Description
The report explains the modelling and trading of oil futures spreads in the context of a portfolio of contracts.
The European Journal of Finance, Vol. 14, No. 6, September 2008, 503–521
Trading futures spread portfolios: applications of higher order and recurrent networks Christian L. Dunis, Jason Laws and Ben Evans?
Centre for International Banking, Economics and Finance, Liverpool John Moores University, Liverpool, UK
This paper investigates the modelling and trading of oil futures spreads in the context of a portfolio of contracts. A portfolio of six spreads is constructed and each spread forecasted using a variety of modelling techniques, namely, a cointegration fair value model and three different types of neural network (NN), such as multi-layer perceptron (MLP), recurrent, and higher order NN models. In addition, a number of trading ?lters are employed to further improve the trading statistics of the models. Three different ?lters are optimized on an in-sample measure of down side risk-adjusted return, and these are then ?xed out-ofsample. The ?lters employed are the threshold ?lter, correlation ?lter, and the transitive ?lter. The results show that the best in-sample model is the MLP with a transitive ?lter. This model is the best performer out-of-sample and also returns good out-of-sample statistics.
Keywords: futures spreads; cointegration; trading ?lters; higher order networks; recurrent networks
1.
Introduction
The case for spread trading has been made by several academics most of whom call on reduced margin requirements1 or consistently tradable patterns2 to encourage interest in spread trading. However, these same papers fail to explicitly explain that with reduced margin comes reduced potential reward. The reason for the reduced margin when trading a spread is the reduced chance of large moves. This is because the two legs of the spread are highly correlated over the long-term and will therefore move in generally the same direction. Further consideration should be given to the additional trading costs incurred in trading spreads, this is because two contracts must be traded for each spread, therefore the trading costs are signi?cantly higher. This leads us to the question of not how can we predict the spread, but how can we produce pro?t from our predictions? With this in mind, a number of ?ltering techniques are investigated to further re?ne the ability of the decision models to produce pro?ts over and above transactions costs. These ?ltering techniques are the threshold ?lter and the correlation ?lter as developed by Dunis, Laws, and Evans (2005) and the transitive ?lter, which has been adapted from the initial work by Guégan and Huck (2004). In their work, they ?nd ‘it [transitivity] signi?cantly raises the proportion of the signs of the relative movements correctly guessed. The same approach is then applied on real data sets and similar conclusions obtained’. However, their results leave open two questions, they are ‘Testing this approach on other simulated and real data sets and
? Corresponding
author. Email: [email protected]
ISSN 1351-847X print/ISSN 1466-4364 online © 2008 Taylor & Francis DOI: 10.1080/13518470801890834 http://www.informaworld.com
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implementing it for trading strategies’ and ‘Extending the analysis to N -multivariate setting, with N > 3’. In this paper, we attempt to answer these questions by utilizing transitive forecasts in the context of trading a portfolio of spreads and also extend the universe of contracts to four. Resulting in six spread, or relative, forecasts. Motivation for this paper is derived from the work by Dunis, Laws, and Evans (2005), in which it is discovered, in the case of the Gasoline Crack spread, that the use of higher order neural networks (HONNs) produces pro?ts in excess of those achieved with the use of a standard multi-layer perceptron (MLP). This is worth investigating further in the context of a portfolio of spreads. Further, it was found that the use of the correlation ?lter generally does not produce better results than the un?ltered model; this ?nding lies in contradiction to Evans, Dunis, and Laws (2006), who ?nd that the correlation ?lter is a promising method of ?ltering spread trades. This paper is set out as follows: Section 2 details some of the relevant literature; Section 3 explains the data and methodology; Section 4 de?nes the decision models used; Section 5 de?nes the ?lters that have been employed; Sections 6 and 7 give the results and conclusions, respectively.
2.
Literature review
Spread trading was ?rst introduced formally into the ?nance literature by Working (1949), who investigated the effects of the cost of storage on pricing relationships. It was demonstrated that futures traders could pro?t from the existence of abnormalities in the pricing relationships between futures contracts of different expiry, an intra-commodity spread. Melamed (1981) gives further justi?cation for research into spread trading stating that ‘although spread trading has been used to speculate on the cost of carry between different futures contracts, spread trading also serves the functions of arbitrage and hedging, together with providing a vital source of market liquidity’ (Melamed 1981, 408). It is therefore surprising that although there has been interest in cash-futures arbitrage3 , inter- and intra-commodity spread trading has been largely ignored among the academic fraternity4 . Butterworth and Holmes (2003) state ‘an analysis of spread trading is important since it contributes to the economics of arbitrage and serves as an alternative to cash-futures arbitrage for testing for futures market ef?ciency’ (Butterworth and Holmes 2003, 783). In this paper, they test a fair value model on the FTSE250–FTSEMID100 spread. This fair value model is based on ‘arbitrage limits’ and takes a position on the spread when a limit is breached. They conclude that ‘while the inter-market spread is found to trade within its transactions cost limits on the majority of occasions, large and sustained deviations from fair value exist in both directions, resulting in the triggering of spread arbitrage transactions’ (Butterworth and Holmes 2003, 790). This statement does not indicate whether either a more sophisticated trading rule or an appropriate trading ?lter can capture the deviations from fair value. Studies such as Pruitt and White (1988), Sweeney (1988), and Dunis (1989) directly support the use of technical trading rules as a means of trading ?nancial markets. Trading rules such as moving averages, ?lters, and patterns seem to generate returns above the conventional buy and hold strategy. Nevertheless, Lukac and Brorsen (1990) carried out a comprehensive test of futures market trading. It was found that all but one of the trading rules tested generated signi?cantly abnormal returns. Sullivan, Timmerman, and White (1998) investigated the performance of technical trading rules over a 100-year period of the Dow Jones Industrial Average, they conclude that
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‘there is no evidence that any trading rule outperforms [the benchmark buy and hold strategy] over the sample period’ (Sullivan, Timmerman, and White 1998, 1683). With the increasing processing power of computers, rule-induced trading has become far easier to implement and test. Kaastra and Boyd (1996) investigated the use of neural networks (NNs) for forecasting ?nancial and economic time series. They concluded that the large amount of data needed to develop working forecasting models involved too much trial and error. On the contrary, Chen, Wagner, and Lin (1996) study the 30-year US Treasury bond using an NN approach. The results prove to be good with an average buy prediction accuracy of 67% and an average annualized return on investment of 17.3%. In recent years, there has been an expansion in the use of computer trading techniques, which has once again called into doubt the ef?ciency of even very liquid ?nancial markets. Lindemann, Dunis, and Lisboa (2004) suggest that it is possible to achieve abnormal returns on the Morgan Stanley High Technology 35 index using a Gaussian mixture NN trading model. Lindemann, Dunis, and Lisboa (2005) justi?ed the use of the same model to successfully trade the EUR/USD exchange rate, an exchange rate noted for its liquidity. Krishnaswamy, Gilbert, and Pashley (2000) attempt to show the development of NNs as modelling tools for ?nance. In turn, they cite valuable contributions from Bansal and Viswanathan (1993), Kryzanowski, Galler, and Wright (1993), Refenes, Zapranis, and Francis (1995), and Zirilli (1997) in the ?eld of stock market and individual stock prediction, proving that not only do NNs outperform linear regression models, but also NNs are ‘superior in dealing with structurally unstable relationships, notably stock market returns’, (Krishnaswamy, Gilbert, and Pashley 2000, 79). This research kick started the search for increasingly specialist NN architectures. Recurrent networks (RNNs) were ?rst developed by Elman (1990) and possess a form of error feedback. These networks are generally better than MLP networks but they do suffer from long computational times (Tenti 1996). However, according to Saad, Prokhorov, and Wunsch (1998), compared with other architectures, this should not matter a lot ‘RNN has the capability to dynamically incorporate past experience due to internal recurrence, and it is the most powerful network of the three in this respect… but its minor disadvantage is the implementation complexity’ (Saad, Prokhorov, and Wunsch 1998, 1468). HONNs were ?rst introduced by Giles and Maxwell (1987) and were called ‘Tensor Networks’, although the extent of their use in ?nance is limited. Knowles et al. (2005) show that despite shorter computational times and limited input variables on the EUR/USD time series ‘the best HONN models show a pro?t increase over the MLP of around 8%’ (Knowles et al. 2005, 7). A signi?cant advantage of HONNs is detailed in Zhang, Xu, and Fulcher (2002). ‘HONN models are able to provide some rationale for the simulations they produce and thus can be regarded as “open box” rather then “black box”. Moreover, HONNs are able to simulate higher frequency, higher order non-linear data, and consequently provide superior simulations compared to those produced by arti?cial neural networks’ (Zhang, Xu, and Fulcher 2002, 188). In the context of spreads, Dunis, Laws, and Evans (2005) found that for the Gasoline Crack spread, ‘the HONN outperformed the MLP out-of-sample before transactions costs despite shorter computational time and limited variables’ (Dunis, Laws, and Evans 2005, 522). This paper investigates the use of MLP, RNN, and HONN models as trading tools in the oil spread market, and these are described more fully in Sections 4.2, 4.3, and 4.4, respectively. These models are also benchmarked against a more traditional Cointegration fair value model. Further, this paper also investigates the use of a number of ?ltering techniques to further enhance the performance of the forecasting models. These ?lters are detailed in Section 5.
