A Parametric Physics-Informed Neural Network for SABR Option Pricing

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 in certain cases, 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 hybrid 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. The trained PINN reproduces reference prices within single-digit basis points of implied volatility on average, with larger errors near the boundary of the trained domain. The PINN generalises to unseen parameter combinations, and outperforms the Hagan asymptotic formula in some of the regimes where the expansion is known to break down. The results suggest that a parametric PINN could serve as a SABR pricer covering a continuous region of the model parameter space, while being more accurate than the standard asymptotic approximation in some of the subdomains where it is known to be unreliable.

Popular Abstract

When pricing options under one of the industry’s models, a common fast approximation becomes unreliable in certain cases. A more precise alternative is a numerical solver, but it is too slow for practical use. Could a neural network which leverages the dynamics of the model offer a compromise, staying accurate without the solver’s heavy cost?

Most people have heard of buying and selling stocks, but a large share of global trading takes place in financial derivatives, contracts whose value is tied to some other asset. A common type of derivative is an option, which gives its holder the right, but not the obligation, to buy or sell an underlying asset, at a set price (its strike price) on a future date. Options are mostly traded by large institutions, which rely on mathematical models to decide what a contract is worth. Rather than providing a raw price, traders usually describe an option through its implied volatility, the market's expectation of how much the underlying asset will move. Implied volatility varies with an option's strike price and its time to maturity, together forming a surface that describes the market.

A widely used model to attempt to recreate this observation is the Stochastic Alpha Beta Rho (SABR) model, which treats volatility itself as random and changing over time, and reproduces how implied volatility varies across maturities and strike prices. The catch is that the SABR model has no formula that returns an option's price directly. One way around this is a shortcut, the Hagan approximation, a formula that returns the implied volatility when given a set of inputs. It is very fast, but because it is only an approximation, it loses accuracy in some situations, for example for options with a long time to maturity.

Our thesis explores using a Physics-Informed Neural Network (PINN) to calculate SABR prices. The PINN is a neural network that is not only fitted to data but also required to obey the dynamics of the SABR model during training. The equation describing these dynamics acts as a constraint that keeps the network's answers consistent with the model. To produce accurate reference values to learn from and compare against, we use a slow but highly precise numerical solver, the kind of brute-force calculation that is too computationally heavy to use in practice. A single trained network can then price options across a continuous range of market conditions without being retrained for each one.

The results are encouraging. Across the range it was trained on, the network's prices stay quite close, on average, to the accurate numerical reference. It also prices well when given inputs with values between the parameter values it was trained on, which suggests it has learned the underlying pricing behaviour rather than memorising values. The network's advantage over the Hagan approximation is largest for strike prices far from the current asset price and for contracts with a long time to maturity. This is expected, since these are exactly the situations where the Hagan approximation is known to lose accuracy. Furthermore, the Hagan approximation is most accurate for short-maturity options with strike prices close to the current asset price, and it beats the network there. A practical consequence is that the two could be used together, each in the domain where it is strongest.

The network was trained on a limited set of the model's parameters. A natural next step would be to let it vary more of the model's parameters, over wider ranges, while keeping the training stable.

• author: Haglund, Carl and Halabi, Pascal
• supervisor: Alexandros Sopasakis
• organization: Mathematics (Faculty of Engineering)
• course: FMAM05 20261
• year: 2026
• type: H2 - Master's Degree (Two Years)
• subject: Mathematics and Statistics
• publication/series: Master’s Theses in Mathematical Sciences 2026:E63
• report number: LUTFMA-3630-2026
• ISSN: 1404-6342
• other publication id: E:63
• language: English
• id: 9234333
• date added to LUP: 2026-06-12 15:09:46
• date last changed: 2026-06-12 15:09:46