Modelling, Forecasting and Trading the Crack: A Sliding Window Approach to Training Neural Networks

Abstract

The aim of this analysis is to expand on earlier work carried out by Dunis et al. (Modelling and trading the gasoline crack spread: A non-linear story. Derivatives Use, Trading & Regulation, 12(1–2), 126–145, 2005) who modelled the Crack Spread from 1 January 1995 to 1 January 2005. This chapter however provides a more sophisticated approach to the non-linear modelling of the ‘Crack’. The selected trading period covers 777 trading days starting on 9 April 2010 and ending on 28 March 2013. The proposed model is a combined PSO (particle swarm optimizer) and an RBF (radial basis function) NN (neural network), which is trained daily using sliding windows of 380 and 500 days. Performance is benchmarked against a multi-layer perceptron (MLP) NN using the same training protocol. In artificial intelligence (AI) outputs from the neural networks provide forecasts and in this case the outputs provide forecasts for one-day-ahead trading simulations. To model the spread an expansive universe of 59 inputs across different asset classes are also used. This empirical application is the first time five autoregressive moving average (ARMA) models and two GARCH (generalized autoregressive conditional heteroscedasticity) volatility models have been included in the input dataset as a mixed model approach to train the NNs. In addition, other significant contributions include a sliding window approach to training and estimation of NN models and the use of two fitness functions.

Experimental results reveal that the sliding window approach to modelling the Crack Spread is effective when using 380- and 500-day training periods. Sliding windows of less than 380 days were found to produce unsatisfactory trading performance and reduced statistical accuracy. The PSO RBF model that was trained over 380 is superior in both trading performance and statistical accuracy when compared to its peers. As each of the unfiltered models’ volatility and maximum drawdown were unattractive a threshold confirmation filter is employed.

Moreover, the threshold confirmation filter only trades when the forecasted returns are greater than an optimized threshold of forecasted returns. As a consequence, only forecasted returns of stronger conviction produce trading signals. This filter attempts to reduce maximum drawdowns and volatility by trading less frequently and only during times of greater predicted change. Ultimately, the confirmation filter improves risk return profiles for each model and transaction costs were also significantly reduced.

Appendix

1.1 Performance Measures

See Table 3.7

Table 3.7 Statistical and trading performance measures

1.2 Supplementary Information

See Table 3.8

Table 3.8 The refiner’s market capitalization

1.3 ARMA Equations and Estimations

Autoregressive moving average (ARMA) models assume that the future value of a time-series is governed by its historical values (the autoregressive component) and on previous residual values (the moving average component). A typical ARMA model takes the form of equation 3.15.

$$ {Y}_t={\phi}_0+{\phi}_1{Y}_{t-1}+{\phi}_2{Y}_{t-2}+\dots +{\phi}_p{Y}_{t-p}+{\varepsilon}_t-{w}_1{\varepsilon}_{t-1}-{w}_2{\varepsilon}_{t-2}-\dots -{w}_q{\varepsilon}_{t-q} $$

(3.15)

Where:

Y _t :

is the dependent variable at time _t

Y _{t − 1} , Y _{t − 2} , and Y _{t − p} :

are the lagged dependent variables

ϕ ₀ , ϕ ₁ , ϕ ₂ , and ϕ _p :

are regression coefficients

ε _t :

is the residual term

ε _{t − 1} , ε _{t − 2} , and ε _{t − p} :

are previous values of the residual

w ₁ , w ₂ , and w _q :

are weights.

Using a correlogram as a guide in the training and the test sub-periods the below restricted ARMA models were selected to trade each spread. All coefficients were found to be significant at a 95 % confidence interval. Therefore, the null hypothesis that all coefficients (except the constant) are not significantly different from zero is rejected at the 95 % confidence interval (Table 3.9).

Table 3.9 ARMA equations

1.3.1 GARCH Equations and Estimations

Each of the GARCH models (16,16) and (15,15) are deemed stable and significant at a 95 % confidence level. Following the initial estimation of significant terms a squared residuals test, Jarque-Bera test and an ARCH test are all conducted to test the reliability of the residuals. For the sake of brevity, outputs from these tests are not included. These can be obtained on request from the corresponding author. Autocorrelation is absent from both models and as a result returns derived from each model were used as inputs during the training of the proposed PSO RBF Neural Network (Table 3.10).

Table 3.10 GARCH model # 1

Observation

The AR(1), AR(2), AR(10), AR(16), MA(1), MA(2), MA(10) and MA(16) terms are all deemed significant at a 95 % confidence level. The model is also deemed stable due to the fact that the sum of GARCH(-1) and RESID(-1)^2 is less than 1. In this case it is, 0.896013 + 0.083038 = 0.979 (Table 3.11).

Table 3.11 GARCH model # 2

Observation

The AR(1), AR(4), AR(15), MA(1), MA(4), and MA(15) terms are all deemed significant at a 95 % confidence level. The model is also deemed stationary due to the fact that the sum of GARCH(-1) and RESID(-1)^2 is less than 1. In this case it is, 0.883565 + 0.091396 = 0.974961.

1.4 PSO Parameters

See Tables 3.12 and 3.13

Table 3.12 PSO RBF parameters

Table 3.13 Neural characteristics

1.5 Best Weights over the Training Windows

See Tables 3.14 and 3.15

Table 3.14 Best weights obtained from the 380 day training window

Table 3.15 Best weights obtained from the 500-day training window