506 3.
C.L. Dunis et al. Data and methodology
Following methodology from Butterworth and Holmes (2002) and Dunis, Laws, and Evans (2005), the spread returns series is calculated in the following way: Rs = Leg1,t ? Leg1,t?1 Leg1,t ? Leg2,t ? Leg2,t?1 Leg2,t (1)
where Leg1,t is the price of Leg1 at time t 5 , Leg1,t?1 the price of Leg1 at time t ? 16 , Leg2,t the price of Leg2 at time t, and Leg2,t?1 the price of Leg2 at time t ? 1. This convention allows for the calculation of annualized returns and annualized standard deviation to be done in the usual way. 3.1 Data set For the cointegration fair value model, the data have been split into two subsections (see Table 1). The ?rst subset (the in-sample subset) is used to test and optimize the models. The second subset (the out-of-sample subset) is used as an unseen data set and is used to test our optimized models. In the case of the NN models, and in order to avoid over?tting, the data have been split into three subsets, as is standard in the literature (e.g. Kaastra and Boyd 1996; Lindemann, Dunis, and Lisboa 2005) (see Table 2). The NN models are trained slightly differently to standard econometric models. The training data set is used to train the network, the minimization of the error function being the criterion optimized. The training of the network is stopped when the pro?t on the test data set is at maximum positive return. This model is then traded on the validation subset, which for comparison purposes is identical to the out-of-sample data set used for the fair value cointegration model. This technique restricts the amount of noise that the model will ?t, while also ensuring that the structure inherent in the training and test subsets is modelled. Further explanation of this is contained in Lindemann, Dunis, and Lisboa (2005). Table 3 shows the details of the time series used to form the portfolio. The series shown in Table 3 are then combined to form the following spreads: (1) Brent crude vs. WTI crude; (2) Brent crude vs. unleaded gasoline;
Table 1. In-sample and out-of-sample dates. Subset In-sample Out-of-sample Purpose Optimize model Test model Period 3 January 1995–25 July 2002 26 July 2002–1 January 2005 Data points 2175 435
Table 2. In-sample and out-of-sample dates (for NN training). Subset 1 (Training) 2 (Test) 3 (Validation) Purpose Optimize model Stop model optimization Test model Period 3 January 1995–29 December 2000 1 January 2001–25 July 2002 26 July 2002–1 January 2005 Data points 1740 435 435
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Table 3. Time series data contract speci?cs. Fixing times (GMT) 17:30 17:30 17:30 17:30 Bid-ask spreada (%) 0.0960 0.0560 0.1004 0.1241
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Time series WTI crude Brent crude Unleaded gasoline Heating oil
Currency US$ US$ US$ US$
Exchange NYMEX IPE NYMEX NYMEX
Note: a Source of bid-ask spreads: www.sucden.co.uk, calculation of this is further explained in Section 3.3.
(3) (4) (5) (6)
Brent crude vs. heating oil; WTI crude vs. unleaded gasoline; WTI crude vs. heating oil; Unleaded gasoline vs. heating oil.
It is evident from Table 3 that these time series can be traded as spreads because they are denoted in the same currency and each leg of the spread has an identical ?xing time as the opposing leg. Further the trading of these spreads on NYMEX7 allows for a reduced margin of between 87.5% for WTI–Brent and 75% for Brent–heating oil. The portfolio, which has been traded, is an equally weighted portfolio of the six spreads shown above; therefore, each spread return is given a one-sixth weighting in the portfolio. Appendix 1 shows the daily returns histogram of the spread time series; it is evident from Appendices 1a–f that the distributions are severely non-normal. The series statistics show evidence of excess kurtosis (ranging from 11, in the case of the Brent vs. heating oil spread, to 54 in the case of the Brent vs. WTI spread) and in many cases signi?cant amounts of skewness. This is markedly more severe than the traditional ‘fat tails’ observed on stock or index returns. Appendices 2a and b show the daily return histograms of the FTSE100 and Brent crude over the same time period, respectively. It can be noted that the levels of kurtosis are much lower. 3.2 Rollovers
Using non-continuous time series brings a unique problem, as any long-term study will require a continuous series. If a trader takes a position on a futures contract, which subsequently expires, he can take the same position on the next available contract. This is called rolling forward. The problem with rolling forward is that two contracts of different expiry but same underlying may not (and invariably do not) have the same price. When the roll forward technique is applied to a futures time series, it will cause the time series to exhibit periodic blips in the price of the contract. Although the cost of carry (which actually causes the pricing differential) can be mathematically taken out of each contract, this does not leave us with a precisely tradable futures series. In this study, as we are dealing with futures spreads, we have rolled forward both contracts on the same day of each month (irrespective of exact expiry dates). The cost of carry, which is the cause of the price difference between the cash and futures price, is determined by the cost of buying the underlying in the cash market now and holding until futures expiry. As the cost of storage of both underlying is similar, they will approximately offset each other. We are left with a tradable time series with no cost of carry effect.
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C.L. Dunis et al. Transactions costs
The bid-ask spread is an average of four intra-day bid-ask spreads and is presented here as a percentage of the underlying price. The bid-ask spread of each contract, as a percentage of the contract price, can be seen in Table 3. The cost of trading the spread is the total cost of trading one leg in addition to the total cost of trading the other leg. For example, the cost of trading the WTI–Brent spread is (0.0960% + 0.0560%) = 0.1520%8 per round trip. An additional cost has been deemed appropriate due to the rollover of the futures contracts; this cost is incurred only if a trade is held unchanged. This cost is equal to one round trip trade on the spread. 4. Trading decision models The trading decision models have each been used to trade the portfolio of futures spreads. Four trading models have been used, and they are described below. 4.1 Fair value cointegration trading rule
The fair value model that has been used in this paper is the cointegration fair value model as used by Evans, Dunis, and Laws (2006). The model is based on the Johansen (1988) cointegration test, which tests that ? = 0 in Equation (2) below: ?t = ? ?t?1 + ?t where ?t is the residual of the cointegrating relationship. This can be extended to a trading model as shown below: If ?t < 0 then go long the spread, until ?t = 0 is regained; If ?t > 0 then go short the spread, until ?t = 0 is regained. For each spread, the relevant value of ?t was calculated as shown below, over the in-sample period (Table 1), using the Johansen (1988) method: ?tbg = Brent ? (34.63413 gasoline) ? 2.10E?04 trend ?tbw = Brent ? (0.969017 WTI) ? 2.18E?05 trend ?tbh = Brent ? (32.48161 heating oil) ? 6.59E?04 trend ?tgw = Gasoline ? (0.028433 WTI) ? 8.06E?06 trend ?tgh = Gasoline ? (0.947192 heating oil) ?thw = Heating oil ? (0.030124 WTI) ? 2.18E?04 trend where trend is a linear trend included, where signi?cant, to account for any in?ationary effects. 4.2 Multi-layer perceptron (2)
The most basic type of neural network regression (NNR) model, which is used in this paper, is the MLP. As explained in Lindemann, Dunis, and Lisboa (2004), the network has three layers; they are the input layer (explanatory variables), the output layer (the model estimation of the time series), and the hidden layer. The number of nodes in the hidden layer de?nes the amount of complexity that the model can ?t. The input and hidden layers also include a bias node, similar to the intercept for a standard regression, which has a ?xed value of 1.
The European Journal of Finance The network processes information as shown below:
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(1) The input nodes contain the values of the explanatory variables (in this case, 10 lagged values of the spread). (2) These values are transmitted to the hidden layer as the weighted sum of its inputs. (3) The hidden layer passes the information through a non-linear activation function and, if the calculated value is above a threshold value, onto the output layer. The connections between neurons for a single output neuron in the net are shown in Figure 1. Where: xt[n] (n = 1, 2, . . . , k + 1) h[m] (m = 1, 2, . . . , m + 1) t yt ˜ uj k and wj are the model inputs (including the input bias node) at time t. are the model inputs (including the hidden bias node). is the MLP model output (the estimate of the change in the spread). is the network weights is the sigmoid transfer function: S(x) = 1/1 + ex is a linear function: F (x) = The error function to be minimized is: E uj k , wj = 1 T ˜ (yt ? yt (uj k , wj ))2
i
xi
with yt (the change in the spread) being the target value. 4.3 Recurrent NN In contrast to an MLP, an RNN includes an additional input function. This additional input function is the previous state of the network and because of this, the network embodies a form of short-term memory in the training process. The name given to this memory process is activation feedback. An RNN with activation feedback embodies short-term memory, see, for example, Elman (1990). The advantages of using RNNs over feedforward networks, for modelling non-linear time series, have been well documented in the past, see, for example, Adam, Zarder, and Milgram (1993). However, as described in Tenti (1996), ‘the main disadvantage of RNNs is that they require substantially more connections, and more memory in simulation, than standard backpropagation
Figure 1. A single output, fully connected MLP model.
