چکیده
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In this paper, we evaluate and discuss different numerical methods to solve the Black–
Scholes equation, including the q-method, the mixed method, the Richardson method, the Du Fort and
Frankel method, and the MADE (modified alternating directional explicit) method. These methods
produce numerical drawbacks such as spurious oscillations and negative values in the solution when
the volatility is much smaller than the interest rate. The MADE method sacrifices accuracy to obtain
stability for the numerical solution of the Black–Scholes equation. In the present work, we improve
the MADE scheme by using non-standard finite difference discretization techniques in which we use
a non-local approximation for the reaction term (we call it the MMADE method). We will discuss the
sufficient conditions to be positive of the new scheme. Also, we show that the proposed method is
free of spurious oscillations even in the presence of discontinuous initial conditions. To demonstrate
how efficient the new scheme is, some numerical experiments are performed at the end.
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