturity (LagM), very slightly lower than the ones of the conventional strategy ("RP Futures (Sellers)") and of the best strategy ("RP Best (Sellers)"). Observe that the RP of sellers for all maturities was 10.9% ("All" in Figure 8), which indicates significant overall profit for sellers under the predictive strategy. Thus, for sellers, the predictive strategy led to profits (in the maturities) very close to the ones of the best strategy. Lastly, observe that the decreases of sellers' profits are much less significant than the improvement in the decreases of buyers' losses.
In general, the proposed predictive trading strategy, using the monthly average spot price forecast, achieved gains or losses depending on several volatility conditions. We concluded that the predictive strategy has interesting potential as a hedging strategy. In the testing period (a representative year with special volatility perturbations in spot and Futures prices), the conditional predictive strategy provides excellent hedging against losses for buyers and it slightly reduces the profits for sellers. In Figures 7  and 8, in an overall balance (RP for all maturities), the conditional predictive strategy provides a positive balance of 10.6% (overall benchmark strategy of buying Futures is 0% by definition), which is quite good compared with the best possible ex post strategy that could obtain a maximum of 12.2%. These percentage values can be calculated by (28) and (29) using the results from Figures 7 and 8. of buyers' losses.
In general, the proposed predictive trading strategy, using the monthly average spot price forecast, achieved gains or losses depending on several volatility conditions. We concluded that the predictive strategy has interesting potential as a hedging strategy. In the testing period (a representative year with special volatility perturbations in spot and Futures prices), the conditional predictive strategy provides excellent hedging against losses for buyers and it slightly reduces the profits for sellers. In Figure 7 and Figure 8, in an overall balance (RP for all maturities), the conditional predictive strategy provides a positive balance of 10.6% (overall benchmark strategy of buying Futures is 0% by definition), which is quite good compared with the best possible ex post strategy that could obtain a maximum of 12.2%. These percentage values can be calculated by (28) and (29) using the results from Figure 7 and Figure 8.
The overall RP of the predictive strategy for all maturities was 10.6% (the limits were −12.2% and 12.2%) which showed that this RP was quite a good (positive) indicator. Thus, the set of reference trading strategies ("best," "worst" and "conventional," formulated in Section 2.2) were efficient for the evaluation of the performance of the proposed predictive trading strategy.

Results of the Predictive Trading Strategy of Physical Futures with a Simpler Forecasting Model
As a benchmark, a simpler forecasting model than the ELM neural network model was selected to be used with the predictive trading strategy. This benchmark model was an ordinary least square forecasting model (OLS model), whose coefficients were adjusted with the data corresponding to the training dataset (using the six input variables described in Section 2.1). This model obtained worse forecasting results than the ELM neural network model in the initial selection phase with the cross-validation procedure, i.e., the average RMSE with the five folds used as testing sets was 7.41 €/MWh for the ELM model and 8.10 €/MWh for the OLS model. However, because of its simplicity, the OLS model was used for comparative purposes in the trading of physical Futures in the MIBEL market.
The forecasting results obtained by the OLS model with the data from the testing dataset gave an RMSE value of 6.15 €/MWh, slightly worse than the 6.12 €/MWh value achieved by the ELM neural network model. Table 3 shows the RMSE values for the different maturities (LagM values). In addition to the forecasting accuracy measurements provided by the RMSE values, the Diebold-Mariano (D-M) test [56] was used to test the statistical significance of the forecasts provided by both models. The D-M test aims to test the null hypothesis of equal accuracy of the forecasts provided by both models against an alternative hypothesis of different accuracy. In order to ensure a better evaluation, we used the modified D-M [57], i.e., an improved version that corrects some shortcomings of the original D-M test. The mean square error was selected as the loss function. A complete information about the D-M test and the modified D-M (mD-M) test can be found in [56,57]. These tests have been used in numerous applications to compare the forecasts provided by two competing models. In the literature related to price forecasting in electricity markets, good application examples can be found in [58][59][60].
The one-sided version of the mD-M test was applied to determine whether the difference in accuracy of both models was statistically significant. In this test, the null hypothesis was the equality of accuracy, whereas the alternative hypothesis corresponded to a better accuracy of the forecasts provided by the ELM neural network model. Table 4 shows the results obtained with the mD-M test, the statistic values and the p-values for a significance level of 5%. Table 4 shows that the forecasts provided by the ELM neural network model were significantly better for maturities of 1 and 2 months (p-values under the significance level imply the rejection of the null hypothesis). Note that the results of this test also suggested a significant difference in accuracy for the maturities of 4 and 5 months, but in this case, favourable to the OLS model. On the other hand, the results obtained with the predictive trading strategy using the forecasts of the OLS model corresponded to a RP value of 0.0% for buyers and a value of 11.2% for sellers. Note that the RP results obtained using the forecasts of the OLS model were slightly better that the achieved using the forecasts of the ELM neural network model. It should be noted that the forecasted values provided by the OLS model were lower than the last settlement Futures prices throughout the entire testing period. Since the decision signal of buying/selling through Futures is based on the comparison of the output of the forecasting model and the last settlement Futures price, this signal was not activated for buyers on any negotiation day of the testing period (obviously, for sellers it was activated every day). Therefore, the trading results for sellers obtained using the forecasts of the OLS forecasting model were similar to those obtained by the conventional strategy.