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Figure 2. Architecture of Elman or Recurrent Neural Network.
networks’ (Tenti 1996, 159), thus resulting in a substantial increase in computational time. However, having said these RNNs can yield better results in comparison with simple MLPs due to the additional memory inputs. The architecture of a simple RNN is shown in Figure 2. The state/hidden layer shown in Figure 2 is updated with external inputs, as in the simple MLP (Section 4.2) but also with activation from previous forward propagation, shown as ‘Previous State’ above. In short, the RNN architecture can provide more accurate outputs because the inputs are (potentially) taken from all previous values. The Elman network in this study uses the sigmoid transfer function, error function, and linear function as described for the MLP architecture in Section 4.2. This has been done in order to be able to draw direct comparisons between the architectures of both models. 4.4 Higher order NN
HONNs were ?rst introduced by Giles and Maxwell (1987) and were called ‘Tensor Networks’. Although they have already experienced some success in the ?eld of pattern recognition and associative recall9 , their use in ?nance is quite limited. The architecture of a three-input secondorder HONN is shown in Figure 3, where: are the model input. is the transfer sigmoid function: S(x) = 1/1 + e?x is the linear function: F (x) = i xi HONNs use joint activation functions; this technique reduces the need to establish the relationships between inputs when training. Furthermore, this reduces the number of free weights and means that HONNs can be faster to train than even MLPs. However, because the number of inputs can be very large for higher order architectures, orders of four and over are rarely used. Another advantage of the reduction of free weights means that the problems of over?tting and local optima affecting the results can be largely avoided.
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Figure 3. Left, MLP with three inputs and two hidden nodes. Right, second order HONN with three inputs.
The HONN in this study uses the sigmoid transfer function, error function, and linear function as described for the MLP architecture in Section 4.2. This has been done in order to be able to draw direct comparisons between the architectures of the models. 4.5 NN training procedure The training of the network is of utmost importance, as it is possible for the network to learn the training subset exactly (commonly referred to as over?tting). For this reason, the network training must be stopped early. This is achieved by dividing the data set into three different components (as shown in Table 2). First, a training subset is used to optimize the model, and the ‘back propagation of errors’algorithm is used to establish optimal weights from the initial random weights. Secondly, a test subset is used to stop the training subset from being over?tted. Optimization of the training subset is stopped when the test subset is at maximum positive return. These two subsets are the equivalent of the in-sample subset for the fair value model. This technique will prevent the model from over?tting the data while also ensuring that any structure inherent in the spread is captured. Finally, the out-of-sample subset is used to simulate future values of the time series, which for comparison is the same as the out-of-sample subset of the fair value model. As the starting point for each network is a set of random weights, a committee of 10 networks has been used to arrive at a trading decision (the average estimate decides on the trading position taken). This helps to overcome the problem of local minima affecting the training procedure. The trading model predicts the change in the spread from one closing price to the next; therefore, the average result of all 10 NN models was used as the forecast of the change in the spread, or St . This training procedure is identical for all the NNs used in this study. 5. Filter methodologies
The ?lter methodologies employed in this paper are described below. 5.1 The threshold ?lter The threshold ?lter is constructed as follows.
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If | St | < Xt , then stay out of the market; otherwise, take the decision of the trading rule, where St is the estimate of the percentage change in the spread and X t is the size of the threshold ?lter, which is optimized in the in-sample period. In the case of the fair value model, the position is held until fair value is regained. 5.2 The correlation ?lter
As well as the application of a threshold ?lter, the trading rules have been ?ltered in terms of their correlation. This methodology is explained in Dunis, Laws, and Evans (2005) and is shown in Figure 4. A rolling 30-day correlation is produced from the two legs of the spread. The change of this series is then calculated. From this a binary output of either 0 if the change in the correlation is above Xc , or 1 if the change in the correlation is below X c , X c being the correlation ?lter level. This is then multiplied by the returns series of the trading model. This helps us focus on periods when the correlation is falling and stay out of the market during periods when the correlation is rising. By using this ?lter, it should also be possible to ?lter out initial moves away from fair value, which is generally harder to predict than moves back to fair value. Figure 4 gives an example of the entry and exit points of the ?lter when X c = 0. Figure 4 shows that a market entry is triggered the day after the drop in correlation. The market exit is triggered the day after the correlation starts to rise. 5.3 The transitive ?lter
The transitive ?lter used here has been adapted from the work by Guégan and Huck (2004), who state that for a given set of spread (or relative) forecasts, there exists the possibility that they may
Figure 4. Operation of the correlation ?lter.
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Table 4. A set of transitive signals. Spread Forecast A–B Up B–C Up A–C Up
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Table 5. A set of non-transitive signals. Spread Forecast A–B Up B–C Up A–C Down
be consistent or inconsistent. That is, we may have a case where one side of a spread is forecasted to be positive against all other possible sides of a spread. To complete the consistency, there should also be another side that is forecasted to be negative relative to all other sides. If we consider a universe of three contracts, (and denote them as A, B, and C), it is possible to construct the following three spreads: A–B, B–C, and A–C. The forecast of these three spreads can be either consistent (transitive) or inconsistent (non-transitive). For example, a set of transitive forecasts is shown in Table 4. It can be seen from Table 4 that the set of trading decisions is transitive; this is evidenced by the fact that if we prefer A to B and also prefer B to C, we must prefer A to C. In this case, the forecasts of assets A and C are deemed transitive and we would take a long position in asset A and a short position in asset C. A set of non-transitive forecasts is shown in Table 5. It can be seen from Table 5 that the set of trading decisions given is non-transitive; this is evidenced by the fact that while we prefer A to B and B to C, the forecast of A–C indicates that we prefer C to A. In this case, the forecasts are not transitive and the decision of the model is ignored. Subsequently, we stay with the trading decision taken at time t ? 1. See Guégan and Huck (2004) for a complete mathematical description of transitivity. In the case of Guégan and Huck (2004), this ?lter is only applied to spreads with a low level of historic correlation (< 0.3) and also just to forecasting relative stock prices. It is our contention that although assets that are highly correlated may not move as much, the added structure provides us with a payoff in terms of predictability. 6. Results
The following section shows the results of the empirical investigation. The ?lters have been optimized in sample in order to maximize the Calmar ratio, de?ned by Jones and Baehr (2003) as: CalmarRatio = Return |MaxDD| (3)
where Return is annualized return of the trading model and MaxDD is the maximum drawdown of the trading model and is de?ned as:
n
MaximumDrawdown = Min
St ? Max
t=1
St
(4)
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Equation (3) is given a high priority as futures are naturally leveraged instruments. This statistic gives a good measure of the amount of return that can be expected for the amount of investment capital required to ?nance a strategy. Furthermore, unlike the Sharpe ratio, which assumes large losses and large gains are equally undesirable, the Calmar ratio de?nes risk as the maximum likely loss and is therefore a more realistic measure of risk-adjusted return. 6.1 Fair value-based portfolio
The in-sample trading statistics inclusive of transactions costs of the cointegration fair value model are shown in Table 610 . Table 6 shows the in-sample results of the fair value cointegration model. It is evident that all ?lters outperform the un?ltered model and therefore can be considered as ‘selected’ to take through to the out-of-sample period. Table 7 shows the out-of-sample results of the fair value cointegration model. It is evident that the best model out-of-sample is the correlation ?lter with a Calmar ratio of 1.9. However, the threshold and transitive ?lters still prove to be good choices with pro?ts over and above the un?ltered model. Therefore, the use of all ?lters is vindicated with respect to this model. 6.2 MLP-based portfolio The in-sample trading statistics inclusive of transactions costs of the MLP model are shown in Table 8. Table 8 shows the in-sample results of the MLP model. It is evident that all ?lters improve on the un?ltered Calmar ratio and therefore can be considered as selected to carry forward to the out-of-sample period. Table 9 shows the out-of-sample results of the MLP model. It is evident that the best-selected ?lter out-of-sample is the transitive ?lter with a Calmar ratio of 2.5. The threshold and correlation ?lters also prove to be good selections resulting in a Calmar ratio of more than that of the un?ltered model.
Table 6. In-sample fair value model results. In-sample Annualized return Annualized standard deviation Maximum drawdown Calmar ratio Annualized trades Un?ltered 3.60% 13.73% ?46.13% 0.0780 24.72 Threshold 7.81% 10.78% ?27.07% 0.2885 34.57 Correlation 7.50% 11.18% ?31.77% 0.2361 27.73 Transitive 8.72% 15.31% ?23.85% 0.3658 3.25
Table 7. Out-of-sample fair value model results. Out-of-sample Annualized return Annualized standard deviation Maximum drawdown Calmar ratio Annualized trades Un?ltered 18.04% 15.58% ?14.98% 1.2046 20.42 Threshold 14.44% 11.36% ?9.13% 1.5813 27.23 Correlation 18.62% 12.95% ?9.60% 1.9404 33.60 Transitive 31.98% 18.16% ?20.10% 1.5913 5.79
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Table 8. In-sample MLP model results. In-sample Annualized return Annualized standard deviation Maximum drawdown Calmar ratio Annualized trades Un?ltered 25.85% 11.92% ?16.11% 1.6046 115.90 Threshold 25.56% 9.70% ?12.19% 2.0958 74.70 Correlation 26.36% 11.91% ?16.00% 1.6473 115.83
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Transitive 46.29% 11.92% ?5.92% 7.8149 49.91
Table 9. Out-of-sample MLP model results. Out-of-sample Annualized return Annualized standard deviation Maximum drawdown Calmar ratio Annualized trades Un?ltered 3.60% 13.73% ?46.13% 0.0780 114.38 Threshold 7.81% 10.78% ?27.07% 0.2885 70.82 Correlation 7.50% 11.18% ?31.77% 0.2361 113.66 Transitive 16.70% 10.87% ?6.79% 2.4580 50.23
Table 10. In-sample RNN model results. In-sample Annualized return Annualized standard deviation Maximum drawdown Calmar ratio Annualized trades Un?ltered 11.55% 11.14% ?23.14% 0.4989 112.10 Threshold 12.05% 11.07% ?23.30% 0.5171 111.90 Correlation 12.52% 11.11% ?23.03% 0.5434 112.15 Transitive 31.42% 11.15% ?12.23% 2.5688 41.24
6.3 RNN-based portfolio The in-sample trading statistics inclusive of transactions costs of the RNN model are shown in Table 10. Table 10 shows the in-sample results of the RNN model. It is evident that all ?lters improve the in-sample Calmar ratio and therefore could potentially have been chosen for the out-of-sample period. Table 11 shows the out-of-sample results of the RNN model. It is evident that the best model out-of-sample is the transitive ?lter with a Calmar ratio of 11.3. The correlation ?lter, while not as good, can be considered a valid choice because it improves the Calmar ratio of the model. In the case of the threshold ?lter, the Calmar ratio is lower than that of the un?ltered model and can with the bene?t of hindsight can be considered a bad choice. 6.4 HONN-based portfolio The in-sample trading statistics inclusive of transactions costs of the HONN model are shown in Table 12. Table 12 shows the in-sample results of the HONN model. It is evident that all ?lters improve the in-sample Calmar ratio of the un?ltered model and can therefore be considered as selected for the out-of-sample period.
