Smile Asymptotics II: Models with Known Moment Generating Function

TL;DR

This paper extends tail asymptotics analysis for implied volatility smiles using known moment generating functions, providing new guarantees for the limiting slope in implied variance plots and applying results to complex financial models.

Contribution

It introduces a method to derive tail asymptotics from the moment generating function, enabling a version of Lee's moment formula with guaranteed limiting slope.

Findings

Derived tail asymptotics using Tauberian theorems

Established a version of Lee's moment formula with a limiting slope

Applied results to time-changed Levy and Heston models

Abstract

In a recent article the authors obtained a formula which relates explicitly the tail of risk neutral returns with the wing behavior of the Black Scholes implied volatility smile. In situations where precise tail asymptotics are unknown but a moment generating function is available we first establish, under easy-to-check conditions, tail asymptoics on logarithmic scale as soft applications of standard Tauberian theorems. Such asymptotics are enough to make the tail-wing formula work and we so obtain a version of Lee's moment formula with the novel guarantee that there is indeed a limiting slope when plotting implied variance against log-strike. We apply these results to time-changed Levy models and the Heston model. In particular, the term-structure of the wings can be analytically understood.