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CATEGORIES:Cambridge Finance Workshop Series
SUMMARY:CF Weekly Workshop - by Professor Alexander Lipton
- Asymptotics for Exponential Lévy Processes and
their Volatility Smile: Survey and New Results -
Professor Alexander Lipton\, Bank of America Merri
ll Lynch\, Imperial College.
DTSTART;TZID=Europe/London:20121009T170000
DTEND;TZID=Europe/London:20121009T180000
UID:TALK40684AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/40684
DESCRIPTION:Exponential Lévy processes can be used to model th
e evolution of various financial variables such as
FX rates\, stock prices\, etc. Considerable effor
ts have been devoted to pricing derivatives writte
n on underliers governed by such processes\, and t
he corresponding implied volatility surfaces have
been analyzed in some detail. In the non-asymptoti
c regimes\, option prices are described by the Lew
is-Lipton formula which allows one to represent th
em as Fourier integrals\; the prices can be trivia
lly expressed in terms of their implied volatility
. Recently\, attempts at calculating the asymptoti
c limits of the implied volatility have yielded se
veral expressions for the short-time\, long-time\,
and wing asymptotics. In order to study the volat
ility surface in required detail\, in this paper w
e use the FX conventions and describe the implied
volatility as a function of the Black-Scholes delt
a. Surprisingly\, this convention is closely relat
ed to the resolution of singularities frequently u
sed in algebraic geometry. In this framework\, we
survey the literature\, reformulate some known fac
ts regarding the asymptotic behaviour of the impli
ed volatility\, and present several new results. W
e emphasize the role of fractional differentiation
in studying the tempered stable exponential Lévy
processes and derive novel numerical methods based
on judicial finite-difference approximations for
fractional derivatives. We also briefly demonstrat
e how to extend our results in order to study impo
rtant cases of local and stochastic volatility mod
els\, whose close relation to the Lévy process bas
ed models is particularly clear when the Lewis-Lip
ton formula is used. Our main conclusion is that s
tudying asymptotic properties of the implied volat
ility\, while theoretically exciting\, is not alwa
ys practically useful because the domain of validi
ty of many asymptotic expressions is small.
LOCATION:Lecture Theatre\, Trinity Hall
CONTACT:Sheryl Anderson
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