A high-order compact method for nonlinear Black–Scholes option pricing equations of American options

Abstract

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.

Keywords:

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

2010 AMS Subject Classifications :

• 65M06
• 65M12
• 65N06

Acknowledgements

The authors thank Ljudmila Bordag of Halmstad University for her hospitality. The authors are grateful to the anonymous referees for their fruitful comments that improved the content of this manuscript.