Calibration of the local volatility in a generalized Black-Scholes model using tikhonov regularization

SIAM Journal on Mathematical Analysis

Volumn 34, Issue 5, 2003, Pages 1183-1206

  Crepey S ^a  

^a Artabel SA   (France)

Author keywords

CalibrationIll posed nonlinear inverse problem; Options; Parameter estimation; Tikhonov regularization; Wp1,2 estimates

Indexed keywords

EID: 0242511629     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141001400202     Document Type: Article

References (39)

1. L. ANDERSEN AND R. BROTHERTON-RATCLIFFE, The equity option volatility smile: An implicit finite difference approach, J. Comput. Finance, 1 (2) (1997), pp. 5-37.
2. M. AVELLANEDA, C. FRIEDMAN, R. HOLMES, AND D. SAMPERI, Calibrating volatility surfaces via relative-entropy minimization, Appl. Math. Finance, 41 (1997), pp. 37-64.
3. A. BENSOUSSAN AND J.-L. LIONS, Applications des Inéquations Variationnelles en Contrôle Stochastique, Dunod, Paris, 1978.
4. H. BERESTYCKI, J. BUSCA, AND I. FLORENT, Asymptotics and calibration of local volatility models, Quant. Finance, 2 (2002), pp. 61-69.
5. A. BINDER, H.W. ENGL, C.W. GROETSCH, A. NEUBAUER, AND O. SCHERZER, Weakly closed nonlinear operators and parameter identification in parabolic equations by Tikhonov regularization, Appl. Anal., 55 (1994), pp. 13-25.
6. F. BLACK AND M. SCHOLES, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), pp. 637-659.
7. J. BODURTHA AND M. JERMAKYAN, Nonparametric estimation of an implied volatility surface, J. Comput. Finance, 2 (4) (1999), pp. 29-60.
8. I. BOUCHOUEV AND V. ISAKOV, Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets, Inverse Problems, 15 (1999), pp. R95-R116.
9. H. BRÉZIS, Analyse fonctionnelle: Théorie et application, Coll. Math. Appl. pour la Maîtrise, Masson, Paris, 1983.
10. L. CAFFARELLI, M.G. CRANDALL, M. KOCAN, AND A. SWIECH, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math., 49 (1996), pp. 365-397.
11. T. COLEMAN, Y. LI, AND A. VERMA, Reconstructing the unknown volatility function, J. Comput. Finance, 2 (3) (1999), pp. 77-102.
12. M. CRANDALL, H. ISHII, AND P.-L. LIONS. User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), pp. 1-67.
13. p-theory for fully nonlinear parabolic equations Comm. Partial Differential Equations, 25 (2000), pp. 1997-2053.
14. S. CRÉEY, Contribution à des méthodes numériques appliquées àla Finance et aux Jeux Différentiels, Ph.D. thesis, Ecole Polytechnique, France, 2001.
15. S. CRÉPEY, Tikhonov Regularization and Calibration of a Local Volatility in Finance - Stability, Convergence and Convergence Rates Issues, CMAP Internal Research Report 474, CMAP, Ecole Polytechnique, Palaiseau, France, 2002.
16. S. CRÉPEY, Calibration of the local volatility in a trinomial tree using Tikhonov regularization, Inverse Problems, 19 (2003), pp. 91-127.
17. E. DERMAN AND I. KANI, Riding on a smile, Risk, 7 (1994), pp. 32-39.
18. E. DERMAN, I. KANI, AND N. CHRISS, Implied trinomial trees of the volatility smile, J. Derivatives, 3 (4) (1996), pp. 7-22.
19. B. DUPIRE, Pricing with a smile, Risk, 7 (1994), pp. 18-20.
20. N. EL KAROUI, Modèles Stochastiques en Finance (Cours de DEA), Lecture Notes, Université Paris VI, 1997.
21. H.W. ENGL, M. HANKE, AND A. NEUBAUER, Regularization of Inverse Problems, Kluwer, Dordrecht, The Netherlands, 1996.
22. H.W. ENGL, K. KUNISCH, AND A. NEUBAUER, Convergence rates for Tikhonov regularization of nonlinear ill-posed problems, Inverse Problems, 5 (1989), pp. 523-540.
23. E. FABES, Singular integrals and partial differential equations of parabolic type, Studia Math., 28 (1966), pp. 6-131.
24. N. JACKSON, E. SÜLI, AND S. HOWISON, Computation of deterministic volatility surfaces, J. Comput. Finance, 2 (2) (1999), pp. 5-32.
25. I. KARATZAS AND S. SHREVE, Brownian Motion and Stochastic Calculus, Springer-Verlag, Berlin, 1988.
26. N.V. KRYLOV, Controlled Diffusion Processes, Springer-Verlag, Berlin, 1980.
27. K. KUNISCH AND G. GEYMAYER, Convergence rates for regularized nonlinear ill-posed problems, in Modelling and Inverse Problems of Control for Distributed Parameter Systems, Lecture Notes in Control and Inform. Sci. 154, Springer-Verlag, Berlin, 1991, pp. 81-92.
28. O. LADYZHENSKAYA, V. SOLONNIKOV, AND N. URAL'TSEVA, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, RI, 1968.
29. R. LAGNADO AND S. OSHER, A technique for calibrating derivative security pricing models: Numerical solution of an inverse problem, J. Comput. Finance, 1 (1) (1997), pp. 13-25.
30. B. LARROUTUROU AND P.L. LIONS, Méthodes mathématiques pour les sciences de l'ingénieur: Optimisation et Analyse Numérique, Lecture Notes, Ecole Polytechnique, 1994.
31. R.C. MERTON, Theory of rational option pricing, Bell J, Econom. and Management Sci., 4 (1973), pp. 141-183.
32. A. NEUBAUER, Tikhonov regularization for nonlinear ill-posed problems: Optimal convergence and finite dimensional approximation, Inverse Problems, 5 (1989), pp. 541-557.
33. A. NEUBAUER AND O. SCHERZER, Finite dimensional approximation of Tikhonov regularised solution of nonlinear ill-posed problems, Numer. Funct. Anal. Optim., 11 (1990), pp. 85-99.
34. R. REBONATO, Volatility and Correlation in the Pricing of Equity, FX and Interest-Rate Options, John Wiley, New York, 1999.
35. M. RUBINSTEIN, Implied binomial trees, J. Finance, 69 (1994), pp. 771-818.
36. O. SCHERZER, The use of Tikhonov regularization in the identification of electrical conductivities from over determined boundary data, Results Math., 22 (1992), pp. 598-618.
37. D.W. STROOCK AND S.R.S. VARADHAN, Multidimensional Diffusion Processes, Springer-Verlag, Berlin, 1979.
38. M. TIKHONOV, Regularization of incorrectly posed problems, Soviet Math. Dokl., 4 (1963), pp. 1624-1627.
39. L. WANG, On the regularity theory of fully nonlinear parabolic equations. I, Comm. Pure Appl, Math., 45 (1992), pp. 27-76.