Forecasting and Trading Commodity Contract Spreads with Gaussian Processes

1. Forecasting and Trading Commodity Contract Spreads with Gaussian Processes Nicolas Chapados and Yoshua Bengio University of Montreal and ApSTAT Technologies Inc.

2. Approach in a Nutshell • Commodity spreads exhibit regularities • Use a flexible regression approach to forecast the complete future price trajectory of a spread • Gaussian Processes • Augmented functional representation of trajectory • From the forecast trajectory, identify profitable opportunities (accounting for risk) • Experiments with a portfolio of 30 spreads • Profitable out-of-sample after transaction costs

3. Preliminary Remarks • Statistical learning algorithms will not make you rich • Overfitting is a central problem in finance • Only one historical trajectory • Extremely low signal-to-noise ratio • The economy is non-stationary • Bias-variance dilemma takes an interesting form • If you use a long history, you reduce variance but introduce bias • Conversely, with a short history you have little bias but high variance • As a result, model selection is difficult • Bayesian approches promise (theoretically) an automatic control of overfitting

4. Portfolio Choice:Conceptual Landscape • One-Period Models • Classical « mean-variance » framework (Markowitz) • Fixed investment horizon (one month, one quarter) • Predict the moments of the next-period asset return distribution (e.g. mean and covariance matrix) • Quadratic programming to find optimal portfolio weights that maximize a utility function: best return subject to risk constraint • Direct models using learning algorithms • Train a (e.g.) neural network to directly make a portfolio allocation decision from input variables • Can use a regression or classification framework • Training criterion: can maximize a financial utility that incorporates risk aversion and the effect of trading costs

5. Commodity Spreads • Price difference between two futures contracts • Example, as of July 24th, 2008: • Closing price for « Wheat, September 2008 »: $787.75 • Closing price for « Wheat, December 2008 »: $811.25 • Difference (Spread) : 787.75 – 811.25 = –23.50 • Objective: forecast these spreads 100 Jul-Dec CME Lean Hogs15-year average (1991-2005) 80 60 40 20 0 Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug

6. Empirical Regularities in Commodity Spreads • Soybeans Crush Spread (Simon, 1999) • Long-run cointegration among the constituents • Short-term mean reversal (5-day horizon) • Simple rules yield in-sample profits after transaction costs • Petroleum Crack Spread (Girma & Paulson, 1998) • Seasonality at both monthly and trading-week levels • Out-of-sample profits after transaction costs • Gold-Silver Spread (Liu & Chou, 2003) • Dunis et al. (2006 a,b) study both the crack and the crush spreads

7. Modeling Objectives • Nonparametrically exploit seasonalities that occur in commodity spreads • Concentrate on the simplest kind:intracommodity calendar spreads • Fixed maturities: e.g. March–July Wheat • Does not require the definition of a roll schedule • Problem is characterized by a large number of separate historical time series (one per trading year in the historical data)

8. What do Gaussian Processes Buy Us? • Rather than forecasting the distribution of the next-period returns, we can model the complete future price trajectory • A classical approach represents P(rt+1|It) • It is the information set available at time t • Example, an AR(1) model: yt+1 = a + byt + e,with e ~ N(0, s2) • A Gaussian Process can represent the joint distribution of all future prices, in particular P(pt+D|It, D), for D>0.

9. Gaussian Processes • General tools for nonlinear regression • Fully Bayesian Treatment • Start with a prior probability distribution on the space of functions • Observe some data • Infer a posterior distribution, given the observed data (from Bayes’ rule)

10. Example

11. Gaussian Processes — Details • Generalization of the normal distribution • Multivariate normal: elements of a vector are related by a covariance matrix • Gaussian process: values of the function at two points are linked by a covariance function • Analytical solution • Not subject to the optimization difficulties of neural networks — simple matrix algebra • Can produce a full covariance matrix between a set of new test points

12. Gaussian Processes — Details 2 • Let k(x,y) be a semidefinitive positive covariance function (kernel) • X — M x d matrix of training inputsy — M-vector of training targetsX* — M’ x d matrix of test inputs • Predictive distribution of test outputs at test inputs is normal with mean and covariance matrix given by • with

13. Historical Data: March–July Wheat Normalized Price Days to Maturity Year

14. Inputs and Target Representation • Time is an independent variable. Split into: • Current series index (e.g. trading year) • Operation time: time at which the forecast is made • Forecast horizon: # of days ahead we are forecasting • Other inputs must be known at operation time • Target is (normalized) spread price • We are learning a model of

15. Example of Forecast given HistoryWheat March–July / 1996

16. Forecasting Performance • AugRQ/all-inp: Reference model • Inputs: augmented time representation • Spread price + term-structure shape • Economic inputs (USDA ending stocks + stock-to-use ratio) • AugRQ/less-inp. Remove USDA inputs • AugRQ/no-inp. Remove price inputs • StdRQ/no-inp. • Linear/all-inp. Bayesian linear regression • AR(1)

17. Evaluation Methodology • Perform comparison using a modified Diebold-Mariano (1995) test that accounts for cross-correlations between test sets.

18. Forecasting Performance

19. From Forecasts to Trading Decisions • Use a forecast of the complete future trajectory (made at time t0) to find best trading opportunity • Information Ratio-like Criterion • Each component is obtainable from the Gaussian process forecast, e.g. • Entry condition: find t1, t2 > t0 which maximize the IR criterion • Exit condition: find exit time t2 which maximizes the IR criterion, given the current position

21. Behavior on a Single Trading YearWheat March–July / 1996 • Re-train model every 25 days • Sequence of decisions: short – neutral – long • Lower panel: Cumulative P&L ($)

22. Portfolio of 30 Spreads • Common Input Variables • Current spread price • Prices of first 3 near contracts • Normalization • Grains and related (SBM, SB, W) • USDA Ending Stocks (YoY difference) • USDA Stocks-to-Use Ratio • Transaction costs • 5 basis points per trade • (Each leg=separate trade)

23. Portfolio Performance 1994–2007

24. Portfolio Performance

25. Correlation Matrix betweenSub-portfolios

26. Conclusions and Future Research • Functional representation of time series • Make (relatively) long-term forecasts • Progressively-revealed information sets • Handle irregularly-sampled data • Trading decisions based on IR-like criterion • Good out-of-sample performance on a portfolio of 30 commodity spreads • Limits of Gaussian processes: computation time grows as O(N3) with the data size • Approximation methods to handle larger data sets