Short-time asymptotics for the implied volatility skew under a stochastic volatility model with L\'evy jumps

TL;DR

This paper develops high-order asymptotic expansions for the implied volatility skew in stochastic volatility models with Lévy jumps, enhancing understanding of short-term option pricing and calibration.

Contribution

It provides novel high-order asymptotic formulas for the implied volatility skew under models with stable-like jumps, including pure-jump cases, and links these to model features.

Findings

Asymptotic expansions fit well for options up to one month maturity.

Results are consistent with S&P 500 options data.

Expansions reveal the impact of leverage and vol-of-vol on skew.

Abstract

The implied volatility skew has received relatively little attention in the literature on short-term asymptotics for financial models with jumps, despite its importance in model selection and calibration. We rectify this by providing high-order asymptotic expansions for the at-the-money implied volatility skew, under a rich class of stochastic volatility models with independent stable-like jumps of infinite variation. The case of a pure-jump stable-like L\'evy model is also considered under the minimal possible conditions for the resulting expansion to be well defined. Unlike recent results for "near-the-money" option prices and implied volatility, the results herein aid in understanding how the implied volatility smile near expiry is affected by important features of the continuous component, such as the leverage and vol-of-vol parameters. As intermediary results we obtain high-order expansions for at-the-money digital call option prices, which furthermore allow us to infer analogous results for the delta of at-the-money options. Simulation results indicate that our asymptotic expansions give good fits for options with maturities up to one month, underpinning their relevance in practical applications, and an analysis of the implied volatility skew in recent S&P500 options data shows it to be consistent with the infinite variation jump component of our models.