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Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 2007, Volume 47, Number 3, Pages 460–480
(Mi zvmmf318)
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This article is cited in 12 scientific papers (total in 12 papers)
Approximation of the solution and its derivative for the singularly perturbed Black–Scholes equation with nonsmooth initial data
S. Lia, G. I. Shishkinb, L. P. Shishkinab a Department of Computational Science, National University of Singapore, Singapore, 117543
b Institute of Mathematics and Mechanics, Ural Division, Russian Academy of Sciences, Yekaterinburg, 620219, Russia
Abstract:
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.
Key words:
Black–Scholes equation, singularly perturbed parabolic equation, nonsmooth initial data, interior layer, difference scheme, additive splitting of singularities, convergence.
Received: 10.07.2006
Citation:
S. Li, G. I. Shishkin, L. P. Shishkina, “Approximation of the solution and its derivative for the singularly perturbed Black–Scholes equation with nonsmooth initial data”, Zh. Vychisl. Mat. Mat. Fiz., 47:3 (2007), 460–480; Comput. Math. Math. Phys., 47:3 (2007), 442–462
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https://www.mathnet.ru/eng/zvmmf318 https://www.mathnet.ru/eng/zvmmf/v47/i3/p460
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