Option pricing under stochastic interest rates and transaction costs using physics-informed neural networks

This paper prices European options under stochastic interest rates and proportional transaction costs, which together give rise to a nonlinear partial differential equation containing curvature-dependent square-root nonlinearities. Classical numerical methods struggle here because the nonlinear transaction-cost terms create strong sensitivity near the payoff transition region. We solve the resulting pricing equation using Physics-Informed Neural Networks (PINNs). The proposed framework combines a hard output transformation enforcing the terminal payoff and stock-price boundary conditions with a Huber residual loss designed to stabilize training near the payoff kink. Transaction costs are introduced progressively through a staged training procedure in order to improve optimization stability for the fully nonlinear problem. The pricing model incorporates a Vasicek short-rate process together with a hedging portfolio consisting of the underlying asset and a zero-coupon bond. Numerical experiments show that the trained PINN satisfies the terminal condition with mean absolute error 5 . 96 × 1 0 − 9 , achieves boundary-condition errors of order 1 0 − 3 , and produces PDE residual root-mean-square errors of order 1 0 − 2 . The learned solution remains monotone in the asset price across 93.9% of the computational domain. The experiments further show that transaction costs systematically reduce option values, with the largest corrections concentrated near the strike price and at intermediate maturities where the curvature terms become most significant. While previous PINN applications in computational finance have focused primarily on linear or frictionless pricing models, the present study addresses the fully nonlinear setting generated by the interaction of stochastic interest rates and proportional transaction costs. The results indicate that PINNs provide a flexible and accurate framework for nonlinear derivative pricing problems in which traditional discretization methods become computationally cumbersome.