Forecasting and Trading Commodity Contract Spreads with Gaussian

Processes

Nicolas Chapados and Yoshua Bengio

University of Montrealand

ApSTAT Technologies Inc.

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

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

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

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

Jul-Dec CME Lean Hogs15-year average (1991-2005)

Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul0

20

40

60

80

100

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

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)

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 + b yt +

with ~ N(0, • A Gaussian Process can represent the

joint distribution of all future prices, in particular P(pt+|It, ), for >0.

Gaussian Processes

• General tools for nonlinear regression• Fully Bayesian Treatment

1. Start with a prior probability distribution on the space of functions

2. Observe some data3. Infer a posterior distribution, given

the observed data (from Bayes’ rule)

Example

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

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

Historical Data: March–July Wheat

NormalizedPrice

Year Days to Maturity

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

Example of Forecast given History

Wheat March–July / 1996

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)

Evaluation Methodology

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

Forecasting Performance

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

Behavior on a Single Trading Year

Wheat March–July / 1996• Re-train model every 25 days• Sequence of decisions: short – neutral –

long• Lower panel: Cumulative P&L ($)

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)

Commodity Maturities (short-long)

Cotton 10–12, 10–5

FeederCattle 11–3, 8–10

Gasoline 1–5

LeanHogs12–2, 2–4, 2–6, 4–6, 7–12, 8–10, 8–12

LiveCattle 2–8

NaturalGas 6–9, 6–12

SoybeanMeal 5–9, 7–9, 7–12, 8–3, 8–12

Soybeans 1–7, 7–11, 7–9, 8–3, 8–11

Wheat 3–7, 3–9, 5–9, 5–12, 7–12

Portfolio Performance 1994–2007

Portfolio Performance

Correlation Matrix betweenSub-portfolios

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