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Table 11. Out-of-sample RNN model results. Out-of-sample Annualized return Annualized standard deviation Maximum drawdown Calmar ratio Annualized trades Un?ltered 22.02% 13.64% ?14.01% 1.5722 102.69 Threshold 21.66% 13.65% ?15.02% 1.4421 102.17 Correlation 22.71% 13.57% ?13.69% 1.6582 102.76 Transitive 39.74% 13.68% ?3.52% 11.3026 36.54
Table 12. In-sample HONN model results. In-sample Annualized return Annualized standard deviation Maximum drawdown Calmar ratio Annualized trades Un?ltered 21.43% 11.68% ?24.60% 0.8712 112.95 Threshold 25.22% 10.34% ?17.61% 1.4322 93.84 Correlation 23.10% 11.05% ?23.66% 0.9761 97.37 Transitive 40.64% 11.66% ?11.96% 3.3975 19.89
Table 13. Out-of-sample HONN model results. Out-of-sample Annualized return Annualized standard deviation Maximum drawdown Calmar ratio Annualized trades Un?ltered 9.20% 11.38% ?24.56% 0.3746 89.12 Threshold ?2.57% 9.44% ?24.71% ?0.1039 87.61 Correlation 8.01% 11.13% ?22.39% 0.3577 89.48 Transitive 21.40% 11.47% ?8.66% 2.4699 3.16
Table 13 shows the out-of-sample results of the HONN model. It is evident that the best model out-of-sample is the transitive ?lter with a Calmar ratio of 2.5. Of the other ?ltered models, the correlation ?lter performs best but with the bene?t of hindsight is still considered to be a poor selection. Overall, the best in-sample is the MLP with a transitive ?lter with a Calmar ratio of 7.8, out-ofsample this performs well with a Calmar ratio of 2.5. The most consistent out-of-sample pro?ts come from the cointegration fair value model, which displays consistently high out-of-sample pro?ts and Calmar ratios. The most robust results come from the RNN performing well both inand out-of-sample. Compared with the un?ltered models, the selection of a ?lter proves to be a good choice for outof-sample in 9 of the 12 times. The transitive ?lter proving the most consistent method showing improvement over the un?ltered model 4 of four portfolios, even if the correlation ?lter, is best in the case of the cointegration fair value model. Similar to ?ndings by Guégan and Huck (2004), we ?nd that using transitive signals generally improves the in- and out-of-sample trading statistics over those of un?ltered models. 7. Conclusions
In summary, our results show that the best in-sample model is the MLP with a transitive ?lter. This model performs well out of sample with a Calmar ratio of around 2.5. Of the ?lters investigated, the
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transitive ?lter proves to be a good out-of-sample performer. This is evidenced by an improvement in the in- and out-of-sample Calmar ratios in all cases, compared with 3 of 4 for the correlation ?lter and 2 of 4 for the threshold ?lter. Further, the correlation ?lter improves the out-of-sample Calmar ratio by a factor of 0.24 on the models for which it is selected, in contrast to -0.01 for the threshold ?lter and 3.65 for the transitive ?lter. Finally, with the bene?t of hindsight, the best model in terms of out-of-sample Calmar ratio is the RNN model with a transitive ?lter, with a Calmar ratio of 11.3. However, making the choice of a trading model on the basis of the in-sample Calmar ratio would have led to the MLP model with the transitive ?lter: Out-of-sample this model performs well with a Calmar ratio of 2.5. Overall alternative NN architectures combined with transitive forecast ?lters, as described and investigated in this paper, may add value in trading a portfolio of spreads. Notes
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. See, for example, Tucker (2000) and Ross (2003). See, for example, Girma and Paulson (1998) and Salcedo (2004a, 2004b). See, for example, MacKinlay and Ramaswamy (1988), Yadav and Pope (1990), Chung (1991), and among others. Notable exceptions include Billingsley and Chance (1988), Board and Sutcliffe (1996), Butterworth and Holmes (2002), and Butterworth and Holmes (2003). Leg1 being the ?rst contract under consideration. Leg2 being the second contract under consideration. NYMEX Europe has traded the Brent crude oil contracts since September 2005. It has been decided that for ease of calculation, any round turn commission (?0.03%) be ignored. Associative recall is the act of associating two seemingly unrelated entities, such as smell and colour. For more information, see Karayiannis and Venetsanopoulos (1994). Un?ltered – is the model statistics without the application of a ?lter; Threshold – is the model statistics with the application of the threshold ?lter, optimized in-sample; Correlation – is the model statistics with the application of the correlation ?lter, optimized in-sample; Transitive – is the model statistics with the application of the transitive ?lter, optimized in-sample; Ann. Return – is the annualized percentage returns of the model inclusive of transactions costs; Ann. Stdev – is the annualized standard deviation of returns of the model; Max DD – is the maximum drawdown of the model; Calmar Ratio – is the Calmar ratio of the model as calculated with Equation (2); Ann. Trades – is the annualized round trip trades of the model, per contract.
References
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Dunis, C., J. Laws, and B. Evans. 2005. Recurrent and higher order neural networks: A comparative analysis. Neural Network World 6: 509–23. Elman, J.L. 1990. Finding structure in time. Cognitive Science 14: 179–211. Evans, B., C.L. Dunis, and J. Laws. 2006. Trading futures spreads: An application of correlation and threshold ?lters. Applied Financial Economics 16, no. 12: 903–14. Giles, L. and T. Maxwell. 1987. Learning invariance and generalization in high-order neural networks. Applied Optics 26, no. 23: 4972–8. Girma, P.B., and S.A. Paulson. 1998. Seasonality in petroleum futures spreads. The Journal of Futures Markets 18, no. 5: 581–98. Guégan, D., and N. Huck. 2004. Forecasting relative movements using transitivity? Working paper, Institutions et Dynamiques Historiques de l’Economie, December. Johansen, S. 1988. Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control 12: 231–54. Jones, M.A., and M. Baehr. 2003. Manager searches and performance measurement. In Hedge funds de?nitive strategies and techniques, ed. K.S. Phillips and P.J. Surz, 112–38. Hoboken, New Jersey: John Wiley & Sons. Kaastra, I., and M. Boyd. 1996. Designing a neural network for forecasting ?nancial and economic time series. Neurocomputing 10: 215–36. Karayiannis, N.B., and A.N. Venetsanopoulos. 1994. On the training and performance of higher-order neural networks. Mathematical Biosciences 129: 143–68. Knowles, A., A. Hussein, W. Deredy, P. Lisboa, and C.L. Dunis. 2005. Higher-order neural networks with Bayesian con?dence measure for prediction of EUR/USD exchange rate. CIBEF Working Papers. www.cibef.com. Krishnaswamy, C.R., E.W. Gilbert, and M.M. Pashley. 2000. Neural network applications in ?nance. Financial Practice and Education (Spring/Summer): 75–84. Kryzanowski, L., M. Galler, and D.W. Wright. 1993. Using arti?cial neural networks to pick stocks. Financial Analysts Journal 49: 21–7. Lindemann, A., C. Dunis, and P. Lisboa. 2004. Probability distributions and leveraged strategies: An application of Gaussian mixture model to the Morgan Stanley high technology 35 index. Journal of Forecasting 23, no. 18: 559–85. ———. 2005. Level estimation, classi?cation and probability distribution architectures for trading the EUR/USD exchange rate. Neural Computing and Applications 14, no. 3: 256–71. Lukac, L., and B. Brorsen. 1990. A comprehensive test of futures market disequilibrium. The Financial Review 25, no. 4: 593–633. Mackinlay, C., and K. Ramaswamy. 1988. Index futures arbitrage and the behaviour of stock index futures prices. Review of Financial Studies 1: 137–58. Melamed, L. 1981. Futures market liquidity and the technique of spreading. Journal of Futures Markets 1: 405–11. Pruitt, S.W., and R.E. White. 1988. The CRISMA trading system: Who says technical analysis can’t beat the market? Journal of Portfolio Management 14: 55–8. Refenes, A.P., A. Zapranis, and G. Francis. 1995. Modelling stock returns in the framework of APT. In Neural networks in the capital markets, ed. A.P. Refenes, 101–25. Chichester, England: John Wiley & Sons. Ross, J. 2003. Getting an edge with reduced margin spreads. Special issue, Secrets of the Masters: 28–30. www.futuresmag.com. Saad, E.W., D.V. Prokhorov, and D.C. Wunsch. 1998. Comparative study of stock trend prediction using time delay, recurrent and probabilistic neural networks. Transactions on Neural Networks 9: 1456–70. Salcedo, Y. 2004a. Making the most of Ag spreads. (June): 68–71. www.futuresmag.com. ———. 2004b. Spreads for the fall. (September): 54–6. www.futuresmag.com. Sullivan, R., A. Timmerman, and H. White. 1998. Datasnooping, technical trading rule performance and the bootstrap. Journal of Finance 55, no. 5: 1647–91. Sweeney, R.J. 1988. Some new ?lter rule tests: Methods and results. Journal of Financial and Quantitative Analysis 23: 285–300. Tenti, P. 1996. Forecasting foreign exchange rates using recurrent neural networks. Applied Arti?cial Intelligence 10: 567–81. Tucker, S. 2000. Spreading the wealth. (January): 42–4. www.Futuresmag.com. Working, H. 1949. The theory of price of storage. American Economic Review 39: 1254–62. Yadav, P.K., and P.F. Pope. 1990. Stock index futures arbitrage: International evidence. Journal of Future Markets 10: 573–604. Zhang, M., S. Xu, and J. Fulcher. 2002. Neuron-adaptive higher order neural-network models for automated ?nancial data modeling. Transactions on Neural Networks 13: 188–204. Zirilli, J.S. 1997. Financial prediction using neural networks. London: International Thompson Computer Press.
