ЖУРНАЛ ВЫЧИСЛИТЕЛЬНОЙ МАТЕМАТИКИ И МАТЕМАТИЧЕСКОЙ ФИЗИКИ, 2007, том 47, < 3, с. 460-480
УДК 519.63
APPROXIMATION OF THE SOLUTION AND ITS DERIVATIVE FOR THE SINGULARLY PERTURBED BLACK-SCHOLES EQUATION
WITH NONSMOOTH INITIAL DATA1)
© 2007 r. S. Li*, G. I. Shishkin**, L. P. Shishkina**
(* Department of Computational Science, National University of Singapore, Singapore 117543;
** Institute of Mathematics and Mechanics, Ural Division, Russian Academy of Sciences,
Yekaterinburg 62019, Russia) e-mail: shishkin@imm.uran.ru Received July 10, 2006
Аппроксимация решения и производной для сингулярно возмущенного уравнения Блэка-Шоулза с негладкими начальными данными. Ш. Ли, Г.И. Шишкин, Л.П. Шишкина. Задача для уравнения Блэка-Шоулза, возникающая в финансовой математике, преобразованием переменных приводится к задаче Коши для сингулярно возмущенного параболического уравнения в переменных x, t с возмущающим параметром е, ее (0, 1]. Эта задача имеет такие особенности, как бесконечная область, ограниченная гладкость начальной функции (ее производная первого порядка по x терпит разрыв I рода в точке x = 0), переходный слой (движущийся во времени), порождаемый кусочно-гладкой начальной функцией при малых значениях параметра е, и др. Рассматривается сеточная аппроксимация решения задачи и его первой производной на конечной области, содержащей переходный слой. На равномерной сетке с использованием метода аддитивного выделения особенности типа переходного слоя строится специальная разностная схема, аппроксимирующая е-равномерно решение задачи и его первую производную по x с порядками скорости сходимости, близкими к 1 и 0.5 соответственно. Эффективность построенной схемы иллюстрируется численными экспериментами. Библ. 6. Фиг. 1. Табл. 7.
Keywords: Black-Scholes equation, singularly perturbed parabolic equation, nonsmooth initial data, interior layer, difference scheme, additive splitting of singularities, convergence.
1. INTRODUCTION
1.1. Mathematical modeling in financial mathematics leads to the Black-Scholes equation (that is backward parabolic) [1] with respect to the value C = C(S, t'), which is a European call option, where S and t are the current values of the underlying asset and time,
^r + 1 a2S2^ + rS^ - rC = 0, (S, t) е x [0, T), (1.1a)
dt 2 дд S dS
with the final condition
C(S, T) = max (S - E, 0), S е (1.1b)
and the boundary conditions at S = 0 and at infinity S =
C(0, t) = 0; C(S, t) — S for S —- t е [0, T). (1.1c)
Here a, E, T and r are some financial parameters (the volatility, exercise price, expiry time and the interest rate, respectively).
For the problem (1.1), in addition to the solution itself, some of the partial derivatives of the solution are of interest (see [1, Ch. 3]).
When studying this problem, a standard approach is a transformation of the equation by the changes of variables.
Работа выполнена при финансовой поддержке Академического исследовательского фонда NUS (грант < R-151-000-025-112), РФФИ (коды проектов < 04-01-00578, 04-01-89007-HB0_a) и частично Нидерландской организации научных исследований NWO (проект < 047.016.008); и Булевского центра исследования по информатике Национального ун-та Ирландии в Корке.
APPROXIMATION OF THE SOLUTION AND ITS DERIVATIVE By the transformations
S = Eexp (x), t = T - Tr_1, C = Ev(x, t)
(1.1d)
and introducing the notation k = 2a 2r, t* = rT, we come to the following problem for the dimensionless parabolic equation in the new variables x, t:
Z v ( + (* -1 ) __ * - * _ |v ( ^ = 0,
(x, т) e U x(0, т* ]
with the initial condition
where
xe
v(x, 0) = 9V(x), 9v(x) = max( exp (x) - 1, 0), x e
(1.2)
(1.3a)
and with the condition at infinity
v(x, t) —► 0 for x —► — v(x, t)—► exp(x) for x
те (0,t*].
(1.3b)
Under the condition T, r = 0(1) and for a taking an arbitrary value from the half-open interval (0, J2r), we come to the Cauchy problem for the singularly perturbed parabolic equation
Zv ( x'T)sfë +( e)dx-1-дтг( x'T) = °'
(x, т) e U x(0, т* ]
(1.4)
with conditions (1.3). Here £ = 2 1a2r_1 is a dimensionless "perturbation" parameter, £ e (0, 1].
