Research ArticleOpen AccessGoogle Scholar indexed
Nonparametric Model Calibration for Derivatives
Laboratory MICS, Centrale Supélec, Chatenay Malabry, France
Laboratory MICS, Centrale Supélec, Chatenay Malabry, France
Laboratory MICS, Centrale Supélec, Chatenay Malabry, France
- 1 Laboratory MICS, Centrale Supélec, Chatenay Malabry, France
- 2 Laboratory MICS, Centrale Supélec, Chatenay Malabry, France
- 3 Laboratory MICS, Centrale Supélec, Chatenay Malabry, France
Journal of Mathematical Finance·Volume 07 (2017)·Pages 571–596·Published 19 June 2017·DOI10.4236/jmf.2017.73030
Copy link · social · email
Abstract
Consistently fitting vanilla option surface is an important issue in derivative modelling. In this paper, we consider three different models: local and stochastic volatility, local correlation, hybrid local volatility with stochastic rates, and address their exact, nonparametric calibration. This calibration process requires solving a nonlinear partial integro-differential equation. A modified alternating direction implicit algorithm is used, and its theoretical and numerical analysis is performed.
KeywordsLocal Stochastic VolatilityCalibrationDerivative PricingPartial Integro-Differential Equations
- Breeden, D. and Litzenberger, R. (1978) State Contingent Prices Implicit in Option Prices. Journal of Business, 51, 621-651. https://doi.org/10.1086/296025
- Dupire, B. (1993) Pricing and Hedging with Smiles. Proceeding AFFI Conference, La Baule.
- Lipton, A. (2002) The Vol Smile Problem. Risk Magazine, 15, 61-65.
- Richtmyer, R. and Morton, K. (1967) Difference Methods for Initial Value Problems.
- Tachet des Combes, R. (2011) Non-Parametric Model Calibration in Finance. Ph.D. Thesis, Ecole Centrale, Paris.
- Gyongy, I. (1986) Mimicking the One-Dimensional Marginal Distributions of Processes Having an Ito Differential. Probability Theory and Related Fields, 71, 501-516. https://doi.org/10.1007/BF00699039
- Abergel, F. and Tachet, R. (2010) A Nonlinear Partial Integrodifferential Equation from Mathematical Finance. Discrete and Continuous Dynamical Systems, 27, 907-917. https://doi.org/10.3934/dcds.2010.27.907
- Brigo, D. and Alfonsi, A. (2005) Credit Default Swap Calibration and Derivatives Pricing with the SSRD Stochastic Intensity Model. Finance and Stochastics, 9, 29-42. https://doi.org/10.1007/s00780-004-0131-x
- Qu, D. (2010) Pricing Basket Options with Skew. Wilmott Magazine, 58-64.
- Jourdain, B. and Sbai, M. (2012) Coupling Index and Stocks. Quantitative Finance, 12, 805-818. https://doi.org/10.1080/14697681003785959
- Douglas, J. and Rachford, H. (1956) On the Numerical Solution of Heat Conduction Problems in Two and Three Space Variables. Transactions of the American Mathematical Society, 82, 421-439. https://doi.org/10.1090/S0002-9947-1956-0084194-4
- Douglas, J. (1962) Alternating Direction Methods for Three Space Variables. Numerische Mathematik, 4, 41-63. https://doi.org/10.1007/BF01386295
- Douglas, J. and Gunn, J. (1964) A General Formulation of Alternating Direction Methods. Numerische Mathematik, 6, 428-453. https://doi.org/10.1007/BF01386093
- Beam, R. and Warming, R. (1980) Alternating Direction Implicit Methods for Parabolic Equations with a Mixed Derivative. SIAM Journal on Scientific and Statistical Computing, 1, 131-159. https://doi.org/10.1137/0901007
- Alibaud, N. (2007) Existence, Uniqueness and Regularity for Nonlinear Parabolic Equations with Nonlocal Terms Equations with Nonlocal Terms. Nonlinear Differential Equations and Applications, 14, 259-289. https://doi.org/10.1007/s00030-007-5029-9