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

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&P500 options data shows it to be consistent with the infinite variation jump component of our models.AMS 2000 subject classifications: 60G51, 60F99, 91G20, 91G60.

Small-Maturity Asymptotics for the At-The-Money Implied Volatility Slope in Lévy Models

“…Note that these results also follow from the concurrent paper ( Figueroa-López and Ólafsson 2015 ), which treats tempered stable-like models.…”

“…The recent preprint (Figueroa-López and Ólafsson 2015 ) is also closely related to our work. There, the Brownian component is generalized to stochastic volatility.…”

“…See Pinter’s (in preparation ) PhD thesis for details. Note that Equations (6.3) and ( 6.4 ) also follow from concurrent work by Figueroa-López and Ólafsson ( 2015 ). For , they also follow from Proposition 8.5 in Andersen and Lipton ( 2013 ), but not for , because then the constant from that proposition vanishes when specializing it to the CGMY model.…”

Estimation of Tempered Stable Lévy Models of Infinite Variation

“…Roughly, the conditions above amount to say that the small jumps of X behave like those of a Y -stable Lévy process. We refer the reader to [12] for further background about this class. The parameter Y is called the index of jump activity and coincides with the Blumenthal-Getoor index, which controls the jump activity of X in that s∈(0,t] |∆X s | γ < ∞ for all γ > Y and t > 0, where ∆X s := X s − X s− is the jump of X at time s. The range of Y considered here (namely, Y ∈ (1, 2)) is the most relevant for financial applications based on several econometric studies of high-frequency financial data (cf.…”

Nonparametric implied Lévy densities

“…The current paper is related to several strands of literature. First, Lévybased approximations of short-dated options have been studied with various degrees of generality in earlier work, see e.g., [1], [6], [15], [16,17] and [24], and the many references therein. The results of these papers are typically derived for a single option with a fixed strike while here we are interested in the approximation across the whole range of strikes that cover the positive real line.…”