Gaussian Process Regression for Swaption Cube Construction under No-Arbitrage Constraints

Abstract

In this paper, we introduce a 3D finite dimensional Gaussian process (GP) regression approach for learning arbitrage-free swaption cubes. Based on the possibly noisy observations of swaption prices, the proposed ‘constrained’ GP regression approach is proven to be arbitrage-free along the strike direction (butterfly and call-spread arbitrages are precluded on the entire 3D input domain). The cube is free from static arbitrage along the tenor and maturity directions if swaption prices satisfy 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. In addition, we compare the performance of the GP approach with the SABR model, which is applied to a data set of payer and receiver out-of-the-money (OTM) swaptions. The constrained GP approach provides better prediction results compared to the SABR approach. In addition, we show that SABR calibration is better when using the GP cube output as new observations (in terms of predictive error and absence of arbitrage). Finally, the GP approach is able to quantify in- and out-of-sample uncertainty through Hamiltonian Monte Carlo simulations, allowing for the computation of model risk Additional Valuation Adjustment (AVA).

Keywords

• swaption cube
• constrained kriging
• no arbitrage
• option pricing
• Hamiltonian Monte Carlo