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.

翻译：期权价格通过隐含密度和隐含波动率编码了市场的集体预期。隐含密度与隐含波动率之间的显式联系将前者的风险中性转化为后者的约束条件，以排除静态套利。尽管二者对等性早已被认识，但数十年来它们一直被孤立研究，直至近期对隐含波动率建模的需求重新激活了这种对等关系。本文提出了一种构建期权隐含信息神经表示的系统性方法。作为预备，我们首先通过一种替代性且极简的视角重新审视隐含密度与隐含波动率之间的显式联系，其中隐含波动率不被视为波动率，而是作为逐点校正器，将Black-Scholes准密度映射为隐含风险中性密度。基于这一视角，我们提出了通过可微校正器融入套利约束的神经表示。以加性逻辑模型作为合成基准，大量实验表明，由于套利约束和神经导数的非线性，更深或更宽的网络结构并不必然提升模型性能。相比之下，具有单隐藏层和特定激活函数的浅层前馈网络能有效逼近隐含密度与隐含波动率。