Abstract

http://ssrn.com/abstract=1542077
 
 

References (32)



 
 

Citations (11)



 


 



Asymptotics of Implied Volatility in Local Volatility Models


Jim Gatheral


Baruch College, CUNY

Elton P. Hsu


Department of Mathematics, Northwestern University

Peter M. Laurence


University of Rome I - Department of Mathematics; Courant Institute, NYU

Cheng Ouyang


Purdue University

Tai-Ho Wang


Baruch College, CUNY

December 23, 2009

Mathematical Finance, Forthcoming

Abstract:     
Using an expansion of the transition density function of a 1-dimensional time inhomogeneous diffusion, we obtain the first and second order terms in the short time asymptotics of European call option prices. The method described can be generalized to any order. We then use these option prices approximations to calculate the first order and second order deviation of the implied volatility from its leading value and obtain approximations which we numerically demonstrate to be highly accurate. The analysis is extended to degenerate diffusion's using probabilistic methods, i.e. the so called principle of not feeling the boundary.

Number of Pages in PDF File: 39

Keywords: Implied volatility, local volatility, asymptotic expansion, heat kernels

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Date posted: January 27, 2010 ; Last revised: July 30, 2010

Suggested Citation

Gatheral, Jim and Hsu, Elton P. and Laurence, Peter M. and Ouyang, Cheng and Wang, Tai-Ho, Asymptotics of Implied Volatility in Local Volatility Models (December 23, 2009). Mathematical Finance, Forthcoming. Available at SSRN: http://ssrn.com/abstract=1542077

Contact Information

Jim Gatheral (Contact Author)
Baruch College, CUNY ( email )
Department of Mathematics
One Bernard Baruch Way
New York, NY 10010
United States
Elton P. Hsu
Department of Mathematics, Northwestern University ( email )
2001 Sheridan Road
Evanston, IL 60208
United States
Peter M. Laurence
University of Rome I - Department of Mathematics ( email )
Roma, I-00185
Italy
Courant Institute, NYU ( email )
Division of Quantitative Finance 251 Mercer Street
New York, NY 10012
United States
212 9983000 (Phone)
212 9954121 (Fax)
Cheng Ouyang
Purdue University ( email )
150 N. University Street
Department of Mathematics
West Lafayette, IN 47907
United States
765-496-3578 (Phone)
Tai-Ho Wang
Baruch College, CUNY ( email )
1 Bernard Baruch Way
New York, NY 10010
United States
+1-646-312-4130 (Phone)
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