Abstract
A degenerate parabolic equation in the zero-coupon bond pricing (ZCBP) is studied. First, we analyze the time discretization of the equation. Involving weighted Sobolev spaces, we develop a variational analysis to describe qualitative properties of the solution. On each time-level we formulate a Petrov-Galerkin FEM, in which each of the basis functions of the trial space is determined by the finite volume difference scheme in [2, 3]. Using this formulation, we establish the stability of the method with respect to a discrete energy norm and show that the error of the numerical solution in the energy norm is O(h), where h denotes the mesh parameter.
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
Preview
Unable to display preview. Download preview PDF.
References
Black, F., Sholes, M.: The pricing of options and corporate liabilities. J. Pol. Econ. 81, 637–659 (1973)
Chernogorova, T., Valkov, R.: A computational scheme for a problem in the zero-coupon bond pricing. Amer. Inst. of Physics 1301, 370–378 (2010)
Chernogorova, T., Valkov, R.: Finite volume difference scheme for a degenerate parabolic equation in the zero-coupon bond pricing. Math. and Comp. Modelling 54(11-12), 2659–2671 (2011)
Chernogorova, T., Valkov, R.: Finite-Volume Difference Scheme for the Black-Scholes Equation in Stochastic Volatility Models. In: Dimov, I., Dimova, S., Kolkovska, N. (eds.) NMA 2010. LNCS, vol. 6046, pp. 377–385. Springer, Heidelberg (2011)
Deng, Z.C., Yu, J.N., Yang, L.: An inverse problem arisen in the zero bond pricing. Nonl. Anal.: Real World Appls 11(3), 1278–1288 (2010)
Huang, C.-S., Hung, C.-H., Wang, S.: A fitted finite volume method for the valuation of options on assets with stochastic volatilities. Computing 77, 297–320 (2006)
Kacur, J.: Method of Rothe in Evolution Equations. BSB Teubner Verlagsges, Leipzig (1985)
Oleinik, O.A., Radkevic, E.V.: Second Order Differential Equation with Non-negative Characteristic Form. Rhode Island and Plenum Press, American Mathematical Society, New York (1973)
Seydel, R.: Tools for Computational Finance, 2nd edn. Springer, Heidelberg (2003)
Stamffi, J., Goodman, V.: The Mathematics of Finance: Modeling and Hedging. Thomas Learning (2001)
Wang, S.: A novel finite volume method for Black-Sholes equation governing option pricing. IMA J. of Numer. Anal. 24, 699–720 (2004)
Wang, I.R., Wan, J.W.I., Forsyth, P.A.: Robust numerical valuation of European and American options under CGMY process. J. Comp. Finance 10(4), 32–69 (2007)
Windeliff, H., Forsyth, P.A., Vezzal, K.R.: Analysis of the stability of the linear boundary conditions for the Black-Sholes equation. J. Comp. Finance 8, 65–92 (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Valkov, R.L. (2012). Petrov-Galerkin Analysis for a Degenerate Parabolic Equation in Zero-Coupon Bond Pricing. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2011. Lecture Notes in Computer Science, vol 7116. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29843-1_75
Download citation
DOI: https://doi.org/10.1007/978-3-642-29843-1_75
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-29842-4
Online ISBN: 978-3-642-29843-1
eBook Packages: Computer ScienceComputer Science (R0)