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August  2010, 27(3): 907-917. doi: 10.3934/dcds.2010.27.907

A nonlinear partial integro-differential equation from mathematical finance

Frederic Abergel 1, and  Remi Tachet 1,

1. 

Grande Voie des Vignes, Chatenay-Malabry, 92295, France, France

Received  July 2009 Revised  January 2010 Published  March 2010

Consistently fitting vanilla option surfaces is an important issue when it comes to modeling in finance. As far as local and stochastic volatility models are concerned, this problem boils down to the resolution of a nonlinear integro-differential pde. The non-locality of this equation stems from the quotient of two integral terms and is not defined for all bounded continuous functions. In this paper, we use a fixed point argument and suitable a priori estimates to prove short-time existence of solutions for this equation.
Keywords: Analysis of pde's, local and stochastic volatility models, nonlinear parabolic partial integro-differential equation, fixed point methods, short-time existence, calibration of vanillas..
Mathematics Subject Classification: Primary: 35A01, 35K55, 35R09, 47G20; Secondary: 60H1.
Citation: Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907
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