Furthermore, the trading results obtained using the OLS forecasting model should be equal to those obtained with any other forecasting model, independently of its forecasting accuracy, under the condition that its forecasts were always lower than the last Futures settlement prices. Consequently, it can be inferred that the success of the predictive trading strategy presented in this article does not only depend on the accuracy of the mid-term monthly average spot price forecasting model but also on the fact that there is an ex ante decision for either buying/selling on the futures market or on the spot market.

Conclusions
This article presents a novel trading strategy for physical monthly futures (Futures) to help electricity agents (buyers and sellers) decide between buying/selling on spot markets or respectively buying/selling on the derivatives markets, several months in advance. The predictive strategy is essentially based on the forecast provided by a mid-term forecasting model of the monthly average spot price, which is also presented in this article. The strategy takes advantage of the forecasting model predictions to define a decision signal (buy/sell electricity through Futures instead of waiting for the spot market). The signal is generated at the decision time, when the forecast of the monthly average spot price is carried out and the last Futures settlement prices are available, several months before the clearing of the spot prices.
The trading strategy only requires access to the electricity derivatives market. This access could be done with online platforms for trading electricity through futures or with any other technological mode of trading that could be established in the different countries or regions that have this type of market. The trading strategy can also be applied to forward electricity contracts on the OTC market. The proposed predictive trading strategy is applicable to any possible negotiation day, open for the trading of physical Futures several months in advance. The trading strategy was studied from the perspectives of electricity buyers and sellers.
The predictive strategy and the mid-term forecasting model of the spot price were successfully applied to the Iberian Electricity Market (MIBEL). They could also be adapted and used in other forward contracts and other electricity derivatives markets. Despite the difficulty in obtaining good performance for the results (forecast) of the monthly average spot price forecasting model, the computer results achieved were promising, proving that it is possible to predict monthly average spot prices with lower errors (deviations) than the ones corresponding to using the Futures settlement prices as the forecast.
The evaluation of the new predictive trading strategy uses the measures of ex post Risk Premium (RP) applied to the following strategies: The new conditional predictive trading strategy ("predictive"), the conventional trading strategy ("conventional"), the ex post best conditional trading strategy ("best"), and the ex post worst conditional trading strategy ("worst"). All RP measures are ex post because profits/losses are measured against the ex post monthly average spot price. The "best" and "worst" strategies lead to RP reference indicators, representing the inner limits of the scale where the RP of the "predictive" strategy performs. The "conventional" strategy is a realistic reference for evaluating the relative performance obtained for the "predictive" strategy. The "predictive," "best" and "worst" strategies are conditional strategies, where at each negotiation moment, the agent decides between trading in futures or on the spot markets. Conversely, for the "conventional" strategy, there is no conditional decision: buying/selling electricity on the futures market instead of on the spot market is always assumed. All the trading strategies ("predictive," "best," "worst" and "conventional") were tested for buyers and sellers. The sum of the RP of buyers and sellers for each trading strategy was used for the purposes of comparing the overall performance of the set of strategies for both perspectives jointly (of buyers and sellers). The average of these types of RP indicators was computed for the testing period (all delivery months) and for each futures maturity. The comprehensive use of these overall RP indicators for comparisons in this article constitutes a structured innovative evaluation approach.
The strategies were evaluated for 1 complete year, a period with exceptionally high price values for futures trading, what tended to economic losses for buyers and profits for sellers under the "conventional" strategy. The results obtained shown that the proposed predictive trading strategy achieved an overall benefit using any of the two mid-term monthly average spot price forecasting models tested in this article, although under a practical perspective, the OLS forecasting model could be preferred due to its easy use and implementation.
As a final conclusion of the article, it can be stated that there are advantages when using the new predictive trading strategy for negotiation in electricity futures markets. It can be effectively used for hedging operations in electricity markets.
Further research is underway in order to improve the benefits of the proposed trading strategy by addressing an ex ante approach and enhancing the forecasting model using more powerful forecasting techniques, a more exhaustive feature selection and using other influence variables external to electricity markets. Funding: This work was supported by the "Ministerio de Ciencia, Innovación y Universidades," Spanish Government, under the research project ENE-2016-78509-C3-3-P, and the European Regional Development Fund, EU.