Numerical solution of differential and integral equations
documentstyle[12pt]{article} itle{On numerical methods for a singularly perturbed Black-Scholes equation with nonsmooth initial data} author{Li Shuiying, Dennis B. Creamer, Grigory I. Shishkin} date{} egin{document} maketitle
We consider an initial-value problem for the Black-Scholes equation in dimensionless form, with the time variable changing on the unit interval. For European call options, the initial condition is $v(x,0)=max (exp(x) -1,0)$, $x in R$, i.e. the first derivative of the initial condition is discontinuous at $x=0$. When $T$, $ au=O(1)$ and $sigma in (0,1]$, where $T$, $ au$ and $sigma$ are the expiry time, interest rate and volatility, respectively, the Black-Scholes equation is a singularly perturbed convection-diffusion equation with the perturbation parameter $varepsilon=sigma^2$ (multiplying the second order space derivative). Additionally, the solution of this initial-value problem grows exponentially as $x o infty$. Thus, we come to a singularly perturbed problem which has other types of singularities. Note that this problem has an analytical solution when the coefficients in the differential equation are constant.
Here, we are focused on an approximation to the solution of the singularly perturbed problem with a nonsmooth initial condition, ignoring other types of singularities. We consider the Dirichlet problem for the singularly perturbed Black-Scholes equation on the bounded domain $overline{G}={(x,t): |x|<1, t in [0,1]}$ with the initial condition having controlled restricted smoothness; the initial function $varphi_0(x)$, $x in [-1,1]$ belongs to a Helder space $H^{alpha}$ with $alpha in (0,2]$. As a rule, a solution of this problem has an additional singularity that is a regular boundary layer. We use the classical finite difference scheme on uniform meshes to solve this problem. We show that the computed solution does not converge on the whole domain, but this solution does converge $varepsilon$-uniformly in a finite neighbourhood of the point $(0,0)$. The order of convergence depends on the smoothness of the initial data. Numerical experiments illustrate these effects.
Thus, the use of condensing meshes is not necessary for $varepsilon$-uniform convergence of the classical scheme in a neighbourhood of the set where initial conditions are not sufficiently smooth. However, the rate of such convergence essentially depends on the value $alpha$, which defines the class $H^{alpha}$.
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
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