# Международная конференция по вычислительной математике МКВМ-2004

21-25 июня 2004 г. Академгородок, Новосибирск, Россия

## Тезисы докладов

Численное решение дифференциальных и интегральных уравнений

## Robust numerical method for a singularly perturbed Black-Scholes equation in a bounded subdomain

### Creamer D.B.

Department of Computational Science,
National University of Singapore (Singapore)

documentstyle[12pt]{article} itle{Robust numerical method for a singularly perturbed Black-Scholes equation in a bounded subdomain} author{Dennis B. Creamer, John J.H. Miller, Grigory I. Shishkin} date{} egin{document} maketitle

We consider the Black-Scholes equation written in the dimensionless form for the case when the time variable \$t\$ changes on the unit interval. Let \$T\$, \$ au=O(1)\$ and \$sigma in (0,1]\$, where \$T\$, \$ au\$ and \$sigma\$ are the expiry time, interest rate and volatility, respectively. Under this condition the Black-Scholes equation is a singularly perturbed one with a perturbation parameter \$varepsilon\$ multiplying the second-order space derivative, where \$varepsilon=sigma^2\$. For European call option, the initial condition is \$v(x,0)=max(exp(x)-1, 0)\$, \$x in R\$. Thus, such a singularly perturbed initial-value problem is defined in a unbounded domain \$overline{G}=R imes [0,1]\$, the first derivative of the initial condition is discontinuous at \$x=0\$, and the solution of this problem grows (exponentially) without bounds as \$x o infty\$.

Suppose that we are interested to find a solution of such a problem but in a bounded subdomain \$overline{G}^{l}=[-l,l] imes [0,1]\$. To have an approximate solution on \$overline{G}^{l}\$, we consider the equation in the larger domain \$overline{G}^{L}=[-L,L] imes [0,1]\$, \$L > l\$ with the exact initial condition on \$[-L,L]\$, and with a some boundary condition that is an ``extension'' of the initial condition. To solve such a subproblem we use the classical finite difference scheme on the uniform mesh \$overline{G}_h^{L}\$ with \$N+1\$ and \$N_0+1 \$ nodes in the intervals \$[-L, L]\$ and \$[0,1]\$, respectively. Let \$u_h^L(x,t)\$, \$(x,t)in overline{G}_h^{L}\$ be a discrete solution of this problem. By \$u_h(x,t)\$, \$(x,t)in overline{G}_h\$, we denote a solution of the above finite difference scheme on an uniform mesh \$overline{G}_h\$ (introduced in \$overline{G}\$) with the same stepsizes in \$x\$ and \$t\$ as in the mesh \$overline{G}_h^{L}\$. We give the condition under which the computed solution \$u_h^L(x,t)\$, \$(x,t)in overline{G}_h^{L}\$ converges to the solution \$u_h(x,t)\$, \$(x,t)in overline{G}_h\$ on the set \$overline{G}_h^{l}\$ \$varepsilon\$-uniformly as \$N, N_0 o infty\$. Such approach allows us to find a solution of the initial-value problem to the Black-Scholes equation in a chosen bounded subdomain \$overline{G}^{l}\$, by solving a discrete problem on the mesh \$overline{G}_h^{L}\$ with the finite number of grid nodes.

We discuss numerical experiments that confirm the derived technique.

Acknowledgements. This research was supported in part by the NUS ARF grant R-151-000-025-112, by the EERSS Programme and by the Russian Foundation for Basic Research under grant 04-01-00578. end{document}

Примечание. Тезисы докладов публикуются в авторской редакции

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