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The AI Black-Scholes: Finance-Informed Neural Network

Amine M. Aboussalah, Xuanze Li, Cheng Chi, Raj Patel

TL;DR

FINN addresses the mismatch between principled PDE-based pricing and data-driven methods by embedding no-arbitrage and dynamic hedging into a neural network training objective. It generalizes to both GBM and Heston dynamics, delivering accurate option prices and reliable Greeks while remaining computationally efficient when analytical solutions are unavailable. The framework is underpinned by a convergence argument for delta-gamma hedging and demonstrated through GBM and Heston experiments, including Delta-Gamma hedging. Overall, FINN provides a finance-informed, interpretable, and versatile tool for pricing and hedging derivatives under diverse market conditions.

Abstract

In the realm of option pricing, existing models are typically classified into principle-driven methods, such as solving partial differential equations (PDEs) that pricing function satisfies, and data-driven approaches, such as machine learning (ML) techniques that parameterize the pricing function directly. While principle-driven models offer a rigorous theoretical framework, they often rely on unrealistic assumptions, such as asset processes adhering to fixed stochastic differential equations (SDEs). Moreover, they can become computationally intensive, particularly in high-dimensional settings when analytical solutions are not available and thus numerical solutions are needed. In contrast, data-driven models excel in capturing market data trends, but they often lack alignment with core financial principles, raising concerns about interpretability and predictive accuracy, especially when dealing with limited or biased datasets. This work proposes a hybrid approach to address these limitations by integrating the strengths of both principled and data-driven methodologies. Our framework combines the theoretical rigor and interpretability of PDE-based models with the adaptability of machine learning techniques, yielding a more versatile methodology for pricing a broad spectrum of options. We validate our approach across different volatility modeling approaches-both with constant volatility (Black-Scholes) and stochastic volatility (Heston), demonstrating that our proposed framework, Finance-Informed Neural Network (FINN), not only enhances predictive accuracy but also maintains adherence to core financial principles. FINN presents a promising tool for practitioners, offering robust performance across a variety of market conditions.

The AI Black-Scholes: Finance-Informed Neural Network

TL;DR

FINN addresses the mismatch between principled PDE-based pricing and data-driven methods by embedding no-arbitrage and dynamic hedging into a neural network training objective. It generalizes to both GBM and Heston dynamics, delivering accurate option prices and reliable Greeks while remaining computationally efficient when analytical solutions are unavailable. The framework is underpinned by a convergence argument for delta-gamma hedging and demonstrated through GBM and Heston experiments, including Delta-Gamma hedging. Overall, FINN provides a finance-informed, interpretable, and versatile tool for pricing and hedging derivatives under diverse market conditions.

Abstract

In the realm of option pricing, existing models are typically classified into principle-driven methods, such as solving partial differential equations (PDEs) that pricing function satisfies, and data-driven approaches, such as machine learning (ML) techniques that parameterize the pricing function directly. While principle-driven models offer a rigorous theoretical framework, they often rely on unrealistic assumptions, such as asset processes adhering to fixed stochastic differential equations (SDEs). Moreover, they can become computationally intensive, particularly in high-dimensional settings when analytical solutions are not available and thus numerical solutions are needed. In contrast, data-driven models excel in capturing market data trends, but they often lack alignment with core financial principles, raising concerns about interpretability and predictive accuracy, especially when dealing with limited or biased datasets. This work proposes a hybrid approach to address these limitations by integrating the strengths of both principled and data-driven methodologies. Our framework combines the theoretical rigor and interpretability of PDE-based models with the adaptability of machine learning techniques, yielding a more versatile methodology for pricing a broad spectrum of options. We validate our approach across different volatility modeling approaches-both with constant volatility (Black-Scholes) and stochastic volatility (Heston), demonstrating that our proposed framework, Finance-Informed Neural Network (FINN), not only enhances predictive accuracy but also maintains adherence to core financial principles. FINN presents a promising tool for practitioners, offering robust performance across a variety of market conditions.

Paper Structure

This paper contains 36 sections, 16 theorems, 112 equations, 7 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature review
  3. Dynamic hedging argument for option pricing
  4. Specification of asset price dynamics
  5. Construction of zero-cost, self-financing portfolio
  6. Construction of Zero-Cost, Self-Financing Portfolio
  7. Enforcing No-Arbitrage Conditions on the Portfolio
  8. Machine Learning Approaches to Option Pricing
  9. Finance-Informed Neural Network (FINN)
  10. Experimental Setup
  11. Results and Discussion
  12. Numerical Example: FINN Implementation with GBM
  13. Initial Market Conditions:
  14. Delta Computation:
  15. Option Price Change:
  16. ...and 21 more sections

Key Result

Theorem 1

Let $g^\theta$ be a FINN model parameterized by $\theta$, trained via a self-supervised approach on data generated from a complete, arbitrage-free market with underlying asset $S_t$ and additional instrument(s) $H_t$. Assume: Then, for any $\epsilon > 0$, there exists a set of parameters $\theta^*$ such that: where $\Delta_g = \partial_S g^*$ and $\Gamma_g = \partial_S^2 g^*$ are the first and s

Figures (7)

  • Figure 1: Overview of the FINN Model Design
  • Figure 2: A one-period illustrative example
  • Figure 3: Comparison of Call Option Prices, Hedge Ratios, and Errors
  • Figure 4: Comparison of Call Option Prices, Hedge Ratios, and Errors (Heston Model)
  • Figure 5: Comparison of Call Option Prices, Hedge Ratios, Gamma, and Errors
  • ...and 2 more figures

Theorems & Definitions (34)

  • Theorem 1: FINN Convergence under Delta-Gamma Hedging
  • Theorem 2: FINN Convergence under Delta-Gamma Hedging
  • proof
  • Definition 1: Market Setting
  • Theorem 3: Existence and uniqueness of Pricing Function
  • proof
  • Proposition 4: PDE Characterization
  • Theorem 5: Regularity of Solution
  • proof
  • Definition 2: Neural Network Parameterization
  • ...and 24 more