Authors: Bryan Kelly, Boris Kuznetsov, Semyon Malamud, Yuan Zhang
Organizations: Yale School of Management, AQR Capital Management, and NBER · AQR Capital Management · NBER · Swiss Finance Institute, EPFL, and CEPR · EPFL · CEPR · AQR · Shanghai University of Finance and Economics
We show that a deep neural network (DNN) trained to construct a stochastic discount factor (SDF) admits an additive decomposition separating nonlinear characteristic discovery from the pricing rule that aggregates them. This decomposition yields a linear factor representation governed by the Portfolio Tangent Kernel (PTK), which summarizes the network's learned features. In population, the implied SDF converges to a ridge-regularized version of the true SDF, with the degree of regularization determined by spectral complexity. Empirically, using U.S. equity data, the PTK representation delivers economically and statistically significant performance gains, while rising spectral complexity imposes tighter limits on finite-sample pricing.