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Quantum Machine Learning methods for Fourier-based distribution estimation with application in option pricing

Fernando Alonso, Álvaro Leitao, Carlos Vázquez

TL;DR

This study targets efficient option pricing by reconstructing Fourier representations of distributions from Parametrized Quantum Circuits. It presents two QML-based approaches (PDF-focused supervised learning and CDF-focused self-supervised learning) plus a QAMC-based benchmark (mRQAE) to extract Fourier coefficients and price derivatives. The methods are evaluated on European puts under Black–Scholes dynamics, showing convergence with increasing data and circuit capacity, with Method II notably competitive using ~10^4 samples and without requiring explicit PDFs. Collectively, the work demonstrates that quantum-classical hybrids can achieve high accuracy in derivatives valuation and offer practical alternatives to traditional QAMC approaches for Fourier-based distribution estimation.

Abstract

The ongoing progress in quantum technologies has fueled a sustained exploration of their potential applications across various domains. One particularly promising field is quantitative finance, where a central challenge is the pricing of financial derivatives-traditionally addressed through Monte Carlo integration techniques. In this work, we introduce two hybrid classical-quantum methods to address the option pricing problem. These approaches rely on reconstructing Fourier series representations of statistical distributions from the outputs of Quantum Machine Learning (QML) models based on Parametrized Quantum Circuits (PQCs). We analyze the impact of data size and PQC dimensionality on performance. Quantum Accelerated Monte Carlo (QAMC) is employed as a benchmark to quantitatively assess the proposed models in terms of computational cost and accuracy in the extraction of Fourier coefficients. Through the numerical experiments, we show that the proposed methods achieve remarkable accuracy, becoming a competitive quantum alternative for derivatives valuation.

Quantum Machine Learning methods for Fourier-based distribution estimation with application in option pricing

TL;DR

This study targets efficient option pricing by reconstructing Fourier representations of distributions from Parametrized Quantum Circuits. It presents two QML-based approaches (PDF-focused supervised learning and CDF-focused self-supervised learning) plus a QAMC-based benchmark (mRQAE) to extract Fourier coefficients and price derivatives. The methods are evaluated on European puts under Black–Scholes dynamics, showing convergence with increasing data and circuit capacity, with Method II notably competitive using ~10^4 samples and without requiring explicit PDFs. Collectively, the work demonstrates that quantum-classical hybrids can achieve high accuracy in derivatives valuation and offer practical alternatives to traditional QAMC approaches for Fourier-based distribution estimation.

Abstract

The ongoing progress in quantum technologies has fueled a sustained exploration of their potential applications across various domains. One particularly promising field is quantitative finance, where a central challenge is the pricing of financial derivatives-traditionally addressed through Monte Carlo integration techniques. In this work, we introduce two hybrid classical-quantum methods to address the option pricing problem. These approaches rely on reconstructing Fourier series representations of statistical distributions from the outputs of Quantum Machine Learning (QML) models based on Parametrized Quantum Circuits (PQCs). We analyze the impact of data size and PQC dimensionality on performance. Quantum Accelerated Monte Carlo (QAMC) is employed as a benchmark to quantitatively assess the proposed models in terms of computational cost and accuracy in the extraction of Fourier coefficients. Through the numerical experiments, we show that the proposed methods achieve remarkable accuracy, becoming a competitive quantum alternative for derivatives valuation.
Paper Structure (19 sections, 3 theorems, 68 equations, 8 figures, 2 tables)

This paper contains 19 sections, 3 theorems, 68 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic concepts on options pricing
  4. PQCs as universal approximators
  5. Quantum accelerated Monte Carlo techniques
  6. Formulation and methodology
  7. Fourier series approximation of PDF and CDF
  8. PDF approximation formulation
  9. CDF approximation formulation
  10. Methodology
  11. Method I: Supervised Learning for PDF approximation
  12. Method II: Self-supervised Learning for CDF approximation
  13. Method III: QAMC with mRQAE for PDF approximation
  14. Numerical results
  15. Experiment setting
  16. ...and 4 more sections

Key Result

Theorem 2.1

(Convergence in $C^0$) Let $\{H_m\}$ be a universal Hamiltonian family, and $\{f_m\}$ the associated quantum model family (quantum_model_family). For all functions $f^{\ast} \in C_{0}(U)$ where $U$ is compactly contained in $[0,2\pi]^N$, and for all $\epsilon > 0$, there exists some $m' \in \mathbb{ with

Figures (8)

  • Figure 1: Scheme of a quantum circuit composed of $L$ layers, where each layer consists of a trainable circuit block $W^{(i)}(\boldsymbol{\theta})$, with $i \in \{1, \ldots, L\}$, and a data-encoding block $S_H(x)$, from Schuld_2021.
  • Figure 2: Approximation of the PDF with training in different intervals.
  • Figure 3: Approximation of the CDF with training in different intervals.
  • Figure 4: Scheme of a single layer of the quantum ansatz used to construct the PQC for 2 qubits, extracted from manzano2024thesis.
  • Figure 5: Convergence results for Method I (left column) and Method II (right column).
  • ...and 3 more figures

Theorems & Definitions (4)

  • Definition 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3