A high-order compact method for nonlinear Black-Scholes option pricing equations of American options. (English) Zbl 1237.91228

Dremkova, E.; Ehrhardt, M.

Int. J. Comput. Math. 88, No. 13, 2782-2797 (2011).

Summary: Due to transaction costs, illiquid markets, large investors or risks from an unprotected portfolio the assumptions in the classical Black-Scholes model become unrealistic and the model results in nonlinear, possibly degenerate, parabolic diffusion-convection equations. Since in general, a closed-form solution to the nonlinear Black-Scholes equation for American options does not exist (even in the linear case), these problems have to be solved numerically. We present from the literature different compact finite difference schemes to solve nonlinear Black-Scholes equations for American options with a nonlinear volatility function. As compact schemes cannot be directly applied to American type options, we use a fixed domain transformation proposed by Ševčovič and show how the accuracy of the method can be increased to order four in space and time.

MSC:

91G60	Numerical methods (including Monte Carlo methods)
91G20	Derivative securities (option pricing, hedging, etc.)
65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65N06	Finite difference methods for boundary value problems involving PDEs

Keywords:

nonlinear Black-Scholes equation; compact finite difference scheme; American options; high-order methods; fixed domain transformation; transaction costs

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