Neural Networks with Local Converging Inputs for Efficient Options Pricing Models
NNLCI (Neural Networks with Local Converging Inputs) is introduced as a novel framework to enhance the efficiency of numerical methods for multi-asset options pricing The method uses a neural network to locally correct solutions from both coarse and refined meshes, requiring only minimal high-fidelity training data Demonstrated on cash-or-nothing options under Black-Scholes (1D, 2D, 3D) and single-asset down-and-out barrier calls under the Heston model (2D PDE in spot price S and variance v) NNL
Analysis
TL;DR
- NNLCI (Neural Networks with Local Converging Inputs) is introduced as a novel framework to enhance the efficiency of numerical methods for multi-asset options pricing
- The method uses a neural network to locally correct solutions from both coarse and refined meshes, requiring only minimal high-fidelity training data
- Demonstrated on cash-or-nothing options under Black-Scholes (1D, 2D, 3D) and single-asset down-and-out barrier calls under the Heston model (2D PDE in spot price S and variance v)
- NNLCI reduces RMSE of refined-mesh solutions by a factor of approximately 4-12 on test sets, even when trained on only a small subset of parameter combinations
- The approach offers low training costs, strong generalization, and significantly reduced computational requirements for high-dimensional problems in real-time trading and risk management
Why It Matters
This work bridges machine learning and computational finance by demonstrating that neural networks can dramatically accelerate options pricing without sacrificing accuracy—a critical need in real-time trading environments where speed and precision are paramount. For AI practitioners, it showcases an elegant hybrid approach where deep learning augments classical numerical methods rather than replacing them, offering a template for applying ML to other PDE-driven financial models.
Technical Details
- Architecture: NNLCI employs a neural network that takes inputs from both a coarse mesh solution and a correspondingly refined mesh solution, learning to produce local corrections that bridge the gap between the two
- Input Format: A concise input format for NNLCI is introduced, designed to maximize convenience and computational efficiency during both training and inference
- Benchmark Problems: (1) Cash-or-nothing options under the Black-Scholes equation in 1, 2, and 3 spatial dimensions; (2) Single-asset down-and-out barrier call options under the Heston stochastic-volatility model, where the pricing PDE is two-dimensional in spot price S and instantaneous variance v
- Performance Metrics: Root-mean-square error (RMSE) reduction by a factor of 4-12 on test sets compared to standard refined-mesh numerical solutions, achieved with training on only a small subset of parameter combinations
- Key Advantages: Low training data requirements, strong generalization across parameter spaces, and scalability to higher-dimensional problems that are computationally prohibitive for traditional numerical methods
Industry Insight
- Financial institutions dealing with real-time options pricing and risk management should explore NNLCI as a drop-in enhancement to existing numerical pipelines, potentially achieving order-of-magnitude speedups without retraining from scratch
- The hybrid ML-classical approach demonstrated here—where neural networks correct rather than replace numerical solvers—offers a pragmatic template for deploying AI in safety-critical domains where interpretability and reliability are essential
- As multi-asset and stochastic-volatility models grow in complexity, methods like NNLCI that decouple accuracy from mesh refinement could become standard tools, reducing infrastructure costs and enabling more sophisticated pricing models in production
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