On the implied volatility extraction and the selection of suitable kernel

At the market, we can identify various kinds of options. Some of them are traded at organized exchanges and are quite liquid. Others are traded only between particular parties. The current market practice is to obtain implied volatility of liquid options as based on Black-Scholes type (BS hereafter) models. Such volatility is subsequently used to price illiquid or even exotic options. It therefore follows that the BS model at one time moment can be related to the whole set of IVs as given by maturity/moneyness relation of tradable options. One can therefore get IV curve or surface (a so called smirk or smile). Since the moneyness and maturity of IV often do not match the data of valuated options, some sort of estimating and local smoothing is necessary. However, it can lead to arbitrage opportunity, if no-arbitrage conditions on state price density (SPD) are ignored. In this paper, using option data on DAX index, we aim on the analyses of the behavior of IV and SPD with respect to different choices of bandwidth parameter h and identification of a set of bandwidths which violates no-arbitrage conditions. We document that the change of h implies interesting changes in the violation interval of moneyness. Finally, we also show the impact of h on the total area of SPD under zero, which can be seen as a degree of no-arbitrage violation.