Robust Implied Volatility Surface Construction via B-Spline-Enhanced Graph Neural Operators

Abstract

The reliable recovery of implied volatility surfaces from sparse option quotes remains challenging, yet it is essential for pricing, hedging, and market risk measurement. In practice, widely-used parametric specifications can be fast and arbitrage-aware, but their stability depends on repeated calibration and can deteriorate when market data are irregular. Operator learning offers a complementary perspective by modelling a mapping from scattered observations to a continuous surface, though many implementations become expensive because they predict values over dense grids and therefore require large training sets.

This thesis develops a graph-based neural operator that targets a low-dimensional spline representation of the surface. A bivariate cubic B-spline is first used to obtain an initial smooth approximation, after which the operator learns residual adjustments to the spline coefficients rather than producing a full grid of volatilities. This coefficient-level formulation embeds smoothness by construction and reduces computational burden. In addition, the analytic derivatives available under the spline representation enable butterfly and calendar no-arbitrage penalties to be imposed without finite-difference noise.

Experiments on daily S&P 500 index options show that the resulting framework is more stable under sparse conditions and achieves improved efficiency relative to common parametric baselines and grid-based neural operator models. These results support spline-prior operator learning as a practical route to constructing smooth, arbitrage-regularised volatility surfaces.