Additive normal tempered stable processes for equity derivatives and power-law scaling

Abstract

We introduce a simple additive process for equity index derivatives. The model generalizes Lévy Normal Tempered Stable processes (e.g. NIG and VG) with time-dependent parameters. It accurately fits the equity index volatility surfaces in the whole time range of quoted instruments, including options with small time-horizon (days) and long time-horizon (years). We introduce the model via its characteristic function. This allows using classical Fourier pricing techniques. We discuss the calibration issues in detail and we show that, in terms of mean squared error, calibration is on average two orders of magnitude better than both Lévy and Sato processes alternatives. We show that even if the model loses the classical stationarity property of Lévy processes, it presents interesting scaling properties for the calibrated parameters.