We show that a deep neural network (DNN) trained to construct a stochastic discount factor (SDF) admits a sharp additive decomposition that separates nonlinear characteristic discovery from the pricing rule that aggregates them. The economically relevant component of this decomposition is governed by a new object, the Portfolio Tangent Kernel (PTK), which captures the features learned by the network and induces an explicit linear factor pricing representation for the SDF. In population, the PTK-implied SDF converges to a ridge-regularized version of the true SDF, with the effective strength of regularization determined by the spectral complexity of the PTK. Using U.S. equity data, we show that the PTK representation delivers large and statistically significant performance gains, while its spectral complexity has risen sharply-by roughly a factor of six since the early 2000s-imposing increasingly tight limits on finite-sample pricing performance.

翻译：我们证明，用于构建随机贴现因子（SDF）的深度神经网络（DNN）存在一种清晰的加性分解，该分解将非线性特征发现与聚合这些特征的定价规则分离开来。该分解的经济学相关部分由一个称为投资组合切线核（PTK）的新对象主导，该核捕获了网络学习到的特征，并为SDF诱导出一个显式的线性因子定价表示。在总体中，PTK隐含的SDF收敛于真实SDF的岭正则化版本，其正则化的有效强度由PTK的谱复杂度决定。利用美国股票数据，我们表明PTK表示带来了巨大且统计上显著的性能提升，同时其谱复杂度自21世纪初以来急剧上升——大约增加了六倍——对有限样本定价性能施加了日益严格的限制。