Fourier Neural Network Approximation of Transition Densities in Finance

ArXiv ID: 2309.03966 “View on arXiv”

Authors: Unknown

Abstract

This paper introduces FourNet, a novel single-layer feed-forward neural network (FFNN) method designed to approximate transition densities for which closed-form expressions of their Fourier transforms, i.e. characteristic functions, are available. A unique feature of FourNet lies in its use of a Gaussian activation function, enabling exact Fourier and inverse Fourier transformations and drawing analogies with the Gaussian mixture model. We mathematically establish FourNet’s capacity to approximate transition densities in the $L_2$-sense arbitrarily well with finite number of neurons. The parameters of FourNet are learned by minimizing a loss function derived from the known characteristic function and the Fourier transform of the FFNN, complemented by a strategic sampling approach to enhance training. We derive practical bounds for the $L_2$ estimation error and the potential pointwise loss of nonnegativity in FourNet for $d$-dimensions ($d\ge 1$), highlighting its robustness and applicability in practical settings. FourNet’s accuracy and versatility are demonstrated through a wide range of dynamics common in quantitative finance, including Lévy processes and the Heston stochastic volatility models-including those augmented with the self-exciting Queue-Hawkes jump process.

Keywords: FourNet, Gaussian activation function, Characteristic function, Heston model, Lévy processes, Equity Derivatives

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail
• Why: The paper demonstrates high mathematical complexity through rigorous $L_2$-analysis, universal approximation theorems, and multi-dimensional error bounds, while maintaining strong empirical rigor with validated demonstrations across diverse financial models (Lévy, Heston, Queue-Hawkes) and practical error controls for real-world applications.

  flowchart TD
    A["Research Goal"] --> B["FourNet Architecture"]
    A --> C["Input Data"]
    subgraph B ["Neural Network Structure"]
        B1["Gaussian Activation"]
        B2["Fourier Transform"]
    end
    subgraph C ["Financial Models"]
        C1["Levy Processes"]
        C2["Heston Model"]
        C3["Queue-Hawkes Jumps"]
    end
    B --> D["Training Process"]
    C --> D
    D --> E["Loss Function"]
    D --> F["Strategic Sampling"]
    E --> G["Minimization"]
    F --> G
    G --> H["Key Outcomes"]
    subgraph H ["Findings"]
        H1["Arbitrary L2 Approximation"]
        H2["Pointwise Error Bounds"]
        H3["Preserves Nonnegativity"]
    end