Semi-nonparametric estimation of the call price surface under strike and time-to-expiry no-arbitrage constraints

We suggest a semi-nonparametric estimator for the entire call price surface based on a tensor-product B-spline. To enforce no-arbitrage constraints in strike and calendar dimensions we establish sufficient no-arbitrage conditions on the control net of the tensor product (TP) B-spline. Since these conditions are independent of the degrees of the underlying polynomials, the estimator can be parametrized with TP B-splines of arbitrary order. We derive consistency and explore the statistical efficiency benefits from surface estimation. As an application, we estimate families of state price densities and a local volatility surface for S&P500 option data.