AUTEURS : ARESKI COUSIN, ADRIEN MISKO, ADRIEN DELEPLACE
In this paper, we introduce a 3D finite dimensional Gaussian process (GP) regression approach for learning arbitrage-free swaption cubes. Based on possibly noisy observations of swaption prices, the proposed ‘constrained’ GP regression approach is proven arbitrage-free along the strike direction (butterfly and call-spread arbitrages are precluded on the entire 3D input domain). From Johnson and Nonas (2009), the cube is free from static arbitrage along the tenor and maturity directions if swaption prices satisfies an infinite set of in-plane triangular inequalities. We empirically demonstrate that, considering a finite-dimensional weaker form of this condition, is enough for the GP to generate swaption cubes with a negligible proportion of violation points, even for a small training set. Finally, we compare the performance of the GP approach with a SABR construction model, when applied to a data set composed of payer and receiver OTM swaptions. The constrained GP approach provides very good calibration and prediction results compared to the SABR approach. In addition, the GP approach is able to quantify in- and out-of-sample uncertainty through Hamiltonian Monte Carlo simulations.