KrigHedge: Gaussian Process Surrogates for Delta Hedging

Abstract

We investigate a machine learning approach to option Greeks approximation based on Gaussian Process (GP) surrogates. Our motivation is to implement Delta hedging in cases where direct computation is expensive, such as in local volatility models, or can only ever be done approximately. The proposed method takes in noisily observed option prices, fits a non-parametric input-output map and then analytically differentiates the latter to obtain the various price sensitivities. Thus, a single surrogate yields multiple self-consistent Greeks. We provide a detailed analysis of numerous aspects of GP surrogates, including choice of kernel family, simulation design, choice of trend function and impact of noise. We moreover connect the quality of the Delta approximation to the resulting discrete-time hedging loss. Results are illustrated with two extensive case studies that consider estimation of Delta, Theta and Gamma and benchmark approximation quality and uncertainty quantification using a variety of statistical metrics. Among our key take-aways are the recommendation to use Matérn kernels, the benefit of including virtual training points to capture boundary conditions, and the significant loss of fidelity when training on stock-path-based datasets.