Short-term asymptotics for the implied volatility skew under a stochastic volatility model with Lévy jumps

José E. Figueroa-López, Sveinn Ólafsson

Research output: Contribution to journalArticlepeer-review

14 Scopus citations

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 Lévy 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&P 500 options data shows it to be consistent with the infinite variation jump component of our models.

Original languageEnglish
Pages (from-to)973-1020
Number of pages48
JournalFinance and Stochastics
Volume20
Issue number4
DOIs
StatePublished - 1 Oct 2016

Keywords

  • ATM digital call option prices
  • ATM implied volatility slope
  • Exponential Lévy models
  • Short-term asymptotics
  • Stochastic volatility models

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