Taming the Greeks: Option Portfolios with Inductive Biases

Accurate option prices do not imply accurate recovery of the latent risk-neutral density. We study this distinction with two complementary benchmarks. A controlled benchmark exposes simulator-truth densities for latent evaluation, while a chronological NIFTY benchmark tests only held-out market prices. A two-component lognormal mixture has the lowest aggregate price, $L^1$, Wasserstein, and fixed-tail errors on the synthetic benchmark. Learned operators retain narrower strengths: DeepONet reduces 1% quantile and variance error by 39.0% and 34.6% relative to the mixture, and a quote transformer reduces $L^1$ by 16.4% on the structurally misspecified Merton family. A numerical conditioning analysis explains why these rankings can differ: after enforcing mass and forward constraints, 95 of 126 pricing directions are numerically null, and two densities separated by $L^1 = 0.061$ produce identical prices on the covered strikes. On 524 held-out NIFTY calls, validation-selected test-time adaptation reduces DeepONet RMSE by 28.3%, but per-expiry mixture and SVI fits remain much more accurate. The evidence supports target-dependent inductive bias, not a universal winner.

We introduce a deep kernel hedging framework that combines the flexibility of deep learning with the structural inductive bias of kernel methods. The hedging functional is restricted to a reproducing kernel Hilbert space whose kernel is parameterized through a neural network embedding of the input features. The framework minimizes a regularized empirical risk under convex loss functions and can accommodate path-dependent information through truncated time-augmented signature features. We derive a generalized representer theorem for the joint hedging problem, reducing the empirical optimization to a finite-dimensional problem. To further reduce the computational cost associated with large kernel matrices, we develop a scalable random Fourier feature approximation and establish convergence guarantees. The random Fourier parameters are sampled once and remain fixed throughout training, while the deep kernel adapts to market data through the learned neural representation. We evaluate the performance of the proposed deep kernel approach on both synthetic and real data and compare it with standard kernel methods and classical deep hedging architectures. Numerical results indicate competitive and robust hedging performance, particularly in low-data regimes, which highlights the benefits of combining expressive neural representations with the inductive bias of kernel methods.

Portfolio optimization in real-world financial markets is notoriously difficult due to non-stationarity, noisy data, and high transaction costs. Standard predict-then-optimize methods first forecast returns and then solve for weights, compounding prediction errors and often failing under regime shifts. We propose an end-to-end framework that directly optimizes differentiable surrogates of key financial metrics (Sharpe ratio, Omega ratio, Conditional Value-at-Risk, and risk parity), allowing neural networks to learn portfolio weights via backpropagation. Our expanding-window walk-forward procedure, applied to 50 S&P 500 stocks from 2007 to 2023, incorporates realistic bid-ask spread costs and rebalances quarterly. On the challenging out-of-sample test period (2022-2023), the best model, an AttentionLSTM with the Omega-CVaR-RiskParity loss, achieves an annualized Sharpe of 0.29 and a total compounded return of +7.86%, while the S&P 500 delivers -4.52% total compounded return and an annualized Sharpe of -0.02. This outperforms the S&P 500 by 12.38 percentage points, while keeping tail risk (CVaR) nearly unchanged. The framework outperforms the equal-weight portfolio, S&P 500, and traditional methods (MVP, HRP, NCO, ERC), demonstrating that embedding financial objectives directly into model training yields robust, economically meaningful outperformance even in adverse market conditions.