Estimation of local volatilities in a generalized Black-Scholes model. (English) Zbl 1079.91022

Summary: This paper studies a parameter estimation problem for a generalized Black-Scholes equation, which is used for option pricing. In estimating the volatility function from a set of market observations, we use an implicit finite difference scheme. The function space parameter estimation convergence (FSPEC) is proved and numerical simulations were performed.


91G20 Derivative securities (option pricing, hedging, etc.)
Full Text: DOI


[1] Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101
[2] Banks, H.T.; Kunish, K., Estimation techniques for distributed parameter systems, (1989), Birkhäuser Boston · Zbl 0695.93020
[3] Black, F.; Scholes, M.S., The pricing of options and corporate liabilities, J. political economy, 81, 637-654, (1973) · Zbl 1092.91524
[4] Bouchouev, I.; Isakov, V., The inverse problem of option pricing, Inverse problems, 13, 11-17, (1997)
[5] Bouchouev, I.; Isakov, V., Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets, Inverse problems, 15, 1-22, (1999) · Zbl 0938.35190
[6] Coleman, T.F.; Li, Y.; Verma, A., Reconstructing the unknown local volatility function, J. computat. finance, 2, 77-102, (1999)
[7] Hull, J.C., Options, futures, and other derivatives, (1997), Prentice-Hall Englewood Cliffs, NJ · Zbl 1087.91025
[8] Jackson, N.; Süli, E.; Howison, S., Computation of deterministic volatility surfaces, J. computat. finance, 2, 5-32, (1999)
[9] Kangro, R.; Nicolaides, R., Far field boundary conditions for black – scholes equations, SIAM J. numer. anal., 38, 1357-1368, (2000) · Zbl 0990.35013
[10] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc. Transl. 23, Providence, RI, 1968
[11] Lishang, J.; Youshan, T., Identifying the volatility of underlying assets from option prices, Inverse problems, 17, 137-155, (2001) · Zbl 0997.91024
[12] Press, W.; Teukolsky, S.; Vetterling, W.; Flannery, B., Numerical recipies in C: the art of scientific computing, (1992), Cambridge University Press New York
[13] Roos, H.G.; Stynes, M.; Tobiska, L., Numerical methods for singularly perturbed differential equations, (1996), Springer-Verlag
[14] Schultz, M., Spline analysis, (1973), Prentice-Hall Englewood Cliffs · Zbl 0333.41009
[15] Zvan, R.; Vetzal, K.; Forsyth, P., Swing high swing low, Risk mag., 97, 71-74, (1998)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.