On the nature of ill-posedness of an inverse problem arising in option pricing

Inverse Problems

Volumn 19, Issue 6, 2003, Pages 1319-1338

  Hein, Torsten ^a   Hofmann, Bernd ^a  

^a CHEMNITZ UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

BENCHMARKING; COMPUTER SIMULATION; CONVERGENCE OF NUMERICAL METHODS; DIFFERENTIATION (CALCULUS); MATHEMATICAL OPERATORS; NONLINEAR EQUATIONS; VECTORS;

BLACK-SCHOLES FUNCTIONS; ILL POSEDNESS; INNER LINEAR CONVOLUTION OPERATOR; NEMYTSKII OPERATOR; OPTION PRICING; TIKHONOV REGULARIZATION;

INVERSE PROBLEMS;

EID: 0347577824     PISSN: 02665611     EISSN: None     Source Type: Journal    
DOI: 10.1088/0266-5611/19/6/006     Document Type: Article

References (33)

1. Ambrosetti A and Prodi G 1993 A Primer of Nonlinear Analysis (Cambridge: Cambridge University Press)
2. Andersen L B G and Brotherton-Ratcliffe R 1998 The equity option volatility smile: an implicit finite-difference approach J. Comput. Finance 1 5-37
3. Avellaneda M, Friedman C, Holmes, R and Samperi D 1997 Calibrating volatility surfaces via relative entropy minimization Appl. Math. Finance 4 3-64
4. Bakushinsky A and Goncharsky A 1994 Ill-Posed Problems: Theory and Applications (Dordrecht: Kluwer)
5. Binder A, Engl H W and Schatz A 2003 Advanced Numerical Techniques for Financial Engineering (Linz: MathConsult)
6. Black F and Scholes M 1973 The pricing of options and corporate liabilities J. Political Econom. 81 637-54
7. Bouchouev I and Isakov V 1997 The inverse problem of option pricing Inverse Problems 13 L11-7
8. Bouchouev I and Isakov V 1999 Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets Inverse Problems 15 R95-116
9. Colton D and Kress R 1992 Inverse Acoustic and Electromagnetic Scattering Theory (Berlin: Springer)
10. Cont R 2003 Inverse Problems in Financial Modeling (Lecture Series) (Berlin: Humboldt-Universität)
11. Crépey S 2003 Calibration of the local volatility in a trinomial tree using Tikhonov regularization Inverse Problems 19 91-127
12. Crépey S 2003 Calibration of the local volatility in a generalized Black-Scholes model using Tikhonov regularization SIAM J. Math. Anal. 34 1183-206
13. Dupire B 1994 Pricing with a smile Risk 7 18-20
14. Engl H W, Hanke M and Neubauer A 1996 Regularization of Inverse Problems (Dordrecht: Kluwer)
15. Engl H W, Kunisch K and Neubauer A 1989 Convergence rates for Tikhonov regularization of non-linear ill-posed problems Inverse Problems 5 523-40
16. Emvudu Y, Hein T and Hofmann B 2001 Some approach for solving an inverse option pricing problem Preprint 2001-7 Technische Universität, Fakultät für Mathematik, Chemnitz
17. Fichtenholz G M 1970 Differential- und Integralrechnung 5th edn, vol 1 (Berlin: Deutsche)
18. Hanke M and Scherzer O 2001 Inverse problems light: numerical differentiation Am. Math. Mon. 108 512-21
19. Hansen P C 1998 Rank-Deficient and Discrete Ill-Posed Problems - Numerical Aspects of Linear Inversion (Philadelphia, PA: SIAM)
20. Hein T 2003 Numerische Studie zu einem inversen Problem der Optionspreisbildung im zeitabhängigen Fall Preprint 2003-2 Technische Universität, Fakultät für Mathematik, Chemnitz
21. Hein T and Hofmann B 2003 On the inverse problem of option pricing in the time-dependent case Preprint 2003-1 Technische Universität, Fakultät für Mathematik. Chemnitz
22. Hofmann B. 1986 Regularization for Applied Inverse and Ill-Posed Problems (Leipzig: Teubner)
23. Hofmann B 1994 On the degree of ill-posedness for nonlinear problems J. Inv. Ill-Posed Problems 2 61-76
24. Hofmann B and Fleischer G 1999 Stability rates for linear ill-posed problems with compact and non-compact operators Z. Anal. Anw. 18 267-86
25. Hofmann B and Scherzer O 1994 Factors influencing the ill-posedness of nonlinear problems Inverse Problems 10 1277-97
26. Hull J C 2000 Options, Futures and Other Derivatives 4th edn (London: Prentice-Hall)
27. Krasnoselskii M A 1995 Asymptotics of Nonlinearities and Operator Equations (Basle: Birkhäuser)
28. Kress R 1989 Linear Integral Equations (Berlin: Springer)
29. Kwok Y K 1998 Mathematical Models of Financial Derivatives (Singapore: Springer)
30. Lishang J and Youshan T 2001 Identifying the volatility of underlying assets from option prices Inverse Problems 17 137-55
31. Merton R C 1973 Theory of rational option pricing Bell J. Econom. Manag. Sci. 4 141-183
32. Natanson I P 1975 Theorie der Funktionen einer reellen Veränderlichen (Berlin: Akademie)
33. Wilmott P, Dewynne J and Howison S 1998 The Mathematics of Financial Derivatives (Cambridge: Cambridge University Press)