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Smile Asymptotics II: Models with Known Moment Generating Functions

Part of: Distribution theory - Probability Mathematical economics

Published online by Cambridge University Press:  14 July 2016

Shalom Benaim  and
Peter Friz
Shalom Benaim*
Affiliation:
University of Cambridge
Peter Friz*
Affiliation:
University of Cambridge
*
∗Postal address: Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK.
∗Postal address: Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK.
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Abstract

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The tail of risk neutral returns can be related explicitly with the wing behaviour of the Black-Scholes implied volatility smile. In situations where precise tail asymptotics are unknown but a moment generating function is available we establish, under easy-to-check Tauberian conditions, tail asymptotics on logarithmic scales. Such asymptotics are enough to make the tail-wing formula (see Benaim and Friz (2008)) work and so we obtain, under generic conditions, a limiting slope when plotting the square of the implied volatility against the log strike, improving a lim sup statement obtained earlier by Lee (2004). We apply these results to time-changed exponential Lévy models and examine several popular models in more detail, both analytically and numerically.

Keywords

Implied volatility smiletail-wing formulaTauberian theory

MSC classification

Secondary: 60E99: None of the above, but in this section 91B70: Stochastic models
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 1 , March 2008 , pp. 16 - 32
Copyright
Copyright © Applied Probability Trust 2008 

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