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Appendix 1b. PDF of WTI–heating oil spread daily percentage returns.
Appendix 1c. PDF of WTI–gasoline spread daily percentage returns.
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Appendix 1d. PDF of gasoline–heating oil spread daily percentage returns.
Appendix 1e. PDF of Brent–heating oil spread daily percentage returns.
Appendix 1f. PDF of Brent–gasoline spread daily percentage returns.
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Appendix 2b.
doc_839217991.pdf
The report explains the modelling and trading of oil futures spreads in the context of a portfolio of contracts.
The European Journal of Finance, Vol. 14, No. 6, September 2008, 503–521
Trading futures spread portfolios: applications of higher order and recurrent networks Christian L. Dunis, Jason Laws and Ben Evans?
Centre for International Banking, Economics and Finance, Liverpool John Moores University, Liverpool, UK
This paper investigates the modelling and trading of oil futures spreads in the context of a portfolio of contracts. A portfolio of six spreads is constructed and each spread forecasted using a variety of modelling techniques, namely, a cointegration fair value model and three different types of neural network (NN), such as multi-layer perceptron (MLP), recurrent, and higher order NN models. In addition, a number of trading ?lters are employed to further improve the trading statistics of the models. Three different ?lters are optimized on an in-sample measure of down side risk-adjusted return, and these are then ?xed out-ofsample. The ?lters employed are the threshold ?lter, correlation ?lter, and the transitive ?lter. The results show that the best in-sample model is the MLP with a transitive ?lter. This model is the best performer out-of-sample and also returns good out-of-sample statistics.
Keywords: futures spreads; cointegration; trading ?lters; higher order networks; recurrent networks
1.
Introduction
The case for spread trading has been made by several academics most of whom call on reduced margin requirements1 or consistently tradable patterns2 to encourage interest in spread trading. However, these same papers fail to explicitly explain that with reduced margin comes reduced potential reward. The reason for the reduced margin when trading a spread is the reduced chance of large moves. This is because the two legs of the spread are highly correlated over the long-term and will therefore move in generally the same direction. Further consideration should be given to the additional trading costs incurred in trading spreads, this is because two contracts must be traded for each spread, therefore the trading costs are signi?cantly higher. This leads us to the question of not how can we predict the spread, but how can we produce pro?t from our predictions? With this in mind, a number of ?ltering techniques are investigated to further re?ne the ability of the decision models to produce pro?ts over and above transactions costs. These ?ltering techniques are the threshold ?lter and the correlation ?lter as developed by Dunis, Laws, and Evans (2005) and the transitive ?lter, which has been adapted from the initial work by Guégan and Huck (2004). In their work, they ?nd ‘it [transitivity] signi?cantly raises the proportion of the signs of the relative movements correctly guessed. The same approach is then applied on real data sets and similar conclusions obtained’. However, their results leave open two questions, they are ‘Testing this approach on other simulated and real data sets and
? Corresponding
author. Email: [email protected]
ISSN 1351-847X print/ISSN 1466-4364 online © 2008 Taylor & Francis DOI: 10.1080/13518470801890834 http://www.informaworld.com
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implementing it for trading strategies’ and ‘Extending the analysis to N -multivariate setting, with N > 3’. In this paper, we attempt to answer these questions by utilizing transitive forecasts in the context of trading a portfolio of spreads and also extend the universe of contracts to four. Resulting in six spread, or relative, forecasts. Motivation for this paper is derived from the work by Dunis, Laws, and Evans (2005), in which it is discovered, in the case of the Gasoline Crack spread, that the use of higher order neural networks (HONNs) produces pro?ts in excess of those achieved with the use of a standard multi-layer perceptron (MLP). This is worth investigating further in the context of a portfolio of spreads. Further, it was found that the use of the correlation ?lter generally does not produce better results than the un?ltered model; this ?nding lies in contradiction to Evans, Dunis, and Laws (2006), who ?nd that the correlation ?lter is a promising method of ?ltering spread trades. This paper is set out as follows: Section 2 details some of the relevant literature; Section 3 explains the data and methodology; Section 4 de?nes the decision models used; Section 5 de?nes the ?lters that have been employed; Sections 6 and 7 give the results and conclusions, respectively.
2.
Literature review
Spread trading was ?rst introduced formally into the ?nance literature by Working (1949), who investigated the effects of the cost of storage on pricing relationships. It was demonstrated that futures traders could pro?t from the existence of abnormalities in the pricing relationships between futures contracts of different expiry, an intra-commodity spread. Melamed (1981) gives further justi?cation for research into spread trading stating that ‘although spread trading has been used to speculate on the cost of carry between different futures contracts, spread trading also serves the functions of arbitrage and hedging, together with providing a vital source of market liquidity’ (Melamed 1981, 408). It is therefore surprising that although there has been interest in cash-futures arbitrage3 , inter- and intra-commodity spread trading has been largely ignored among the academic fraternity4 . Butterworth and Holmes (2003) state ‘an analysis of spread trading is important since it contributes to the economics of arbitrage and serves as an alternative to cash-futures arbitrage for testing for futures market ef?ciency’ (Butterworth and Holmes 2003, 783). In this paper, they test a fair value model on the FTSE250–FTSEMID100 spread. This fair value model is based on ‘arbitrage limits’ and takes a position on the spread when a limit is breached. They conclude that ‘while the inter-market spread is found to trade within its transactions cost limits on the majority of occasions, large and sustained deviations from fair value exist in both directions, resulting in the triggering of spread arbitrage transactions’ (Butterworth and Holmes 2003, 790). This statement does not indicate whether either a more sophisticated trading rule or an appropriate trading ?lter can capture the deviations from fair value. Studies such as Pruitt and White (1988), Sweeney (1988), and Dunis (1989) directly support the use of technical trading rules as a means of trading ?nancial markets. Trading rules such as moving averages, ?lters, and patterns seem to generate returns above the conventional buy and hold strategy. Nevertheless, Lukac and Brorsen (1990) carried out a comprehensive test of futures market trading. It was found that all but one of the trading rules tested generated signi?cantly abnormal returns. Sullivan, Timmerman, and White (1998) investigated the performance of technical trading rules over a 100-year period of the Dow Jones Industrial Average, they conclude that
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‘there is no evidence that any trading rule outperforms [the benchmark buy and hold strategy] over the sample period’ (Sullivan, Timmerman, and White 1998, 1683). With the increasing processing power of computers, rule-induced trading has become far easier to implement and test. Kaastra and Boyd (1996) investigated the use of neural networks (NNs) for forecasting ?nancial and economic time series. They concluded that the large amount of data needed to develop working forecasting models involved too much trial and error. On the contrary, Chen, Wagner, and Lin (1996) study the 30-year US Treasury bond using an NN approach. The results prove to be good with an average buy prediction accuracy of 67% and an average annualized return on investment of 17.3%. In recent years, there has been an expansion in the use of computer trading techniques, which has once again called into doubt the ef?ciency of even very liquid ?nancial markets. Lindemann, Dunis, and Lisboa (2004) suggest that it is possible to achieve abnormal returns on the Morgan Stanley High Technology 35 index using a Gaussian mixture NN trading model. Lindemann, Dunis, and Lisboa (2005) justi?ed the use of the same model to successfully trade the EUR/USD exchange rate, an exchange rate noted for its liquidity. Krishnaswamy, Gilbert, and Pashley (2000) attempt to show the development of NNs as modelling tools for ?nance. In turn, they cite valuable contributions from Bansal and Viswanathan (1993), Kryzanowski, Galler, and Wright (1993), Refenes, Zapranis, and Francis (1995), and Zirilli (1997) in the ?eld of stock market and individual stock prediction, proving that not only do NNs outperform linear regression models, but also NNs are ‘superior in dealing with structurally unstable relationships, notably stock market returns’, (Krishnaswamy, Gilbert, and Pashley 2000, 79). This research kick started the search for increasingly specialist NN architectures. Recurrent networks (RNNs) were ?rst developed by Elman (1990) and possess a form of error feedback. These networks are generally better than MLP networks but they do suffer from long computational times (Tenti 1996). However, according to Saad, Prokhorov, and Wunsch (1998), compared with other architectures, this should not matter a lot ‘RNN has the capability to dynamically incorporate past experience due to internal recurrence, and it is the most powerful network of the three in this respect… but its minor disadvantage is the implementation complexity’ (Saad, Prokhorov, and Wunsch 1998, 1468). HONNs were ?rst introduced by Giles and Maxwell (1987) and were called ‘Tensor Networks’, although the extent of their use in ?nance is limited. Knowles et al. (2005) show that despite shorter computational times and limited input variables on the EUR/USD time series ‘the best HONN models show a pro?t increase over the MLP of around 8%’ (Knowles et al. 2005, 7). A signi?cant advantage of HONNs is detailed in Zhang, Xu, and Fulcher (2002). ‘HONN models are able to provide some rationale for the simulations they produce and thus can be regarded as “open box” rather then “black box”. Moreover, HONNs are able to simulate higher frequency, higher order non-linear data, and consequently provide superior simulations compared to those produced by arti?cial neural networks’ (Zhang, Xu, and Fulcher 2002, 188). In the context of spreads, Dunis, Laws, and Evans (2005) found that for the Gasoline Crack spread, ‘the HONN outperformed the MLP out-of-sample before transactions costs despite shorter computational time and limited variables’ (Dunis, Laws, and Evans 2005, 522). This paper investigates the use of MLP, RNN, and HONN models as trading tools in the oil spread market, and these are described more fully in Sections 4.2, 4.3, and 4.4, respectively. These models are also benchmarked against a more traditional Cointegration fair value model. Further, this paper also investigates the use of a number of ?ltering techniques to further enhance the performance of the forecasting models. These ?lters are detailed in Section 5.
