The Black–Scholes equation has been used for modeling option pricing extensively.
When the volatility of financial markets creates irregularities, the model equation is
difficult to solve numerically; for this reason nonuniform grids are often
used for greater accuracy. This paper studies the numerical consistency of
popular explicit, implicit and leapfrog finite difference schemes for solving the
Black–Scholes equation when nonuniform meshes are utilized. Mathematical tools
including Taylor expansions are used throughout our analysis. The consistency
ensures the basic reliability of the finite difference schemes based on choices of
temporal and variable spatial derivative approximations. Truncation error
terms are derived and discussed, and numerical experiments using C, C++
and Matlab are given to illustrate our discussions. We show that, though
orders of accuracy are lower compared with their peers on uniform grids,
nonuniform algorithms are easy to implement and use for turbulent financial
markets.
