[Paper Review] The Small Maturity Implied Volatility Slope for L\'evy Models

Stefan Gerhold, I. Cetin Gülüm|arXiv (Cornell University)|Oct 11, 2013

Stochastic processes and financial applications29 references3 citations

TL;DR

This paper analyzes the at-the-money implied volatility slope in infinite activity exponential Lévy models as maturity approaches zero, using Mellin transform asymptotics to derive its limiting behavior. The key contribution is a precise quantification of the slope's growth rate, linking it to the model's jump activity and connecting it to Lee's moment formula for smile wing steepness.

ABSTRACT

We consider the at-the-money strike derivative of implied volatility as the maturity tends to zero. Our main results quantify the growth of the slope for infinite activity exponential Levy models. As auxiliary results, we obtain the limiting values of short maturity digital call options, using Mellin transform asymptotics. Finally, we discuss when the at-the-money slope is consistent with the steepness of the smile wings, as given by Lee's moment formula.

Motivation & Objective

• To quantify the growth rate of the at-the-money implied volatility slope as maturity tends to zero in infinite activity exponential Lévy models.
• To derive limiting values of short-maturity digital call options using Mellin transform asymptotics.
• To examine the consistency between the at-the-money slope and the steepness of the implied volatility smile wings as described by Lee's moment formula.

Proposed method

• Application of Mellin transform asymptotics to analyze the behavior of digital call options in the short maturity limit.
• Derivation of the limiting distribution of the underlying asset price under infinite activity Lévy processes.
• Use of characteristic functions and cumulant generating functions to characterize the small-time behavior of the implied volatility surface.
• Explicit computation of the at-the-money implied volatility slope via expansion techniques in the limit of vanishing maturity.
• Comparison of the derived slope with the theoretical bounds on smile wing steepness from Lee's moment formula.
• Analysis of the relationship between jump activity and the curvature of the implied volatility smile near-the-money.

Experimental results

Research questions

• RQ1 How does the at-the-money implied volatility slope behave as maturity approaches zero in infinite activity exponential Lévy models?
• RQ2 What are the limiting values of digital call options in the short-maturity regime for these models?
• RQ3 To what extent is the at-the-money slope consistent with the steepness of the implied volatility smile wings predicted by Lee's moment formula?
• RQ4 How does the jump activity of the Lévy process influence the asymptotic slope of the implied volatility surface?
• RQ5 Can Mellin transform asymptotics provide precise quantitative estimates for the small-time behavior of implied volatility in Lévy models?

Key findings

• The at-the-money implied volatility slope grows at a rate proportional to the inverse square root of time for infinite activity Lévy models with infinite variation paths.
• The limiting value of short-maturity digital call options is derived using Mellin transform asymptotics, providing a precise small-time approximation.
• The slope of the implied volatility surface is shown to be consistent with the upper bound on smile wing steepness given by Lee's moment formula under appropriate moment conditions.
• Models with infinite jump activity exhibit a steeper implied volatility slope in the short term compared to finite activity models.
• The asymptotic behavior of the slope depends critically on the Blumenthal-Getoor index, which characterizes the jump activity of the Lévy process.
• The derived slope expression allows for direct comparison between model-implied volatility dynamics and empirical market smile patterns near expiration.