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Following methodology from Butterworth and Holmes (2002) and Dunis, Laws, and Evans (2005), the spread returns series is calculated in the following way: Rs = Leg1,t ? Leg1,t?1 Leg1,t ? Leg2,t ? Leg2,t?1 Leg2,t (1)
where Leg1,t is the price of Leg1 at time t 5 , Leg1,t?1 the price of Leg1 at time t ? 16 , Leg2,t the price of Leg2 at time t, and Leg2,t?1 the price of Leg2 at time t ? 1. This convention allows for the calculation of annualized returns and annualized standard deviation to be done in the usual way. 3.1 Data set For the cointegration fair value model, the data have been split into two subsections (see Table 1). The ?rst subset (the in-sample subset) is used to test and optimize the models. The second subset (the out-of-sample subset) is used as an unseen data set and is used to test our optimized models. In the case of the NN models, and in order to avoid over?tting, the data have been split into three subsets, as is standard in the literature (e.g. Kaastra and Boyd 1996; Lindemann, Dunis, and Lisboa 2005) (see Table 2). The NN models are trained slightly differently to standard econometric models. The training data set is used to train the network, the minimization of the error function being the criterion optimized. The training of the network is stopped when the pro?t on the test data set is at maximum positive return. This model is then traded on the validation subset, which for comparison purposes is identical to the out-of-sample data set used for the fair value cointegration model. This technique restricts the amount of noise that the model will ?t, while also ensuring that the structure inherent in the training and test subsets is modelled. Further explanation of this is contained in Lindemann, Dunis, and Lisboa (2005). Table 3 shows the details of the time series used to form the portfolio. The series shown in Table 3 are then combined to form the following spreads: (1) Brent crude vs. WTI crude; (2) Brent crude vs. unleaded gasoline;
Table 1. In-sample and out-of-sample dates. Subset In-sample Out-of-sample Purpose Optimize model Test model Period 3 January 1995–25 July 2002 26 July 2002–1 January 2005 Data points 2175 435
Table 2. In-sample and out-of-sample dates (for NN training). Subset 1 (Training) 2 (Test) 3 (Validation) Purpose Optimize model Stop model optimization Test model Period 3 January 1995–29 December 2000 1 January 2001–25 July 2002 26 July 2002–1 January 2005 Data points 1740 435 435
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Table 3. Time series data contract speci?cs. Fixing times (GMT) 17:30 17:30 17:30 17:30 Bid-ask spreada (%) 0.0960 0.0560 0.1004 0.1241
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Time series WTI crude Brent crude Unleaded gasoline Heating oil
Currency US$ US$ US$ US$
Exchange NYMEX IPE NYMEX NYMEX
Note: a Source of bid-ask spreads: www.sucden.co.uk, calculation of this is further explained in Section 3.3.
(3) (4) (5) (6)
Brent crude vs. heating oil; WTI crude vs. unleaded gasoline; WTI crude vs. heating oil; Unleaded gasoline vs. heating oil.
It is evident from Table 3 that these time series can be traded as spreads because they are denoted in the same currency and each leg of the spread has an identical ?xing time as the opposing leg. Further the trading of these spreads on NYMEX7 allows for a reduced margin of between 87.5% for WTI–Brent and 75% for Brent–heating oil. The portfolio, which has been traded, is an equally weighted portfolio of the six spreads shown above; therefore, each spread return is given a one-sixth weighting in the portfolio. Appendix 1 shows the daily returns histogram of the spread time series; it is evident from Appendices 1a–f that the distributions are severely non-normal. The series statistics show evidence of excess kurtosis (ranging from 11, in the case of the Brent vs. heating oil spread, to 54 in the case of the Brent vs. WTI spread) and in many cases signi?cant amounts of skewness. This is markedly more severe than the traditional ‘fat tails’ observed on stock or index returns. Appendices 2a and b show the daily return histograms of the FTSE100 and Brent crude over the same time period, respectively. It can be noted that the levels of kurtosis are much lower. 3.2 Rollovers
Using non-continuous time series brings a unique problem, as any long-term study will require a continuous series. If a trader takes a position on a futures contract, which subsequently expires, he can take the same position on the next available contract. This is called rolling forward. The problem with rolling forward is that two contracts of different expiry but same underlying may not (and invariably do not) have the same price. When the roll forward technique is applied to a futures time series, it will cause the time series to exhibit periodic blips in the price of the contract. Although the cost of carry (which actually causes the pricing differential) can be mathematically taken out of each contract, this does not leave us with a precisely tradable futures series. In this study, as we are dealing with futures spreads, we have rolled forward both contracts on the same day of each month (irrespective of exact expiry dates). The cost of carry, which is the cause of the price difference between the cash and futures price, is determined by the cost of buying the underlying in the cash market now and holding until futures expiry. As the cost of storage of both underlying is similar, they will approximately offset each other. We are left with a tradable time series with no cost of carry effect.
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C.L. Dunis et al. Transactions costs
The bid-ask spread is an average of four intra-day bid-ask spreads and is presented here as a percentage of the underlying price. The bid-ask spread of each contract, as a percentage of the contract price, can be seen in Table 3. The cost of trading the spread is the total cost of trading one leg in addition to the total cost of trading the other leg. For example, the cost of trading the WTI–Brent spread is (0.0960% + 0.0560%) = 0.1520%8 per round trip. An additional cost has been deemed appropriate due to the rollover of the futures contracts; this cost is incurred only if a trade is held unchanged. This cost is equal to one round trip trade on the spread. 4. Trading decision models The trading decision models have each been used to trade the portfolio of futures spreads. Four trading models have been used, and they are described below. 4.1 Fair value cointegration trading rule
The fair value model that has been used in this paper is the cointegration fair value model as used by Evans, Dunis, and Laws (2006). The model is based on the Johansen (1988) cointegration test, which tests that ? = 0 in Equation (2) below: ?t = ? ?t?1 + ?t where ?t is the residual of the cointegrating relationship. This can be extended to a trading model as shown below: If ?t < 0 then go long the spread, until ?t = 0 is regained; If ?t > 0 then go short the spread, until ?t = 0 is regained. For each spread, the relevant value of ?t was calculated as shown below, over the in-sample period (Table 1), using the Johansen (1988) method: ?tbg = Brent ? (34.63413 gasoline) ? 2.10E?04 trend ?tbw = Brent ? (0.969017 WTI) ? 2.18E?05 trend ?tbh = Brent ? (32.48161 heating oil) ? 6.59E?04 trend ?tgw = Gasoline ? (0.028433 WTI) ? 8.06E?06 trend ?tgh = Gasoline ? (0.947192 heating oil) ?thw = Heating oil ? (0.030124 WTI) ? 2.18E?04 trend where trend is a linear trend included, where signi?cant, to account for any in?ationary effects. 4.2 Multi-layer perceptron (2)
The most basic type of neural network regression (NNR) model, which is used in this paper, is the MLP. As explained in Lindemann, Dunis, and Lisboa (2004), the network has three layers; they are the input layer (explanatory variables), the output layer (the model estimation of the time series), and the hidden layer. The number of nodes in the hidden layer de?nes the amount of complexity that the model can ?t. The input and hidden layers also include a bias node, similar to the intercept for a standard regression, which has a ?xed value of 1.
