Abstract
We present and analyze a splitting numerical scheme for two non-linear models of mathematical finance. Each of the problems is split into two parts: a hyperbolic equation solved numerically by using a flux limiter technique and a parabolic equation computed by implicit-explicit finite difference scheme. We show that the presented splitting numerical schemes are convergent and positivity preserving. Numerical results are also discussed.
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
Agliardi, R., Popivanov, P., Slavova, A.: Nonhypoellipticity and comparison principle for partial differential equations of Black-Scholes type. Nonlinear Anal. Real Word Appl. 12, 1429–1436 (2011)
Company, R., Navarro, E., Pintos, J., Ponsoda, E.: Numerical solution of linear and nonlinear Black-Scholes option pricing equations. Int. J. Comp. Math. Appl. 56, 813–821 (2008)
Ehrhardt, M. (ed.): Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing. Nova Science Publishers, New York (2008)
Gerisch, A., Griffiths, D.F., Weiner, R., Chaplain, M.A.J.: A positive splitting method for mixed hyperbolic-parabolic systems. Numer. Meth. PDEs 17(2), 152–168 (2001)
Hundsdorfer, W., Verwer, J.: Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer, Heidelberg (2003)
Koleva, M.N.: Positivity preserving numerical method for non-linear Black-Scholes models. In: Dimov, I., Faragó, I., Vulkov, L. (eds.) NAA 2012. LNCS, vol. 8236, pp. 363–370. Springer, Heidelberg (2013)
Koleva, M.N., Vulkov, L.G.: A numerical study of a parabolic Monge-Ampère equation in mathematical finance. In: Dimov, I., Dimova, S., Kolkovska, N. (eds.) NMA 2010. LNCS, vol. 6046, pp. 461–468. Springer, Heidelberg (2011)
Koleva, M., Vulkov, L.: Quasilinearization numerical scheme for fully nonlinear parabolic problems with applications in models of mathematical finance. Math. Comput. Model. 57(9–10), 2564–2575 (2013)
Lai, C.-H., Parrott, A.K., Rout, S.: A distributed algorithm for European options with nonlinear volatility. Comput. Math. Appl. 49, 885–894 (2005)
Le Roux, A.-Y., Le Roux, M.-N.: Numerical solution of a Cauchy problem for nonlinear reaction process. J. Comput. Appl. Math. 214, 90–110 (2008)
Le Roux, M.-N.: Numerical solution of a parabolic problem arising in finance. Appl. Numer. Math. 62, 815–832 (2012)
Songzhe, L.: Existence of solution to initial value problem for a parabolic Monge-Ampère equation and application. Nonlinear Anal. 65, 59–78 (2006)
Wilmott, P., Howison, S., Dewynne, J.: The Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press, Cambridge (1995)
Yong, J.: Introduction to mathematical finance. In: Yong, J., Cont, R. (eds.) Mathematical Finance-Theory and Applications. High Education Press, Beijing (2000)
Acknowledgements
This research was supported by the European Union under Grant Agreement number 304617 (FP7 Marie Curie Action Project Multi-ITN STRIKE - Novel Methods in Computational Finance) and the Bulgarian National Fund of Science under Project DID 02/37-2009.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Koleva, M.N., Vulkov, L.G. (2014). A Splitting Numerical Scheme for Non-linear Models of Mathematical Finance. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2013. Lecture Notes in Computer Science(), vol 8353. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43880-0_69
Download citation
DOI: https://doi.org/10.1007/978-3-662-43880-0_69
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-43879-4
Online ISBN: 978-3-662-43880-0
eBook Packages: Computer ScienceComputer Science (R0)