Approximation of the solution and its derivative for the singularly perturbed Black–Scholes equation with nonsmooth initial data

S. Li ; G. I. Shishkin ; L. P. Shishkina

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 3, pp. 460-480

Résumé

A problem for the Black–Scholes equation that arises in financial mathematics is reduced, by a transformation of variables, to the Cauchy problem for a singularly perturbed parabolic equation with the variables $x$, $t$ and a perturbation parameter $\varepsilon$, $\varepsilon\in(0,1]$. This problem has several singularities such as the unbounded domain, the piecewise smooth initial function (its first-order derivative in $x$ has a discontinuity of the first kind at the point $x=0$), an interior (moving in time) layer generated by the piecewise smooth initial function for small values of the parameter $\varepsilon$, etc. In this paper, a grid approximation of the solution and its first-order derivative is studied in a finite domain including the interior layer. On a uniform mesh, using the method of additive splitting of a singularity of the interior layer type, a special difference scheme is constructed that allows us to $\varepsilon$-uniformly approximate both the solution to the boundary value problem and its first-order derivative in $x$ with convergence orders close to 1 and 0.5, respectively. The efficiency of the constructed scheme is illustrated by numerical experiments.