Mathematics of Differential Machine Learning in Derivative Pricing and Hedging

ArXiv ID: 2405.01233 “View on arXiv”

Authors: Unknown

Abstract

This article introduces the groundbreaking concept of the financial differential machine learning algorithm through a rigorous mathematical framework. Diverging from existing literature on financial machine learning, the work highlights the profound implications of theoretical assumptions within financial models on the construction of machine learning algorithms. This endeavour is particularly timely as the finance landscape witnesses a surge in interest towards data-driven models for the valuation and hedging of derivative products. Notably, the predictive capabilities of neural networks have garnered substantial attention in both academic research and practical financial applications. The approach offers a unified theoretical foundation that facilitates comprehensive comparisons, both at a theoretical level and in experimental outcomes. Importantly, this theoretical grounding lends substantial weight to the experimental results, affirming the differential machine learning method’s optimality within the prevailing context. By anchoring the insights in rigorous mathematics, the article bridges the gap between abstract financial concepts and practical algorithmic implementations.

Keywords: Differential machine learning, Derivative pricing, Neural networks, Hedging, Theoretical finance

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail
  • Why: The paper presents a rigorous mathematical foundation using Hilbert spaces, Hahn-Banach theorem, and generalized function theory, yet it includes experimental sections comparing neural network and monomial bases on Black-Scholes, with hedging PnL results.
  flowchart TD
    A["Research Goal: Validate DML for<br>Derivative Pricing & Hedging"] --> B["Methodology: Rigorous Mathematical<br>Framework"]
    B --> C{"Data/Inputs: Market Scenarios<br>&amp; Model Parameters"}
    C --> D["Computational Process:<br>Neural Network Training w/ Differential Constraints"]
    D --> E["Key Findings: Unified Theory,<br>Optimal Hedging, Reduced Pricing Error"]
    E --> F["Outcome: Bridge between<br>Theoretical Finance &amp; Practical AI"]