REGULAR VARIATION AND SMILE ASYMPTOTICS

By Shalom Benaim, Peter K. Friz. 2006

125 citations recorded by the source corpus.

We consider risk‐neutral returns and show how their tail asymptotics translate directly to asymptotics of the implied volatility smile, thereby sharpening Roger Lee's celebrated moment formula. The theory of regular variation provides the ideal mathematical framework to formulate and prove such results. The practical value of our formulae comes from the vast literature on tail asymptotics and our conditions are often seen to be true by simple inspection of known results.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6d2e80b5ee9d3daa06747fa274cbd00cb7a8f757

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Citations in the corpus

A NEW LOOK AT SHORT‐TERM IMPLIED VOLATILITY IN ASSET PRICE MODELS WITH JUMPS (2012, background): “lly, for a given modelling framework, the implied volatility ˙b(t;k) is not available in closed form. Hence the study of the asymptotic behaviour in a variety of asymptotic regimes (e.g. xed tand k!1 [14, 8, 11]; t!1with kconstant [22] or proportional [13] to t; t!0 and kconstant [18, 21, 7] etc.) h”
A General Valuation Framework for SABR and Stochastic Local Volatility Models (2018, background): “It is well known that the implied volatility expansion of Hagan et al. (2002) is less accurate when the time to maturity is large or when the option is deep in or out of the money (see Paulot (2015) and Benaim and Friz (2009) for related discussions).”
Forecasting with Option-Implied Information (2012, unknown): “2 Jumps and Stochastic Volatility While the Black and Scholes (1973) and stochastic volatility option pricing models are often used to extract volatility, the study of higher moments calls for different models.”
Marginal Density Expansions for Diffusions and Stochastic Volatility I: Theoretical Foundations (2011, background): “We note that the square-root growth of implied volatility, in terms of log-strike, is actually a very general feature of models with moment explosions, [38, 7] which includes many stochastic volatility models [24, 39, 8].”
SMILE ASYMPTOTICS II: MODELS WITH KNOWN MOMENT GENERATING FUNCTIONS (2006, background): “In particular, the term-structure of the wings can be analytically understood.”
Asymptotics for Exponential Levy Processes and Their Volatility Smile: Survey and New Results (2012, background): “follow from the moment formulas of [69], [15], [16] and [55]; see [116] for a brief discussion.”
Pricing and Hedging in Exponential Lévy Models: Review of Recent Results (2011, methodology): “We start with the large and small strike behavior which was first analyzed by Roger Lee [35]; this analysis was subsequently extended and made more precise by Benaim and Friz [24, 23].”