Outperforming a Benchmark with α-Bregman Wasserstein divergence

Battle Count: 72

The paper is highly relevant to quantitative portfolio management, particularly for active fund managers who need to outperform a benchmark while controlling tracking error. The α-BW divergence provides a flexible framework for asymmetric penalization of under- vs. outperformance, which is directly applicable to performance fee structures and benchmark-relative investing. The quantile formulation provides explicit optimal strategies. However, the theoretical nature, complete market assumption, and lack of empirical validation limit immediate practical implementation. The framework is most relevant for institutional portfolio managers and quantitative strategists working on benchmark-constrained optimization.

The theoretical framework is mathematically sophisticated, requiring knowledge of optimal transport, convex analysis, isotonic regression, and Lagrangian optimization. The optimal quantile function involves isotonic projections, which require specialized numerical algorithms. The case analysis for under- vs. outperformance adds complexity. The GBM numerical example is tractable, but generalization to realistic market models with stochastic parameters, transaction costs, and multiple constraints would significantly increase implementation difficulty. The Power family Bregman generator simplifies computation but the general case requires careful numerical handling of the indicator functions in the α-BW divergence.

The paper provides a complete mathematical framework with explicit formulas for the optimal quantile functions, the GBM market model parameters, and the Power family Bregman generator. Numerical illustrations include specific parameter choices (R=1, Γ=2, Ψ=0.8, x₀=1, c=0.9, γ=1/2, ε=0.5). However, no code repository is provided, and the numerical implementation details (e.g., how isotonic projections are computed numerically) are not fully specified. The theoretical results are self-contained with all proofs included.