Paper: arXiv 2512.11731

Authors: Andrea Conti, Giacomo Morelli

Abstract

The estimation of the Risk Neutral Density (RND) implicit in option prices is challenging, especially in illiquid markets. We introduce the Deep Log-Sum-Exp Neural Network, an architecture that leverages Deep and Transfer learning to address RND estimation in the presence of irregular and illiquid strikes. We prove key statistical properties of the model and the consistency of the estimator. We illustrate the benefits of transfer learning to improve the estimation of the RND in severe illiquidity conditions through Monte Carlo simulations, and we test it empirically on SPX data, comparing it with popular estimation methods. Overall, our framework shows recovery of the RND in conditions of extreme illiquidity with as few as three option quotes.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 8.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents an innovative Deep Log-Sum-Exp Neural Network architecture combined with transfer learning to estimate Risk-Neutral Densities under extreme illiquidity. It provides rigorous mathematical proofs for universal approximation and consistency alongside extensive Monte Carlo simulations and SPX empirical tests.

Research Flowchart

  flowchart TD
    A[Goal: RND Estimation in Illiquid Markets] --> B[Input: SPX Option Data & Simulated Strikes]
    B --> C[Model: Deep Log-Sum-Exp Neural Network]
    C --> D[Method: Apply Transfer Learning Framework]
    D --> E[Evaluation: Monte Carlo Simulations & SPX Empirical Tests]
    E --> F[Findings: Robust RND Recovery with as few as 3 Quotes]

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