Solving high-dimensional Partial Differential Equations (PDEs) is a significant challenge in quantitative finance, particularly for pricing complex derivatives. Traditional numerical methods often suffer from the curse of dimensionality. Neural networks offer a promising alternative.
A financial institution required a method to efficiently solve high-dimensional Black-Scholes type PDEs for derivative pricing. The existing methods were too slow or inaccurate for the dimensionality of the problems they were facing, especially when dealing with options on multiple underlying assets.
I explored and implemented a deep learning-based approach, specifically using Physics-Informed Neural Networks (PINNs) or similar architectures tailored for solving PDEs. The neural network was trained to satisfy the PDE, boundary, and terminal conditions. This method leverages the universal approximation theorem of neural networks and the efficiency of automatic differentiation, providing accurate solutions even for high-dimensional problems.