Paper: arXiv 2403.14231

Authors: Sébastien Bossu, Stéphane Crépey, Hoang-Dung Nguyen

Abstract

We propose a distributional formulation of the spanning problem of a multi-asset payoff by vanilla basket options. This problem is shown to have a unique solution if and only if the payoff function is even and absolutely homogeneous, and we establish a Fourier-based formula to calculate the solution. Financial payoffs are typically piecewise linear, resulting in a solution that may be derived explicitly, yet may also be hard to numerically exploit. One-hidden-layer feedforward neural networks instead provide a natural and efficient numerical alternative for discrete spanning. We test this approach for a selection of archetypal payoffs and obtain better hedging results with vanilla basket options compared to industry-favored approaches based on single-asset vanilla hedges.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 5.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper introduces an innovative theoretical framework using Fourier transforms and neural network activations (ReLUs) to solve the multi-asset payoff spanning problem. While the mathematical derivations are rigorous and novel, the empirical section focuses on standard archetypal payoffs rather than extensive market backtests.

Research Flowchart

  flowchart TD
    A["Research Goal: Span Multi-Asset Payoffs<br/>using Vanilla Basket Options"] --> B["Distributional Formulation &<br/>Fourier-Based Solution"]
    
    B --> C{"Conditions Met?<br/>(Even & Absolutely Homogeneous)"}
    
    C -- "Theoretical Exact Solution" --> D["Piecewise Linear Payoffs<br/>(Hard to exploit numerically)"]
    
    D --> E["1-Hidden-Layer Neural Network<br/>(Efficient Discrete Spanning)"]
    
    F["Data Inputs: Archetypal Financial<br/>Multi-Asset Payoffs"] --> E
    
    E --> G["Key Outcome: Superior Hedging Performance<br/>vs. Single-Asset Industry Approaches"]

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1 of 69 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.