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

TL;DR

This paper introduces an additive normal tempered stable process model for equity derivatives that improves calibration accuracy and captures scaling properties, extending traditional Lévy models with time-dependent parameters.

Contribution

The paper presents a novel additive process model with time-dependent parameters that fits implied volatility surfaces more accurately than existing Lévy models.

Findings

Calibration error is two orders of magnitude lower than traditional models.

The model accurately fits a wide range of implied volatility surfaces.

It exhibits interesting scaling properties despite losing stationarity.

Abstract

We introduce a simple model for equity index derivatives. The model generalizes well known L\`evy Normal Tempered Stable processes (e.g. NIG and VG) with time dependent parameters. It accurately fits Equity index implied volatility surfaces in the whole time range of quoted instruments, including small time horizon (few days) and long time horizon options (years). We prove that the model is an Additive process that is constructed using an Additive subordinator. This allows us to use classical L\`evy-type 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 L\`evy processes and Self-similar alternatives. We show that even if the model loses the classical stationarity property of L\`evy processes, it presents interesting scaling properties for the calibrated parameters.