Main Article Content
Differential equations play significant role in the world of finance since most problems in these areas are modeled by differential equations. Majority of these problems are sometimes nonlinear and are normally solved by the use of numerical methods. This work takes a critical look at Nonlinear Black-Scholes model with special reference to the model by Guy Barles and Halil Mete Soner. The resulting model is a nonlinear Black-Scholes equation in which the variable volatility is a function of the second derivative of the option price. The nonlinear equation is solved by a special class of numerical technique, called, the meshfree approximation using radial basis function. The numerical results are presented in diagrams and tables.
During, B. Black-Scholes type equations: Mathematical analysis, parameter identification & numerical solution. Unpublished doctoral dissertation, am Fachbereich Physik, Mathematik und Informatik der Johannes Gutenberg-Universitat Mainz; 2005.
Kansa E. Multiquadrics- A scattered data approximation scheme with applications to computational fluid-dynamics. Computers & Mathematics with Applications. 1990;19:127-145.
Sharan M, Kansa E, Gupta S. Application of the multiquadric method for numerical solution of elliptic partial differential equations. Applied Mathematics and Computation. 1997;84:275-302.
Goto Y, Fei Z, Kan S, Kita E. Option valuation by using radial basis function. Engineering Analysis with Boundary Elements. 2007;31:836-843.
Milovanovic S, Shcherbakov V. Pricing Derivatives under multiple stochastic factors by localized radial basis function methods, Uppsala University, Sweden; 2018.
Ankudinova J. The numerical solution of nonlinear Black-Scholes equations. Unpublished doctoral dissertation, Technische Universitat Berlin; 2008.
Leland HE. Option pricing and replication with transaction costs. The Journal of Finance. 1985;40: 1283-1301.
Hodges S, Neuberger A. Optimal replication of contingent claims under transaction costs. The Review of Futures Markets. 1989; 8, 222-239.
Boyle, P. P., & Vorst, T. Option replication in discrete time with transaction costs. The Journal of Finance. 1992;47(1):271-293.
Barles G, Soner HM. Option pricing with transaction costs and a nonlinear Black-Scholes equation. Finance and Stochastics. 1998;2:369-397.
Fasshauer GE. Meshfree methods. Department of Applied Mathematics, Illinois Institute of Technology IL 60616, USA; 2006.
Duffy DJ. Finite difference methods in financial engineering a partial differential equation approach. John Wiley & Sons Ltd; 2006.
Guarin A, Liu X, Ng WL. Valuation of american options with meshfree methods. (Centre for Computational Finance and Economic Agents, Working Paper Series; 2012.
Gonzalez-Gaxiola O, Gonzalez-Perez PP. Nonlinear Black-Scholes equation through radial basis function. Journal of Applied Mathematics and Bioinformatics. 2014;4(3):75-86.
Belova A, Shmidt T. Meshfree methods in option pricing. Unpublished master’s thesis, School of Information Science, Computer and Electrical Engineering Halmstad University, School of Information Science, Computer and Electrical Engineering Halmstad University; 2011. (In press).
Mawah B. Option pricing with transaction costs and a non-linear Black-Scholes equation. Unpublished Master’s Thesis, Department of Mathematics, Uppsala University; 2007.