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Adaptively trained Physics-informed Radial Basis Function Neural Networks for Solving Multi-asset Option Pricing Problems

Yan Ma, Yumeng Ren

TL;DR

The paper tackles high-dimensional multi-asset option pricing by solving the Black-Scholes terminal-boundary value problem with a physics-informed radial basis function neural network (PIRBFNN). The method combines a two-layer RBF network, where centers, shapes, and weights are trainable, with a residual-based adaptive training strategy that adds neurons where PDE residuals are large, treating time as an additional spatial dimension. Numerical experiments on 1D European puts, 2D exchange options, and 4D basket calls demonstrate accurate pricing (RMSE around 10^{-3} to a few 10^{-3}) and faster convergence compared with PINNs, while effectively handling non-smooth payoffs without pre-smoothing. The approach offers a robust, meshless alternative for high-dimensional PDEs in finance, with potential for real-time pricing and extension to deeper architectures and broader derivative products.

Abstract

The present study investigates the numerical solution of Black-Scholes partial differential equation (PDE) for option valuation with multiple underlying assets. We develop a physics-informed (PI) machine learning algorithm based on a radial basis function neural network (RBFNN) that concurrently optimizes the network architecture and predicts the target option price. The physics-informed radial basis function neural network (PIRBFNN) combines the strengths of the traditional radial basis function collocation method and the physics-informed neural network machine learning approach to effectively solve PDE problems in the financial context. By employing a PDE residual-based technique to adaptively refine the distribution of hidden neurons during the training process, the PIRBFNN facilitates accurate and efficient handling of multidimensional option pricing models featuring non-smooth payoff conditions. The validity of the proposed method is demonstrated through a set of experiments encompassing a single-asset European put option, a double-asset exchange option, and a four-asset basket call option.

Adaptively trained Physics-informed Radial Basis Function Neural Networks for Solving Multi-asset Option Pricing Problems

TL;DR

The paper tackles high-dimensional multi-asset option pricing by solving the Black-Scholes terminal-boundary value problem with a physics-informed radial basis function neural network (PIRBFNN). The method combines a two-layer RBF network, where centers, shapes, and weights are trainable, with a residual-based adaptive training strategy that adds neurons where PDE residuals are large, treating time as an additional spatial dimension. Numerical experiments on 1D European puts, 2D exchange options, and 4D basket calls demonstrate accurate pricing (RMSE around 10^{-3} to a few 10^{-3}) and faster convergence compared with PINNs, while effectively handling non-smooth payoffs without pre-smoothing. The approach offers a robust, meshless alternative for high-dimensional PDEs in finance, with potential for real-time pricing and extension to deeper architectures and broader derivative products.

Abstract

The present study investigates the numerical solution of Black-Scholes partial differential equation (PDE) for option valuation with multiple underlying assets. We develop a physics-informed (PI) machine learning algorithm based on a radial basis function neural network (RBFNN) that concurrently optimizes the network architecture and predicts the target option price. The physics-informed radial basis function neural network (PIRBFNN) combines the strengths of the traditional radial basis function collocation method and the physics-informed neural network machine learning approach to effectively solve PDE problems in the financial context. By employing a PDE residual-based technique to adaptively refine the distribution of hidden neurons during the training process, the PIRBFNN facilitates accurate and efficient handling of multidimensional option pricing models featuring non-smooth payoff conditions. The validity of the proposed method is demonstrated through a set of experiments encompassing a single-asset European put option, a double-asset exchange option, and a four-asset basket call option.
Paper Structure (18 sections, 29 equations, 16 figures, 2 tables)

This paper contains 18 sections, 29 equations, 16 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Black-Scholes model for multi-asset option pricing problem
  3. Physics-informed radial basis function neural network (PIRBFNN) for solving terminal-boundary value problem
  4. Architecture of radial basis function neural network (RBFNN)
  5. Generation of the training points for various constraints
  6. Formulation of the loss function
  7. Initialization of the network parameters
  8. Residual-based adaptive training dynamics
  9. Evaluation of the applicability of the network
  10. Numerical experiments
  11. Example 1: Single-asset European put option
  12. PIRBFNN with a fixed number of neurons
  13. PIRBFNN with adaptively added neurons
  14. Investigation about the initial strategy
  15. Comparative analysis of PIRBFNN with PINN and traditional RBF collocation method
  16. ...and 3 more sections

Figures (16)

  • Figure 1: The architecture of an PIRBFNN for solving the TBVP \ref{['Gov_equation_general']}--\ref{['Bou_condition_general']}.
  • Figure 2: Example 1: The results obtained from PIRBFNNs using different activation functions with respect to $128$ seed values. (a): Gaussian function, (b): Inverse quadratic function, (c): Inverse multiquadric function.
  • Figure 3: Example 1: The results obtained from PIRBFNNs using different activation functions with respect to single seed value. Top panel: Gaussian function, middle panel: Inverse quadratic function, bottom panel: Inverse multiquadric function.
  • Figure 4: Example 1: Adaptive training of the PIRBFNN for solving single-asset European put option: loss value history (blue curve), RMSE value history (red curve).
  • Figure 5: Example 1: The results obtained from PIRBFNNs using varying numbers of RBF hidden neurons. First row: initial $650$ neurons, second row: increase of $50$ neurons, Third row: increase of $50$ neurons.
  • ...and 11 more figures