Kriging For Implied Volatility Surface

Implied volatility surface is of crucial interest for risk management and exotic option pricing models. Its construction is usually carried out in accordance with the arbitrage-free principle. This condition leads to shape restrictions on the option prices such as monotonicity with respect to maturities and convexity with respect to strike prices. In this paper, we propose a new arbitrage-free construction method that extends classical spline techniques by additionally allowing for quantification of uncertainty. The proposed method extends the constrained kriging techniques developed in [MB16] and [CMR16] to the context of volatility surface construction. Assuming a Gaussian process prior, the posterior price surface becomes a truncated Gaussian field given shape constraints and market observations. Prices of illiquid instruments can also be incorporated when considered as noisy observations. Starting from a suitable finite-dimensional approximation of the Gaussian process prior, the no-arbitrage condition on the entire input domain is characterized by a finite number of linear inequality constraints. We define the most likely response surface and the most-likely noise values as the solution of a quadratic optimization problem. We use Hamiltonian Monte Carlo technics to simulate the posterior truncated Gaussian surface and build pointwise confidence bands. The Gaussian process hyper-parameters are estimated using maximum likelihood. The method is illustrated on Euro Stoxx 50 option prices by building no-arbitrage volatility surfaces and their corresponding confidence bands.