FYS5429 — PINNs for Option Pricing

Can a neural network learn to price options by understanding the physics?

University of Oslo · Spring 2026 · Egil Furnes

The idea

Option pricing is a PDE problem. The Black-Scholes equation is structurally identical to the heat equation — a fact Fischer Black and Myron Scholes exploited in 1973 to derive their celebrated closed form. Heston's 1993 extension adds a second stochastic dimension for volatility, breaking the closed form but preserving the PDE structure.

Physics-Informed Neural Networks (PINNs) turn this into a learning problem: instead of discretising the domain on a mesh, a neural network is trained to satisfy the PDE everywhere simultaneously. The PDE residual — computed via automatic differentiation — enters directly into the loss function. No labelled data required.

Why it's interesting

It connects two fields. Financial PDEs and deep learning rarely meet this cleanly. The Black-Scholes equation under a log-price change is a diffusion equation — the same one that governs heat flow, Brownian motion, and quantum mechanics in imaginary time.

The inverse problem is the real prize. Markets give you prices, not parameters. Calibrating the Heston model — recovering κ, θ, ξ, ρ from observed option prices — is an ill-posed nonlinear optimisation problem. A PINN that has internalised the Heston PDE can be repurposed as a differentiable pricer, making gradient-based calibration natural.

Mesh-free scales. Classical finite-difference methods on a 2D (S, v) grid are expensive. PINNs sample collocation points randomly and scale to higher dimensions without a grid — relevant for multi-asset and path-dependent extensions.

Activation functions matter. The smoothness of the solution — and the sharpness of the terminal payoff — make the choice of activation non-trivial. This project benchmarks tanh, Swish, GELU, Softplus, and SIREN against each other.

The models

Black-Scholes

$$\frac{\partial V}{\partial \tau} = \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV, \qquad \tau = T - t$$

Closed-form solution available — used as ground truth to validate the PINN.

Heston

$$\frac{\partial V}{\partial \tau} = \frac{1}{2}vS^2 V_{SS} + \rho\xi v S, V_{Sv} + \frac{1}{2}\xi^2 v, V_{vv} + rS, V_S + \kappa(\theta - v), V_v - rV$$

Five parameters: mean-reversion speed κ, long-run variance θ, vol-of-vol ξ, spot-vol correlation ρ, initial variance v₀. No closed form — numerical integration (characteristic functions) serves as benchmark.

Loss function

$$\mathcal{L} = \lambda_\text{pde},\mathcal{L}_\text{pde} + \lambda_\text{ic},\mathcal{L}_\text{ic} + \lambda_\text{bc},\mathcal{L}_\text{bc}$$

All three terms computed via PyTorch autograd — no finite differences anywhere in the training loop.

Department of Physics · University of Oslo