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

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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 ɛ, ɛ ∈ (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 ɛ, 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 ɛ-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.