Consistently fitting vanilla option surfaces is an important issue when it comes to modelling in finance. In three different models: local and stochastic volatility, local correlation and hybrid local volatility with stochstic rates, this calibration boils down to the resolution of a nonlinear partial integro-differential equation. In a first part, we give existence results of solutions for the calibration equation. They are based upon fixed point methods in Hölder spaces and short-time a priori estimates. We then apply those existence results to the three models previously mentioned and give the calibration obtained when solving the pde numerically. At last, we focus on the algorithm used for the resolution: an ADI predictor/corrector scheme that needs to be modified to take into account the nonlinear term. We also study an instability phenomenon that occurs in certain cases for the local and stochastic volatility model. Using Hadamard's theory, we try to offer a theoretical explanation to the instability
| Identifer | oai:union.ndltd.org:CCSD/oai:tel.archives-ouvertes.fr:tel-00658766 |
| Date | 06 October 2011 |
| Creators | Tachet des combes, Rémi |
| Publisher | Ecole Centrale Paris |
| Source Sets | CCSD theses-EN-ligne, France |
| Language | English |
| Detected Language | English |
| Type | PhD thesis |
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