A novel stochastic framework for pricing European options on crude oil futures

This paper extends the existing two-factor Schwartz model into a comprehensive three-factor framework that incorporates stochastic volatility through the Heston model, enabling more precise pricing of European options on crude oil futures. In this enhanced model, we account for key variables including market volatility and yield changes, which significantly influence futures pricing dynamics. We present an analytical formula for the pricing of futures contracts, facilitating robust calibration against actual market data. To solve the associated partial differential equations for option pricing, we implement the Deep Galerkin Method (DGM), a novel approach leveraging deep neural networks that proves to be efficient and accurate, particularly in high-dimensional settings. The obtained results demonstrate that the DGM method outperforms traditional numerical techniques, offering substantial improvements in both accuracy and computational efficiency.