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(1) The input nodes contain the values of the explanatory variables (in this case, 10 lagged values of the spread). (2) These values are transmitted to the hidden layer as the weighted sum of its inputs. (3) The hidden layer passes the information through a non-linear activation function and, if the calculated value is above a threshold value, onto the output layer. The connections between neurons for a single output neuron in the net are shown in Figure 1. Where: xt[n] (n = 1, 2, . . . , k + 1) h[m] (m = 1, 2, . . . , m + 1) t yt ˜ uj k and wj are the model inputs (including the input bias node) at time t. are the model inputs (including the hidden bias node). is the MLP model output (the estimate of the change in the spread). is the network weights is the sigmoid transfer function: S(x) = 1/1 + ex is a linear function: F (x) = The error function to be minimized is: E uj k , wj = 1 T ˜ (yt ? yt (uj k , wj ))2
i
xi
with yt (the change in the spread) being the target value. 4.3 Recurrent NN In contrast to an MLP, an RNN includes an additional input function. This additional input function is the previous state of the network and because of this, the network embodies a form of short-term memory in the training process. The name given to this memory process is activation feedback. An RNN with activation feedback embodies short-term memory, see, for example, Elman (1990). The advantages of using RNNs over feedforward networks, for modelling non-linear time series, have been well documented in the past, see, for example, Adam, Zarder, and Milgram (1993). However, as described in Tenti (1996), ‘the main disadvantage of RNNs is that they require substantially more connections, and more memory in simulation, than standard backpropagation
Figure 1. A single output, fully connected MLP model.
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Figure 2. Architecture of Elman or Recurrent Neural Network.
networks’ (Tenti 1996, 159), thus resulting in a substantial increase in computational time. However, having said these RNNs can yield better results in comparison with simple MLPs due to the additional memory inputs. The architecture of a simple RNN is shown in Figure 2. The state/hidden layer shown in Figure 2 is updated with external inputs, as in the simple MLP (Section 4.2) but also with activation from previous forward propagation, shown as ‘Previous State’ above. In short, the RNN architecture can provide more accurate outputs because the inputs are (potentially) taken from all previous values. The Elman network in this study uses the sigmoid transfer function, error function, and linear function as described for the MLP architecture in Section 4.2. This has been done in order to be able to draw direct comparisons between the architectures of both models. 4.4 Higher order NN
HONNs were ?rst introduced by Giles and Maxwell (1987) and were called ‘Tensor Networks’. Although they have already experienced some success in the ?eld of pattern recognition and associative recall9 , their use in ?nance is quite limited. The architecture of a three-input secondorder HONN is shown in Figure 3, where: are the model input. is the transfer sigmoid function: S(x) = 1/1 + e?x is the linear function: F (x) = i xi HONNs use joint activation functions; this technique reduces the need to establish the relationships between inputs when training. Furthermore, this reduces the number of free weights and means that HONNs can be faster to train than even MLPs. However, because the number of inputs can be very large for higher order architectures, orders of four and over are rarely used. Another advantage of the reduction of free weights means that the problems of over?tting and local optima affecting the results can be largely avoided.
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Figure 3. Left, MLP with three inputs and two hidden nodes. Right, second order HONN with three inputs.
The HONN in this study uses the sigmoid transfer function, error function, and linear function as described for the MLP architecture in Section 4.2. This has been done in order to be able to draw direct comparisons between the architectures of the models. 4.5 NN training procedure The training of the network is of utmost importance, as it is possible for the network to learn the training subset exactly (commonly referred to as over?tting). For this reason, the network training must be stopped early. This is achieved by dividing the data set into three different components (as shown in Table 2). First, a training subset is used to optimize the model, and the ‘back propagation of errors’algorithm is used to establish optimal weights from the initial random weights. Secondly, a test subset is used to stop the training subset from being over?tted. Optimization of the training subset is stopped when the test subset is at maximum positive return. These two subsets are the equivalent of the in-sample subset for the fair value model. This technique will prevent the model from over?tting the data while also ensuring that any structure inherent in the spread is captured. Finally, the out-of-sample subset is used to simulate future values of the time series, which for comparison is the same as the out-of-sample subset of the fair value model. As the starting point for each network is a set of random weights, a committee of 10 networks has been used to arrive at a trading decision (the average estimate decides on the trading position taken). This helps to overcome the problem of local minima affecting the training procedure. The trading model predicts the change in the spread from one closing price to the next; therefore, the average result of all 10 NN models was used as the forecast of the change in the spread, or St . This training procedure is identical for all the NNs used in this study. 5. Filter methodologies
The ?lter methodologies employed in this paper are described below. 5.1 The threshold ?lter The threshold ?lter is constructed as follows.
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If | St | < Xt , then stay out of the market; otherwise, take the decision of the trading rule, where St is the estimate of the percentage change in the spread and X t is the size of the threshold ?lter, which is optimized in the in-sample period. In the case of the fair value model, the position is held until fair value is regained. 5.2 The correlation ?lter
As well as the application of a threshold ?lter, the trading rules have been ?ltered in terms of their correlation. This methodology is explained in Dunis, Laws, and Evans (2005) and is shown in Figure 4. A rolling 30-day correlation is produced from the two legs of the spread. The change of this series is then calculated. From this a binary output of either 0 if the change in the correlation is above Xc , or 1 if the change in the correlation is below X c , X c being the correlation ?lter level. This is then multiplied by the returns series of the trading model. This helps us focus on periods when the correlation is falling and stay out of the market during periods when the correlation is rising. By using this ?lter, it should also be possible to ?lter out initial moves away from fair value, which is generally harder to predict than moves back to fair value. Figure 4 gives an example of the entry and exit points of the ?lter when X c = 0. Figure 4 shows that a market entry is triggered the day after the drop in correlation. The market exit is triggered the day after the correlation starts to rise. 5.3 The transitive ?lter
The transitive ?lter used here has been adapted from the work by Guégan and Huck (2004), who state that for a given set of spread (or relative) forecasts, there exists the possibility that they may
Figure 4. Operation of the correlation ?lter.
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Table 4. A set of transitive signals. Spread Forecast A–B Up B–C Up A–C Up
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Table 5. A set of non-transitive signals. Spread Forecast A–B Up B–C Up A–C Down
be consistent or inconsistent. That is, we may have a case where one side of a spread is forecasted to be positive against all other possible sides of a spread. To complete the consistency, there should also be another side that is forecasted to be negative relative to all other sides. If we consider a universe of three contracts, (and denote them as A, B, and C), it is possible to construct the following three spreads: A–B, B–C, and A–C. The forecast of these three spreads can be either consistent (transitive) or inconsistent (non-transitive). For example, a set of transitive forecasts is shown in Table 4. It can be seen from Table 4 that the set of trading decisions is transitive; this is evidenced by the fact that if we prefer A to B and also prefer B to C, we must prefer A to C. In this case, the forecasts of assets A and C are deemed transitive and we would take a long position in asset A and a short position in asset C. A set of non-transitive forecasts is shown in Table 5. It can be seen from Table 5 that the set of trading decisions given is non-transitive; this is evidenced by the fact that while we prefer A to B and B to C, the forecast of A–C indicates that we prefer C to A. In this case, the forecasts are not transitive and the decision of the model is ignored. Subsequently, we stay with the trading decision taken at time t ? 1. See Guégan and Huck (2004) for a complete mathematical description of transitivity. In the case of Guégan and Huck (2004), this ?lter is only applied to spreads with a low level of historic correlation (< 0.3) and also just to forecasting relative stock prices. It is our contention that although assets that are highly correlated may not move as much, the added structure provides us with a payoff in terms of predictability. 6. Results
The following section shows the results of the empirical investigation. The ?lters have been optimized in sample in order to maximize the Calmar ratio, de?ned by Jones and Baehr (2003) as: CalmarRatio = Return |MaxDD| (3)
where Return is annualized return of the trading model and MaxDD is the maximum drawdown of the trading model and is de?ned as:
n
MaximumDrawdown = Min
St ? Max
t=1
St
(4)
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Equation (3) is given a high priority as futures are naturally leveraged instruments. This statistic gives a good measure of the amount of return that can be expected for the amount of investment capital required to ?nance a strategy. Furthermore, unlike the Sharpe ratio, which assumes large losses and large gains are equally undesirable, the Calmar ratio de?nes risk as the maximum likely loss and is therefore a more realistic measure of risk-adjusted return. 6.1 Fair value-based portfolio
The in-sample trading statistics inclusive of transactions costs of the cointegration fair value model are shown in Table 610 . Table 6 shows the in-sample results of the fair value cointegration model. It is evident that all ?lters outperform the un?ltered model and therefore can be considered as ‘selected’ to take through to the out-of-sample period. Table 7 shows the out-of-sample results of the fair value cointegration model. It is evident that the best model out-of-sample is the correlation ?lter with a Calmar ratio of 1.9. However, the threshold and transitive ?lters still prove to be good choices with pro?ts over and above the un?ltered model. Therefore, the use of all ?lters is vindicated with respect to this model. 6.2 MLP-based portfolio The in-sample trading statistics inclusive of transactions costs of the MLP model are shown in Table 8. Table 8 shows the in-sample results of the MLP model. It is evident that all ?lters improve on the un?ltered Calmar ratio and therefore can be considered as selected to carry forward to the out-of-sample period. Table 9 shows the out-of-sample results of the MLP model. It is evident that the best-selected ?lter out-of-sample is the transitive ?lter with a Calmar ratio of 2.5. The threshold and correlation ?lters also prove to be good selections resulting in a Calmar ratio of more than that of the un?ltered model.
