Source-linked AI summary
Neural networks for option pricing and hedging: a literature review
Johannes Ruf, Weiguan Wang
TL;DR
Neural-network research for option pricing and hedging spans more than one hundred papers, but its coverage and evaluation practices vary. This review synthesizes the literature and identifies implementation guidance, including stationary inputs, suitable benchmarks, and time-aware data partitioning.
Problem
More than one hundred papers study neural networks for option pricing and hedging, motivating a review of their methods and evaluation practices.
Method
The review compares literature on inputs, outputs, benchmarks, performance measures, data partitioning, and underlying assets, while discussing model-based neural-network pricing.
Results
Most papers estimate option prices; around 15 papers, or 10%, estimate implied volatilities, while very few address hedging directly.
Takeaways & Limitations
The review advises using stationary features, appropriate benchmarks, and time-series-preserving training and test partitions when implementing neural networks.
Takeaways & Limitations
The review excludes comparisons of parameter-estimation methods and neural-network architectures, whose specifications vary substantially across papers.
Abstract
from arXiv · showhide
Neural networks have been used as a nonparametric method for option pricing and hedging since the early 1990s. Far over a hundred papers have been published on this topic. This note intends to provide a comprehensive review. Papers are compared in terms of input features, output variables, benchmark models, performance measures, data partition methods, and underlying assets. Furthermore, related work and regularisation techniques are discussed.
1 Introduction
This introduction frames artificial neural networks as flexible approximators for nonlinear option-pricing and hedging relationships and positions the paper as a review of their use in this literature. It outlines how the reviewed studies compare models, data, performance measures, and applications.
- More than one hundred papers study artificial neural networks for option pricing and hedging, motivating this literature review.
- ANNs map input features to outputs through compositions of linear and nonlinear functions, with training selecting weights using a loss function.
- Universal approximation results support using ANNs to capture nonlinear dependencies between option-related inputs and outputs.
- In option applications, ANNs can estimate prices, implied-volatility surfaces, or hedging ratios from features such as underlying price, strike, and other contract characteristics.
- The review compares studies using simulated or real datasets, performance measures, benchmark models, and training-data choices.
- A model-based use of ANNs learns computationally intensive pricing formulas directly from data-generating processes involving methods such as partial differential equations or Monte Carlo simulation.
2 ANN based option pricing and hedging in the literature
The review classifies ANN option-pricing and hedging studies by inputs, outputs, benchmarks, performance measures, data partitions, and underlyings, while highlighting recurring methodological concerns. It finds that pricing dominates the literature, conclusions depend on volatility features and benchmarks, and time-series-aware partitioning is essential.
- Scope and classification: More than 150 papers are compared by features, outputs, benchmarks, data partitioning, underlyings, and time spans.The review also uses tables to define the relevant abbreviations and performance measures.
- Scope and limitations: The review excludes systematic comparisons of parameter-estimation methods and ANN architectures because specifications vary substantially across papers.It notes only that newer studies tend to use more complex architectures as computational resources improve.
- Literature patterns: Option-price estimation dominates the literature, while about 10% of listed papers estimate implied volatility and very few address hedging directly.The review recommends a separate selection of papers for readers seeking a smaller entry point.
- Features: Moneyness is increasingly used instead of separate stock and strike prices, and the review reports that it can improve generalisation and reduce overfitting.The review characterizes moneyness as stationary relative to the separate price inputs.
- Features: Reported conclusions often depend on volatility-feature choices: historical, implied, and GARCH volatility produce different ANN or benchmark comparisons.One study reports ANN outperformance over Black-Scholes with historical volatility but underperformance with implied volatility, while other studies favor implied or GARCH forecasts.
- Benchmarks: Contract-specific implied volatility makes Black-Scholes unsuitable as a pricing benchmark because it prices those options with zero error, although it remains valid for hedging.Other parametric benchmarks include stochastic-volatility models, and a benchmark incorporating delta and vega hedging can eliminate ANN outperformance in one cited comparison.
- Data partitioning: Randomly partitioning time-series data leaks information from training into testing and underestimates ANN generalisation error.The review therefore advises preserving time-series structure when splitting training and test sets.
3 Recommended papers
The review highlights foundational and influential papers that introduced direct hedging, financial-knowledge-aware architectures, no-arbitrage constraints, hybrid pricing, and market-friction frameworks.
- Hutchinson et al. [1994] introduced a multi-period methodology for evaluating hedging performance that many later papers adopted.
- Lajbcygier and Connor [1997a] were among the first to learn the difference between model prices and observed market option prices.
- Garcia and Genc¸ay [2000] embedded financial domain knowledge into ANN construction through a homogeneity hint.
- Carverhill and Cheuk [2003] were the first to propose an ANN that outputs hedging strategies directly rather than option prices.
- Dugas et al. [2009] first designed an ANN architecture enforcing no-arbitrage conditions such as option-price convexity.
- Buehler et al. [2019a] developed a market-friction framework using convex risk measures and allowing pricing and hedging without observed option prices.
4 Related papers
Related work extends ANN use beyond direct nonparametric estimation to model calibration, PDEs, dynamic programming, market frictions, risk measures, reinforcement learning, and generative simulation.
- 4.1 Calibration: Calibration ANNs map option prices to parametric model parameters, moving computationally heavy calibration offline and accelerating option pricing.
- 4.1 Calibration: Calibration studies cover Vasicek, Hull–White, stochastic-volatility, and rough-volatility models, while Itkin [2019] proposed resolutions to calibration pitfalls.
- 4.2 PDEs: ANNs have been used to solve Black–Scholes and high-dimensional semilinear parabolic PDEs through Galerkin or backward-stochastic-differential-equation formulations.
- 4.3 Dynamic programming and frictions: Other applications approximate dynamic-programming value functions, learn optimal stopping rules, and solve pricing and hedging under transaction costs.
- 4.4 Other directions: Further directions include ANN quantile regression for XVA risk measures, reinforcement learning under market frictions, and generative simulation for stock or option prices.
5 Digression: regularisation techniques
Regularisation combines standard training controls with financial-domain constraints embedded in architecture, data, losses, and specialised training algorithms.
- Standard regularisation techniques include L2 regularisation, dropout, and early stopping, whose importance has grown with larger ANNs.
- Architecture design: A homogeneity hint splits the ANN into components controlled by moneyness and time-to-maturity.
- Architecture design: Shape-restricted architectures enforce no-arbitrage properties such as monotonicity and convexity in ANN pricing functions.
- Training methods: Data augmentation creates synthetic options, while loss penalties encode no-arbitrage conditions such as penalties for calendar arbitrage.
- Training methods: Novel ANN training algorithms applied to option hedging include the extended Kalman filter, sequential Monte Carlo, and evolutionary algorithms.