风险最优资产再配置的微分几何框架
A Differential-Geometric Framework for Risk-Optimal Asset Reallocation
- National and Kapodistrian University of Athens（雅典国家卡波季斯特里安大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出微分几何框架，将投资组合再配置风险建模为黎曼长度，以测地线实现最小风险转移，实验表明在拥挤区域收益显著。
AI中文摘要：
我们发展了一个用于风险最优投资组合再配置的微分几何框架。一个纯多头投资组合被表示为概率单纯形上的一个点，并赋予一个正定黎曼度量，该度量结合了市场协方差风险与依赖于头寸的集中风险。再平衡轨迹的累积风险被等同于其黎曼长度，因此当前配置与马科维茨目标之间的最小风险转移是一条测地线。我们将这条路径与直接线性再平衡和投影梯度上升进行比较。当仅对市场协方差风险定价时，度量是常值且平坦的，测地线恰好是直线路径。一旦引入依赖于头寸的风险，几何变得弯曲，测地线在相同端点的竞争路径中弱占优。一个Fisher-Rao集中项产生适度但系统的节省，而一个内生拥挤度量产生非凸风险脊，测地线可通过临时分散化绕过这些脊。数值实验、蒙特卡洛转移和回归分析表明，当直接路径穿越强拥挤区域时，收益最大。该框架提供了转移管理的一般几何表述，并能容纳更丰富的风险度量和交易成本结构。
英文摘要：
We develop a differential-geometric framework for risk-optimal portfolio reallocation. A long-only portfolio is represented as a point of the probability simplex, endowed with a positive-definite Riemannian metric combining market covariance risk with position-dependent concentration risk. The cumulative risk of a rebalancing trajectory is identified with its Riemannian length, so the least-risk transition between a current allocation and a Markowitz target is a geodesic. We compare this route with direct linear rebalancing and projected gradient ascent. When only market covariance risk is priced, the metric is constant and flat, and the geodesic is exactly the straight-line path. Once position-dependent risk is introduced, the geometry becomes curved and the geodesic weakly dominates competing paths with the same endpoints. A Fisher-Rao concentration term produces modest but systematic savings, while an endogenous crowding metric creates non-convex risk ridges that geodesics can bypass through temporary diversification. Numerical experiments, Monte Carlo transitions, and regression analysis show that the largest gains occur when the direct path crosses strongly crowded regions. The framework provides a general geometric formulation of transition management and can accommodate richer risk metrics and transaction-cost structures.
