Short-time implied volatility of additive normal tempered stable processes

TL;DR

This paper analyzes the short-time implied volatility behavior of additive normal tempered stable processes, showing they match empirical equity market features when specific parameters are set.

Contribution

It provides a theoretical proof linking the parameters of additive processes to observed short-term implied volatility patterns in equity markets.

Findings

Implied volatility matches empirical skew when β=1 and δ=-1/2

Calibration achieves excellent fit to volatility surfaces

Theoretical validation of parameter choices for market consistency

Abstract

Empirical studies have emphasized that the equity implied volatility is characterized by a negative skew inversely proportional to the square root of the time-to-maturity. We examine the short-time-to-maturity behavior of the implied volatility smile for pure jump exponential additive processes. An excellent calibration of the equity volatility surfaces has been achieved by a class of these additive processes with power-law scaling. The two power-law scaling parameters are $\beta$, related to the variance of jumps, and $\delta$, related to the smile asymmetry. It has been observed, in option market data, that $\beta=1$ and $\delta=-1/2$. In this paper, we prove that the implied volatility of these additive processes is consistent, in the short-time, with the equity market empirical characteristics if and only if $\beta=1$ and $\delta=-1/2$.