Numerical solution of differential and integral equations
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}
Note. Abstracts are published in author's edition
Mail to Webmaster |
|Home Page| |English Part| |
Go to Home |