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Mathematics of Differential Machine Learning in Derivative Pricing and Hedging

Pedro Duarte Gomes

TL;DR

The paper develops a rigorous, Hilbert-space–based framework for pricing and hedging derivative claims under a risk-neutral measure, highlighting the importance of unbiased differential labels in differential machine learning. It unifies two approaches—Least Squares Monte Carlo (LSMC) and differential ML—via a loss that combines payoff error and weak-derivative error within Sobolev space $H^1$, enabling delta estimation even for non-smooth payoffs. By leveraging generalized function theory, it derives unbiased delta estimators and shows how neural-network bases can serve as flexible function spaces, with depth and width affecting approximation power. Numerical experiments in a Black–Scholes setting demonstrate that differential ML with neural-network bases achieves smaller hedging errors and more favorable PnL distributions than traditional LSMC, establishing a mathematically grounded path from theory to improved derivative valuation and hedging in practice.

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.

Mathematics of Differential Machine Learning in Derivative Pricing and Hedging

TL;DR

The paper develops a rigorous, Hilbert-space–based framework for pricing and hedging derivative claims under a risk-neutral measure, highlighting the importance of unbiased differential labels in differential machine learning. It unifies two approaches—Least Squares Monte Carlo (LSMC) and differential ML—via a loss that combines payoff error and weak-derivative error within Sobolev space , enabling delta estimation even for non-smooth payoffs. By leveraging generalized function theory, it derives unbiased delta estimators and shows how neural-network bases can serve as flexible function spaces, with depth and width affecting approximation power. Numerical experiments in a Black–Scholes setting demonstrate that differential ML with neural-network bases achieves smaller hedging errors and more favorable PnL distributions than traditional LSMC, establishing a mathematically grounded path from theory to improved derivative valuation and hedging in practice.

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.
Paper Structure (22 sections, 9 theorems, 56 equations, 5 figures, 1 table)

This paper contains 22 sections, 9 theorems, 56 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Set up
  3. From Classical Results into Differential Machine Learning
  4. Risk Neutral Valuation Approach
  5. Differential Machine learning: building the loss function
  6. Example: Digital Options
  7. Choice of Basis
  8. Limitations of the Fixed-basis
  9. Parametric Basis: Neural Networks
  10. Depth
  11. Width
  12. Simulation-European Call Option
  13. Black-Scholes
  14. Hedging Experiment
  15. Least Squares Monte Carlo Algorithm
  16. ...and 7 more sections

Key Result

Proposition 4.1

The arbitrage-free price of a claim, whose payoff at time $T$ is $X=g_T(Z(.,\omega))$, is given by: where $Q$ is the risk-neutral measure, and $g_{0,t}(Z)$ denotes the expected time $t$ value of $X$.

Figures (5)

  • Figure 1: Left: LSMC price estimated with OLS. Right: $\Delta$ hedging function estimated by differentiating the pricing functional (3).
  • Figure 2: Left: LSMC with multi-layer feed-forward neural network. Right: $\Delta$ hedging from LSMC with multi-layer feed-forward neural network.
  • Figure 3: Left: Prices by differential approach with multi-layer feed-forward neural network. Right: $\Delta$ hedging by differential approach with a multi-layer feed-forward neural network.
  • Figure 4: Left: LSMC with polynomial basis PnL distribution against Black Scholes. Right: LSMC with feed-forward neural network basis PnL distribution against Black Scholes.
  • Figure 5: Differential approach with neural networks basis Pnl distribution against Black Scholes.

Theorems & Definitions (13)

  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Proposition 4.4
  • Definition 4.1
  • Example 5.1
  • Proposition 6.1
  • Definition 6.1
  • Theorem 6.1
  • Corollary 6.1.1
  • ...and 3 more