The deep parametric PDE method and applications to option pricing

Abstract

We propose, formalise and analyse the deep parametric PDE method to solve high-dimensional parametric partial differential equations with a focus on financial applications. A single neural network approximates the solution of a whole family of PDEs after being trained without the need of sample solutions. As a practical application, we compute option prices and Greeks in the multivariate Black–Scholes model as there is an urgent need for highly efficient methods. After a single training phase, the prices and sensitivities for different times, states and model parameters are available in milliseconds. Exploiting the PDE framework and incorporating a-priori knowledge of no-arbitrage bounds improves the performance significantly. We evaluate the accuracy in the price, the Greeks and the implied volatility with examples of up to 25 dimensions. A comparison with alternative machine learning methods confirms the effectiveness of the new approach and reveals advantages of the underlying PDE formulation.

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