Carl Haglund and Pascal Halabi and present their master thesis "Physics-Informed Neural Networks for Option Pricing”

Examinator: Erik Lindström
Advisors: Alexandros Sopasakis 

Abstract. 
The Stochastic Alpha Beta Rho (SABR) stochastic volatility model is widely used to price European options, but the lack of an exact closed-form solution leaves practitioners choosing between an analytical asymptotic approximation that loses accuracy for certain parameter choices, and grid-based numerical solvers that must be rerun for every parameter set. This thesis investigates the application of a parametric Physics-Informed Neural Network (PINN) to option pricing under the SABR model. We train a single PINN on the SABR partial differential pricing equation, supervising it with reference prices computed by an accurate Alternating-Direction Implicit (ADI) solver across a grid of model parameters. Two free SABR model parameters enter the network as additional inputs, such that one trained model spans a continuous region of the parameter space rather than a single configuration. The supervised data loss is complemented with a residual term which enforces the governing dynamics in addition to the training labels alone. The trained PINN reproduces reference prices to within a few basis points of implied volatility across the parameter regimes it was trained on. The PINN generalises to unseen parameter combinations, and outperforms the Hagan asymptotic formula in the regimes where the expansion is known to break down. These results suggest that a parametric PINN could serve as a SABR pricer covering a continuous region of the model parameter space, including the regimes where the standard asymptotic approximation is unreliable.

Keywords: physics-informed neural network, SABR model, stochastic volatility, option pricing, Hagan approximation.