Table 6. In-sample fair value model results. In-sample Annualized return Annualized standard deviation Maximum drawdown Calmar ratio Annualized trades Un?ltered 3.60% 13.73% ?46.13% 0.0780 24.72 Threshold 7.81% 10.78% ?27.07% 0.2885 34.57 Correlation 7.50% 11.18% ?31.77% 0.2361 27.73 Transitive 8.72% 15.31% ?23.85% 0.3658 3.25
Table 7. Out-of-sample fair value model results. Out-of-sample Annualized return Annualized standard deviation Maximum drawdown Calmar ratio Annualized trades Un?ltered 18.04% 15.58% ?14.98% 1.2046 20.42 Threshold 14.44% 11.36% ?9.13% 1.5813 27.23 Correlation 18.62% 12.95% ?9.60% 1.9404 33.60 Transitive 31.98% 18.16% ?20.10% 1.5913 5.79
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Table 8. In-sample MLP model results. In-sample Annualized return Annualized standard deviation Maximum drawdown Calmar ratio Annualized trades Un?ltered 25.85% 11.92% ?16.11% 1.6046 115.90 Threshold 25.56% 9.70% ?12.19% 2.0958 74.70 Correlation 26.36% 11.91% ?16.00% 1.6473 115.83
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Transitive 46.29% 11.92% ?5.92% 7.8149 49.91
Table 9. Out-of-sample MLP model results. Out-of-sample Annualized return Annualized standard deviation Maximum drawdown Calmar ratio Annualized trades Un?ltered 3.60% 13.73% ?46.13% 0.0780 114.38 Threshold 7.81% 10.78% ?27.07% 0.2885 70.82 Correlation 7.50% 11.18% ?31.77% 0.2361 113.66 Transitive 16.70% 10.87% ?6.79% 2.4580 50.23
Table 10. In-sample RNN model results. In-sample Annualized return Annualized standard deviation Maximum drawdown Calmar ratio Annualized trades Un?ltered 11.55% 11.14% ?23.14% 0.4989 112.10 Threshold 12.05% 11.07% ?23.30% 0.5171 111.90 Correlation 12.52% 11.11% ?23.03% 0.5434 112.15 Transitive 31.42% 11.15% ?12.23% 2.5688 41.24
6.3 RNN-based portfolio The in-sample trading statistics inclusive of transactions costs of the RNN model are shown in Table 10. Table 10 shows the in-sample results of the RNN model. It is evident that all ?lters improve the in-sample Calmar ratio and therefore could potentially have been chosen for the out-of-sample period. Table 11 shows the out-of-sample results of the RNN model. It is evident that the best model out-of-sample is the transitive ?lter with a Calmar ratio of 11.3. The correlation ?lter, while not as good, can be considered a valid choice because it improves the Calmar ratio of the model. In the case of the threshold ?lter, the Calmar ratio is lower than that of the un?ltered model and can with the bene?t of hindsight can be considered a bad choice. 6.4 HONN-based portfolio The in-sample trading statistics inclusive of transactions costs of the HONN model are shown in Table 12. Table 12 shows the in-sample results of the HONN model. It is evident that all ?lters improve the in-sample Calmar ratio of the un?ltered model and can therefore be considered as selected for the out-of-sample period.
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Table 11. Out-of-sample RNN model results. Out-of-sample Annualized return Annualized standard deviation Maximum drawdown Calmar ratio Annualized trades Un?ltered 22.02% 13.64% ?14.01% 1.5722 102.69 Threshold 21.66% 13.65% ?15.02% 1.4421 102.17 Correlation 22.71% 13.57% ?13.69% 1.6582 102.76 Transitive 39.74% 13.68% ?3.52% 11.3026 36.54
Table 12. In-sample HONN model results. In-sample Annualized return Annualized standard deviation Maximum drawdown Calmar ratio Annualized trades Un?ltered 21.43% 11.68% ?24.60% 0.8712 112.95 Threshold 25.22% 10.34% ?17.61% 1.4322 93.84 Correlation 23.10% 11.05% ?23.66% 0.9761 97.37 Transitive 40.64% 11.66% ?11.96% 3.3975 19.89
Table 13. Out-of-sample HONN model results. Out-of-sample Annualized return Annualized standard deviation Maximum drawdown Calmar ratio Annualized trades Un?ltered 9.20% 11.38% ?24.56% 0.3746 89.12 Threshold ?2.57% 9.44% ?24.71% ?0.1039 87.61 Correlation 8.01% 11.13% ?22.39% 0.3577 89.48 Transitive 21.40% 11.47% ?8.66% 2.4699 3.16
Table 13 shows the out-of-sample results of the HONN model. It is evident that the best model out-of-sample is the transitive ?lter with a Calmar ratio of 2.5. Of the other ?ltered models, the correlation ?lter performs best but with the bene?t of hindsight is still considered to be a poor selection. Overall, the best in-sample is the MLP with a transitive ?lter with a Calmar ratio of 7.8, out-ofsample this performs well with a Calmar ratio of 2.5. The most consistent out-of-sample pro?ts come from the cointegration fair value model, which displays consistently high out-of-sample pro?ts and Calmar ratios. The most robust results come from the RNN performing well both inand out-of-sample. Compared with the un?ltered models, the selection of a ?lter proves to be a good choice for outof-sample in 9 of the 12 times. The transitive ?lter proving the most consistent method showing improvement over the un?ltered model 4 of four portfolios, even if the correlation ?lter, is best in the case of the cointegration fair value model. Similar to ?ndings by Guégan and Huck (2004), we ?nd that using transitive signals generally improves the in- and out-of-sample trading statistics over those of un?ltered models. 7. Conclusions
In summary, our results show that the best in-sample model is the MLP with a transitive ?lter. This model performs well out of sample with a Calmar ratio of around 2.5. Of the ?lters investigated, the
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transitive ?lter proves to be a good out-of-sample performer. This is evidenced by an improvement in the in- and out-of-sample Calmar ratios in all cases, compared with 3 of 4 for the correlation ?lter and 2 of 4 for the threshold ?lter. Further, the correlation ?lter improves the out-of-sample Calmar ratio by a factor of 0.24 on the models for which it is selected, in contrast to -0.01 for the threshold ?lter and 3.65 for the transitive ?lter. Finally, with the bene?t of hindsight, the best model in terms of out-of-sample Calmar ratio is the RNN model with a transitive ?lter, with a Calmar ratio of 11.3. However, making the choice of a trading model on the basis of the in-sample Calmar ratio would have led to the MLP model with the transitive ?lter: Out-of-sample this model performs well with a Calmar ratio of 2.5. Overall alternative NN architectures combined with transitive forecast ?lters, as described and investigated in this paper, may add value in trading a portfolio of spreads. Notes
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. See, for example, Tucker (2000) and Ross (2003). See, for example, Girma and Paulson (1998) and Salcedo (2004a, 2004b). See, for example, MacKinlay and Ramaswamy (1988), Yadav and Pope (1990), Chung (1991), and among others. Notable exceptions include Billingsley and Chance (1988), Board and Sutcliffe (1996), Butterworth and Holmes (2002), and Butterworth and Holmes (2003). Leg1 being the ?rst contract under consideration. Leg2 being the second contract under consideration. NYMEX Europe has traded the Brent crude oil contracts since September 2005. It has been decided that for ease of calculation, any round turn commission (?0.03%) be ignored. Associative recall is the act of associating two seemingly unrelated entities, such as smell and colour. For more information, see Karayiannis and Venetsanopoulos (1994). Un?ltered – is the model statistics without the application of a ?lter; Threshold – is the model statistics with the application of the threshold ?lter, optimized in-sample; Correlation – is the model statistics with the application of the correlation ?lter, optimized in-sample; Transitive – is the model statistics with the application of the transitive ?lter, optimized in-sample; Ann. Return – is the annualized percentage returns of the model inclusive of transactions costs; Ann. Stdev – is the annualized standard deviation of returns of the model; Max DD – is the maximum drawdown of the model; Calmar Ratio – is the Calmar ratio of the model as calculated with Equation (2); Ann. Trades – is the annualized round trip trades of the model, per contract.
References
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Appendix 1b. PDF of WTI–heating oil spread daily percentage returns.
Appendix 1c. PDF of WTI–gasoline spread daily percentage returns.
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Appendix 1d. PDF of gasoline–heating oil spread daily percentage returns.
Appendix 1e. PDF of Brent–heating oil spread daily percentage returns.
Appendix 1f. PDF of Brent–gasoline spread daily percentage returns.
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