Shallow Representation of Option Implied Information

TL;DR

This paper introduces a shallow neural network approach to efficiently model option implied information, linking implied density and volatility while respecting arbitrage constraints, and demonstrates its effectiveness over deeper models.

Contribution

The paper presents a novel shallow neural network framework that captures option implied density and volatility with arbitrage constraints, challenging the assumption that deeper networks are always better.

Findings

Shallow networks outperform deeper ones in modeling implied information.

The proposed method effectively incorporates arbitrage constraints.

Neural derivatives reveal nonlinearity effects on model performance.

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.