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SUMMARY:CF Weekly Workshop - by Professor Alexander Lipton - Asymptotics f
 or Exponential Lévy Processes and their Volatility Smile: Survey and New 
 Results  - Professor Alexander Lipton\, Bank of America Merrill Lynch\, Im
 perial College.
DTSTART:20121009T160000Z
DTEND:20121009T170000Z
UID:TALK40684@talks.cam.ac.uk
CONTACT:Sheryl Anderson
DESCRIPTION:Exponential Lévy processes can be used to model the evolution
  of various financial variables such as FX rates\, stock prices\, etc. Con
 siderable efforts have been devoted to pricing derivatives written on unde
 rliers governed by such processes\, and the corresponding implied volatili
 ty surfaces have been analyzed in some detail. In the non-asymptotic regim
 es\, option prices are described by the Lewis-Lipton formula which allows 
 one to represent them as Fourier integrals\; the prices can be trivially e
 xpressed in terms of their implied volatility. Recently\, attempts at calc
 ulating the asymptotic limits of the implied volatility have yielded sever
 al expressions for the short-time\, long-time\, and wing asymptotics. In o
 rder to study the volatility surface in required detail\, in this paper we
  use the FX conventions and describe the implied volatility as a function 
 of the Black-Scholes delta. Surprisingly\, this convention is closely rela
 ted to the resolution of singularities frequently used in algebraic geomet
 ry. In this framework\, we survey the literature\, reformulate some known 
 facts regarding the asymptotic behaviour of the implied volatility\, and p
 resent several new results. We emphasize the role of fractional differenti
 ation in studying the tempered stable exponential Lévy processes and deri
 ve novel numerical methods based on judicial finite-difference approximati
 ons for fractional derivatives. We also briefly demonstrate how to extend 
 our results in order to study important cases of local and stochastic vola
 tility models\, whose close relation to the Lévy process based models is 
 particularly clear when the Lewis-Lipton formula is used. Our main conclus
 ion is that studying asymptotic properties of the implied volatility\, whi
 le theoretically exciting\, is not always practically useful because the d
 omain of validity of many asymptotic expressions is small.
LOCATION:Lecture Theatre\, Trinity Hall
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