Finite volume schemes for solving nonlinear partial differential equations in financial mathematics. (English) Zbl 1246.91150

Kútik, Pavol; Mikula, Karol

Fořt, Jaroslav (ed.) et al., Finite volumes for complex applications VI: Problems and perspectives. FVCA 6, international symposium, Prague, Czech Republich, June 6–10, 2011. Vol. 1 and 2. Berlin: Springer (ISBN 978-3-642-20670-2/hbk; 978-3-642-20671-9/ebook). Springer Proceedings in Mathematics 4, 643-651 (2011).

Summary: In order to estimate a fair value of financial derivatives, various generalizations of the classical linear Black-Scholes parabolic equation have been made by adjusting the constant volatility to be a function of the option price itself. We present a second order numerical scheme, based on the finite volume method discretization, for solving the so-called Gamma equation of the Risk Adjusted Pricing Methodology (RAPM) model. Our new approach is based on combination of the fully implicit and explicit schemes where we solve the system of nonlinear equations by iterative application of the semi-implicit approach. Presented numerical experiments show its second order accuracy for the RAPM model as well as for the test with exact Barenblatt solution of the porous-medium equation which has a similar character as the Gamma equation.
For the entire collection see [Zbl 1220.76004].

MSC:

91G60	Numerical methods (including Monte Carlo methods)
65M08	Finite volume methods for initial value and initial-boundary value problems involving PDEs
35Q91	PDEs in connection with game theory, economics, social and behavioral sciences
35K20	Initial-boundary value problems for second-order parabolic equations
35K55	Nonlinear parabolic equations

Keywords:

finite volume method; second-order scheme; financial mathematics