The initial function in condition (1.3a) is continuous; its first derivative in x has a discontinuity of the first kind at the point x = 0
dx ^ ( 0 )
= 1,
where the jump of the derivative is defined by the relation
( 0 )
= ""K'Pv( x ) ( x
x\i 0
The initial function and the solution itself for this problem grow (exponentially) without bound as x —► If the parameter £ = 1 then the problem (1.4), (1.3) becomes the one of reaction-diffusion type, and for £ < 1, it is of convection-diffusion type. For small values of the parameter £, an interior (moving in time) layer with the
typical width of £1/2 appears in a neighbourhood of the characteristic (of the operator L1 = (1 - £) d— 1 - dT)
passing through the point (0, 0).
Thus, the Cauchy problem (1.4), (1.3) is a singularly perturbed problem with different types of singularities. In the present paper we are interested in approximations to both the solution and its first order derivative in a finite subdomain that contains the singularity of the interior layer type.
1.2. Boundary value problems in bounded domains for singularly perturbed parabolic reaction-diffusion equations with a discontinuous initial condition have been considered in [2]-[7]. To construct schemes that converge £-uniformly, the method of condensing meshes (in a neighbourhood of boundary layers), and also either the fitted operator method [2]-[5] or the method of additive splitting of a singularity [6], [7] (in a neighbourhood of the points at which the initial function is discontinuous) were applied.
In [2]-[6], approximations to the normalized derivatives £0/3x)u(x, t), i.e., the first order spatial derivative multiplied by the parameter £, were considered. For this purpose, the method of additive splitting of
оо
the singularity generated by the discontinuity of the initial function was used; however, the approximation of the derivative (9/9x)u(x, t) itself was not considered.
A boundary value problem on a segment for singularly perturbed parabolic convection-diffusion equations with a piecewise smooth initial condition has been considered in [8], [9]. In [9], by using the method of special meshes that condense in a neighbourhood of the boundary layer and the method of the additive splitting of a singularity of the interior layer type, special difference schemes are constructed that make it possible to approximate е-uniformly the solution of the problem on the entire set under consideration, the normalized derivative on the entire set except for the discontinuity point (0, 0), and the first spatial derivative on the same set but outside a small neighbourhood of the boundary layer.
In the present paper, instead of the Cauchy problem (1.4), (1.3), we consider a singularly perturbed boundary value problem for equation (1.4) with a non-smooth initial condition similar to (1.3), namely, the problem (2.2), (2.1) (see the formulation of this problem in Section 2). The technique from [9] is used for studying the problem (2.2), (2.1). Note that in a problem of the type (2.2), (2.1) considered in a finite domain, except for the interior layer, an additional singularity appears, namely, a boundary layer with the typical width of е. The singularity of the boundary layer is stronger than that of the interior layer, which makes it difficult to construct special numerical methods suitable for the adequate description of the singularity of the interior layer type. In contrast to [9], here conditions are defined that allow us to investigate each singularity of the problem separately. For the boundary value problem (2.2), (2.1), we construct a finite difference scheme that approximates the solution and its first order derivative in x. To construct е-uniform approximations for the solution and its first derivative in a finite subdomain including only the interior-layer singularity, it suffices to use a uniform mesh and the method of the additive splitting of the singularity of the interior layer type. The efficiency of the scheme constructed in this paper is verified with numerical experiments.
The numerical method constructed for problem (2.2), (2.1), after the transformation to the original variables S, t and the function C (see the change (1.1d)), allows us to approximate the solution of problem (1.1) and its first derivative (9/9S)C(S, t') in a finite neighbourhood of the point (E, T) (the point of discontinuity of the derivative in condition (1.1b)), including the interior layer (appearing for small values of the dimen-sionless quantity aV1). Errors in the approximation of the solution and derivative (for (S, t') ^ (E, T)) are independent of the value of a2r~x; these errors (in the maximum norm) are defined only by the number of nodes in the mesh used for the numerical solution of the discrete problem.
About Contents. Formulation of the boundary value problem is given in Section 2. Difficulties involved in the approximation of the solution and derivatives on the basis of classical finite difference schemes are discussed in Section 3. A priori estimates used in the constructions are presented in Section 4. Classical difference approximations of the problem on uniform and piecewise uniform meshes are considered in Section 5. A difference scheme (approximating the solution and its first order derivative), which is constructed using the method of the additive splitting of the singular component of the solution generated by the discontinuity of the derivative of the initial function, is given in Section 6. In the same place, conditions are defined under which a certain singularity of the solution can be split off and investigated separately. Numerical experiments are analyzed in Section 7.
2. PROBLEM FORMULATION. AIM OF RESEARCH
2.1. On the set G with the boundary S,
G = G u S, G = D x (0, T], D = {x : x е(-d, d)}, (2.1)
we consider the Dirichlet problem for the singularly
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