Path Portfolio Optimization: Defect, Lift, and the Price of Path Complexity

Battle Count: 79

Directly addresses portfolio optimization with path-dependent strategies. Provides exact algebraic framework for signature-linear portfolios, identifies excess growth as geometricity defect, establishes lift-invariance of directional signals (lead-lag), quantifies estimation cost of path complexity, and provides practical guidance on shrinkage and model-consistent estimation. The dimensional trade-off analysis (11x gain at d=2, 60x at d=20, but requiring 6+ observations per parameter) is directly actionable for quantitative portfolio managers. The activity-vs-direction decomposition of cross-area trading is relevant for pairs trading and lead-lag strategies.

Requires deep expertise in rough path theory, tensor algebra, shuffle products, and expected signatures. Implementation involves: (1) computing truncated signatures up to level m in d dimensions (p=d+d^2 parameters at level 2), (2) assembling the defect form via shuffle products, (3) solving the linear system for optimal weights, (4) computing lift gaps between Marcus and forward conventions, (5) implementing model-consistent estimation via tensor exponential and shuffle reconstruction, (6) handling Hawkes state-weight augmentation for self-exciting drivers. The algebraic structure is elegant but computationally demanding, especially at higher truncation levels or cross-sectional widths. No reference implementation or library is provided.

The paper provides complete analytical formulas, explicit calibrations (drift, covariance, risk aversion), simulation parameters (number of paths, steps, replications), and verification tolerances (machine precision checks). However, no code or repository is provided. All parameters are stated but must be re-implemented from scratch. The tensor algebra and shuffle product computations require specialized libraries. No market data is used, so results are fully reproducible in principle but computationally intensive.