Deep kernel hedging

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.

Deep neural networks dominate modern machine learning, while alternative function approximators remain comparatively underexplored at scale. In this work, we revisit kernel methods as drop-in components for standard deep learning pipelines. We introduce \emph{Sparse Kernels} (SKs), a differentiable, localized, and lazy variant of kernel ridge regression (KRR) that defers training to inference time and reduces to the solution of small local systems. We integrate SKs into PyTorch as modular layers that preserve end-to-end trainability, and we show that they expose three distinct sets of parameters -- feature representations, target values, and evaluation points -- each of which can be fixed or learned. This decomposition broadens the design space available to practitioners, enabling, in particular, training-free transfer, nonlinear probing, and hybrid kernel-neural models. Across convolutional networks, vision transformers, and reinforcement learning, SK-based modules serve two complementary roles: in some settings, they match the performance of trained neural readouts with substantially less training; in others, they augment existing models and improve their performance when used as additional components. Our results suggest that kernel methods, once made scalable and differentiable, can be readily integrated with deep learning rather than treated as a separate paradigm.

We present an end-to-end deep learning framework for systematic options trading that directly embeds hedging behavior through explicit control of portfolio-level risk exposures. While neural networks trained to optimize risk-adjusted performance have been shown to outperform traditional rules-based strategies, such approaches remain agnostic to the sensitivities of the resulting portfolios with respect to specific underlying risk factors. We propose a general training objective that combines a performance-driven loss with a differentiable risk-sensitivity penalty, enforcing neutrality to selected risk dimensions. Unlike reinforcement learning methods that approximate optimal hedging policies via simulated market dynamics, our framework operates entirely on historical data and jointly optimizes risk-adjusted returns and targeted risk constraints in a single learning problem. We instantiate the framework on static delta-neutral straddle portfolios with the penalty directed at first-order directional exposure, and evaluate two penalty variants -- an exposure-normalized penalty and a Greek-ratio drift penalty. Empirical results on Nasdaq 100 equity options demonstrate that appropriately calibrated regularization simultaneously improves out-of-sample risk-adjusted performance relative to an unregularized baseline while reducing realized directional exposure.