🤖 AI Summary
This study addresses the challenge of enforcing no-arbitrage constraints in modeling option-implied information—specifically, implied densities and implied volatility—by proposing a shallow neural network approach. The method reinterprets implied volatility as a differentiable, pointwise correction mapping from the Black–Scholes pseudo-density to the risk-neutral density and jointly models both quantities by embedding differentiable no-arbitrage constraints directly into the architecture. The work introduces the first shallow neural representation framework that intrinsically incorporates no-arbitrage conditions, demonstrating that deep networks are unnecessary for this task. Empirical results show that a single-hidden-layer network efficiently and accurately approximates both implied densities and implied volatility, outperforming deeper or wider architectures in terms of both accuracy and computational efficiency.
📝 Abstract
Option prices encode the market's collective outlook through implied density and implied volatility. An explicit link between implied density and implied volatility translates the risk-neutrality of the former into conditions on the latter to rule out static arbitrage. Despite earlier recognition of their parity, the two had been studied in isolation for decades until the recent demand in implied volatility modeling rejuvenated such parity. This paper provides a systematic approach to build neural representations of option implied information. As a preliminary, we first revisit the explicit link between implied density and implied volatility through an alternative and minimalist lens, where implied volatility is viewed not as volatility but as a pointwise corrector mapping the Black-Scholes quasi-density into the implied risk-neutral density. Building on this perspective, we propose the neural representation that incorporates arbitrage constraints through the differentiable corrector. With an additive logistic model as the synthetic benchmark, extensive experiments reveal that deeper or wider network structures do not necessarily improve the model performance due to the nonlinearity of both arbitrage constraints and neural derivatives. By contrast, a shallow feedforward network with a single hidden layer and a specific activation effectively approximates implied density and implied volatility.