Abstract
We prove the existence and uniqueness theorems for inverse problem of determination of the lower coefficient in the Black-Scholes type equation with additional condition of integral observation. These results are based on the investigation of unique solvability of corresponding direct problem which is of independent interest. We give the example of the inverse problem for which the conditions of the theorems proved are fulfilled.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Kruzhkov, S.N.: Quasilinear parabolic equations and systems with two independent variables. Trudy Sem. im. I.G. Petrovskogo 5, 217–272 (1979)
Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–659 (1973)
Hull, J.: Options, Futures and Other Derivatives. Prentice Hall, Upper Saddle River (2005)
Fichera, G.: Sulle equazioni differenziali lineari ellitico-paraboliche del secondo ordine. Atti Accad. Nazionale dei Lincei. Mem. Cl. Sci. Fis. Mat. Natur. Ser. I(8) 5, 1–30 (1956)
Oleǐnik, O.A., Radkevič, E.A.: Second Order Differential Equations with Nonnegative Characteristic Form. AMS, Rhode Island and Plenum Press, New York (1973)
Deng, Z.C., Yang, L.: An inverse problem of identifying the coefficient of first-order in a degenerate parabolic equation. J. Comput. Appl. Math. 235, 4404–4417 (2011)
Deng, Z.C., Yang, L.: An inverse problem of identifying the radiative coefficient in a degenerate parabolic equation. Chin. Ann. Math. Ser. B. 35B(3), 355–382 (2014)
Bouchouev, I., Isakov, V.: Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets. Inverse Prob. 15(3), 95–116 (1999)
Lishang, J., Yourshan, T.: Identifying the volatibility of underlying assets from option prices. Inverse Prob. 17(1), 137–155 (2001)
Lishang, J., Qihong, C., Lijun, W., Zhang, J.E.: A new well-posed algorithm to recover implied local volatibility. Quant. Financ. 3(6), 451–457 (2003)
Prilepko, A.I., Kamynin, V.L., Kostin, A.B.: Inverse source problem for parabolic equation with the condition of integral observation in time. J. Inverse III-posed Prob. 26(4), 523–539 (2018)
Bukharova, T.I., Kamynin, V.L.: Inverse problem of determining the absorption coefficient in the multidimensional heat equation with unlimited minor coefficients. Comput. Math. Math. Phys. 55(7), 1183–1195 (2015)
Lyusternik, L.A., Sobolev, V.I.: Kratkii Kurs Functcional’nogo Analiza (Brief Course of Functional Analysis). Vysshaya Shkola, Moscow (1982)
Acknowledgements
This work was partially supported by the Program of competitiveness increase of the National Research Nuclear University MEPhI (Moscow Engineering Physics Institute); contract No. 02.a03.21.0005, 27.08.2013.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Kamynin, V.L., Bukharova, T.I. (2019). On Inverse Problem of Determination of the Coefficient in the Black-Scholes Type Equation. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Finite Difference Methods. Theory and Applications. FDM 2018. Lecture Notes in Computer Science(), vol 11386. Springer, Cham. https://doi.org/10.1007/978-3-030-11539-5_35
Download citation
DOI: https://doi.org/10.1007/978-3-030-11539-5_35
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-11538-8
Online ISBN: 978-3-030-11539-5
eBook Packages: Computer ScienceComputer Science (R0)