Asymptotics for Exponential Levy Processes and Their Volatility Smile: Survey and New Results

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

“…This is a special case of Corollary 3.3 in Figueroa-López and Ólafsson ( 2015 ). Note that Proposition 8.5 in Andersen and Lipton ( 2013 ) is not applicable here, because the constant from this proposition vanishes for the CGMY model, and so the leading term of the slope is not obtained. Theorem 1 (iv) from our Section 2 is not useful, either; it gives the correct digital call limit price , but does not provide the second-order term necessary to get slope asymptotics.…”

“…The case and need not be discussed, as it is a special case of Proposition 8.5 in Andersen and Lipton ( 2013 ). Our Proposition 3 could also be applied, as the CGMY process has finite variation in this case.…”

“…Recent years have seen an explosion of the literature on asymptotics of option prices and implied volatilities (see, e.g., Andersen and Lipton 2013 ; Friz et al 2011 for further details). Such results are of practical relevance for fast model calibration, qualitative model assessment and parametrization design.…”

“…One of the few other works dealing with small-time Lévy slope asymptotics is the comprehensive recent paper by Andersen and Lipton ( 2013 ). Besides many other problems on various models and asymptotic regimes, they study the small-maturity ATM digital price and volatility slope for the tempered stable model (Propositions 8.4 and 8.5 in Andersen and Lipton 2013 ). This includes the CGMY model as a special case (see Example 10 for details).…”

Small Noise Expansion for the l'EVY Perturbed Vasicek Model

“…Our analysis is based on [3], in the setting proposed in [9,Sec.6.2]. Let us underline that during recent years a wide range of small noise expansion techniques have been developed, particularly with respect to the so called Loval Volatility Models (LVMs), see, e.g., [4,8,11,13]. LVMs are commonly used to analyse options markets where the underlying volatility strongly depends on the level of the underlying itself, therefore LVMs are also widely accepted as tools to model interest-rate derivatives as is the case for the Vasicek model.…”