We reveal a geometric structure underlying both hedging and investment products. The structure follows from a simple formula expressing investment risks in terms of returns. This informs optimal product designs. Optimal pure hedging (including cost-optimal products) and hybrid hedging (where a partial hedge is built into an optimal investment product) are considered. Duality between hedging and investment is demonstrated with applications to optimal risk recycling. A geometric interpretation of rationality is presented.

翻译：我们揭示了风险对冲与投资产品背后共有的几何结构，该结构源于一个用收益表达投资风险的简洁公式，这为最优产品设计提供了依据。本文探讨了纯对冲（包含成本最优产品）与混合对冲（将部分对冲嵌入最优投资产品中）两类情形。通过最优风险循环的应用案例，论证了对冲与投资之间的对偶性，并给出了理性概念的几何诠释。