Volatility Calibration Using Spline and High Dimensional Model Representation Models

The local volatility function is approximated using two different models: a bicubic spline and High Dimensional Model Representation (HDMR) model. For a bicubic spline the local volatility values σi at a chosen discrete set of knots are determined by minimizing the least square error between market and model option prices. Model option prices are found using the one-factor diffusion process for the underlying asset. For the HDMR model, the parameters to be optimized are the parameters of the HDMR model. Smoothness of the approximation functions is crucial to overcome the ill-posedness of the volatility calibration problem. Twomodels are compared using two test cases. It is shown that the HDMRmodel can produce more accurate results than the cubic spline model, and it is also cheaper to run. It is demonstrated that the considered approach can accurately reproduce not only option prices but Greeks as well.