Quantitative Models

Mathematical and statistical models for understanding and predicting market behavior.

Key Points

• Factor models (Fama-French, Carhart) explain most of the cross-sectional variation in stock returns; alpha is what’s left after factor exposure
• Cointegration (Engle-Granger, Johansen) is the rigorous foundation for pairs trading; correlation alone is not enough
• GARCH(1,1) is the workhorse for volatility forecasting; realized vol captures intra-day moves that close-to-close GARCH misses
• Machine learning in finance: tree-based models (XGBoost) for tabular features, LSTMs for sequences, transformers for alternative data; all are easy to overfit
• Regularization (L1/L2) prevents overfit; out-of-time validation is more important than in-fold cross-validation
• The “no free lunch” theorem says any consistent alpha must be paired with a real risk premium or behavioral edge

Strategies

Statistical Arbitrage via Cointegration

Build a cointegrated basket using Johansen test on historical price ratios. Trade z-score deviations of the spread, with mean-reversion decay factor.

Factor-Tilted Portfolio

Long high-quality, low-volatility names; short the opposite. Captures risk-premium without taking market beta.

ML-Based Cross-Sectional Alpha

Train a regularized gradient-boosted tree on fundamental + price features. Use as a stock-rank signal, fade the worst decile, long the best.

Metrics & Formulas

• Sharpe of factors: SR_factor over SR_market benchmark
• Information ratio: α / tracking error
• GARCH(1,1): σ²(t) = ω + α × ε²(t-1) + β × σ²(t-1)
• Johansen trace statistic — for cointegration rank testing

Tools & Resources

• statsmodels — Time series and econometric models in Python
• PyFlux / arch — GARCH and stochastic volatility
• XGBoost / LightGBM — Gradient-boosted trees for tabular ML
• Zipline — Built-in factor model library