Download Introduction - University of Guelph

Forecasting and Trading Commodity
Contract Spreads with Gaussian
Processes
Nicolas Chapados and Yoshua Bengio
University of Montreal
and
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
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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 Hogs
15-year average (1991-2005)
Aug Sep Oct
Nov Dec Jan Feb Mar Apr May Jun Jul
100
80
60
40
20
0
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 + 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.
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 data
3. 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 inputs
y — M-vector of training targets
X* — 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
Normalized
Price
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-touse ratio)
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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
Commodity
Maturities (short-long)
Cotton
10–12, 10–5
FeederCattle
11–3, 8–10
Gasoline
1–5
LeanHogs
12–2, 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
• 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)
Portfolio Performance 1994–2007
Portfolio Performance
Correlation Matrix between
Sub-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