Fixed Income Quant Trading

The Setup

Eurodollar (ED) futures are among the most liquid instruments in fixed income markets. Their constant-maturity rates at different tenors (2y, 3y, 4y, 5y) move together but not identically — the spreads between them carry information about the term structure and mean-revert over time. A butterfly trade exploits this by going long the wings and short the belly (or vice versa), betting that a temporary distortion in the curve will correct itself.

The question is how to weight the legs of the butterfly so the resulting spread is stationary, and how to build a forecasting model that tells you when to enter and exit.

Cointegrated Butterflies

The first step is constructing butterfly spreads that are genuinely cointegrated — not just correlated. We formulated three butterfly combinations from constant-maturity ED rates:

The belly weight is fixed at 1. The wing weights are estimated using the Box-Tiao canonical decomposition (Canonical Correlation Analysis between levels and their lags), which finds the linear combination with maximum mean-reversion speed. This is more principled than ad-hoc weight selection — the CCA eigenvector corresponding to the largest eigenvalue gives the combination that reverts fastest.

Weights were estimated on Sample A (2010-2014), then held fixed for all subsequent analysis.

Signal Construction

With the butterfly spreads defined, we built three forecasting signals on Sample B (2014-2016), each capturing a different view of the spread’s deviation from equilibrium.

Define z(t, λ) as the butterfly level minus its exponential moving average with decay parameter λ:

Signal 1 (λ = 0) — AR(1) fitted to the raw butterfly spread (constant mean case). This is the simplest model: the spread deviates from its historical mean and reverts.

Signal 2 (λ = 0.05) — AR(1) fitted to the spread minus a slow-moving EMA. The EMA absorbs gradual structural shifts in the curve, so the signal focuses on shorter-term dislocations.

Signal 3 (λ = 0.1) — AR(1) fitted to the spread minus a faster EMA. More aggressive detrending — captures only the highest-frequency mean-reversion.

Each AR(1) model is estimated in a rolling 6-month window and produces a 5-day-ahead forecast. The rolling estimation allows the model parameters to adapt as market dynamics shift, while the fixed horizon provides a consistent signal for trading decisions.

Signal Combination

Signals 1-3 each capture different frequencies of mean-reversion. Rather than choosing one, we combined them:

Signal 4 = w₁ · Signal 1 + w₂ · Signal 2 + w₃ · Signal 3

The weights w₁, w₂, w₃ were optimized over the range [0, 1] on Sample B to minimize forecast error. The combined signal outperformed any individual signal by blending the structural (low λ) and tactical (high λ) views of the spread.

Signal Quality Metrics

We evaluated all four signals using multiple quality metrics applied across different samples to test both in-sample fit and out-of-sample generalization:

Forecast accuracy — RMSE, MAE, and MSE between predicted and realized butterfly values. All signals achieved RMSE below 0.1, with close alignment between training and test performance — indicating the models generalize rather than overfit.

Stationarity — Augmented Dickey-Fuller tests on the butterfly spreads confirm that the cointegrated combinations are genuinely stationary, validating the Box-Tiao weight estimation.

Mean-reversion speed — Half-life of mean reversion, measuring how quickly the spread returns to equilibrium after a deviation. Shorter half-life means faster reversion and more trading opportunities.

Distribution analysis — Examining the distribution of signal residuals across all butterflies and signal types to check for skewness, fat tails, or regime-dependent behavior.

Metrics were applied on the last year of Sample B (in-sample) and on Sample C (2016-2018, out-of-sample). The half-life increased in Sample C relative to Sample B, indicating slower mean-reversion in the later period — a regime shift that the adaptive EMA-based signals (2 and 3) handled better than the constant-mean Signal 1.

Key Takeaways

The project demonstrates that classical statistical methods — cointegration, CCA, AR(1) with rolling estimation — remain highly effective for fixed income relative value when properly applied. The Box-Tiao decomposition provides theoretically grounded butterfly weights, the EMA-parameterized signals separate structural from tactical deviations, and the signal combination framework lets you blend multiple time horizons without overfitting. The out-of-sample analysis on Sample C confirms that these methods generalize across market regimes, though with degraded mean-reversion speed that the quality metrics clearly flag.