|
|||||
|
|
| Analytic and Experimental Studies of the Errors in Numerical Methods for the Valuation of Options | |
| 作者单位： | Department of Mathematics National University of Singapore，Department of Mathematics Trinity College Dublin，Institute for Mathematics and Mechanics Russian Academy of Science Ekaterinburg，Singapore.，Ireland.，Russia. |
| 基金项目： | Russian Foundation for Basic Research，Singapore Academic Research Funds，Boole Centre for Research in Informatics at the National University of Ireland;Cork，Mathematics Applications Consortium for Science and Industry in Ireland (MACSI)，Science Foundation Ireland Mathematics Initiative |
| 摘 要： | The value of a European option satisfies the Black-Scholes equation with appropriately specified final and boundary conditions.We transform the problem to an initial boundary value problem in dimensionless form.There are two parameters in the coefficients of the resulting linear parabolic partial differential equation.For a range of values of these parameters,the solution of the problem has a boundary or an initial layer.The initial function has a discontinuity in the first-order derivative,which leads to the appearance of an interior layer.We construct analytically the asymptotic solution of the equation in a finite domain.Based on the asymptotic solution we can determine the size of the artificial boundary such that the required solution in a finite domain in x and at the final time is not affected by the boundary.Also,we study computationally the behaviour in the maximum norm of the errors in numerical solutions in cases such that one of the parameters varies from finite (or pretty large) to small values,while the other parameter is fixed and takes either finite (or pretty large) or small values. Crank-Nicolson explicit and implicit schemes using centered or upwind approximations to the derivative are studied.We present numerical computations,which determine experimentally the parameter-uniform rates of convergence.We note that this rate is rather weak,due probably to mixed sources of error such as initial and boundary layers and the discontinuity in the derivative of the solution. |
| 关 键 词： | 选件 奇点 有限微分法 单调鲁棒性 |
Analytic and Experimental Studies of the Errors in Numerical Methods for the Valuation of Options |
|
| Authors: | PLin J J H Miller G I Shishkin |
| Institution: | 1. Department of Mathematics, National University of Singapore, Singapore 2. Department of Mathematics, Trinity College Dublin, Ireland 3. Institute for Mathematics and Mechanics, Russian Academy of Science, Ekaterinburg, Russia |
| Abstract: | The value of a European option satisfies the Black-Scholes equation with appropriately specified final and boundary conditions. We transform the problem to an initial boundary value problem in dimensionless form. There are two parameters in the coefficients of the resulting linear parabolic partial differential equation. For a range of values of these parameters, the solution of the problem has a boundary or an initial layer. The initial function has a discontinuity in the first-order derivative, which leads to the appearance of an interior layer. We construct analytically the asymptotic solution of the equation in a finite domain. Based on the asymptotic solution we can determine the size of the artificial boundary such that the required solution in a finite domain in x and at the final time is not affected by the boundary. Also, we study computationally the behaviour in the maximum norm of the errors in numerical solutions in cases such that one of the parameters varies from finite (or pretty large) to small values, while the other parameter is fixed and takes either finite (or pretty large) or small values. Crank-Nicolson explicit and implicit schemes using centered or upwind approximations to the derivative are studied. We present numerical computations, which determine experimentally the parameter-uniform rates of convergence. We note that this rate is rather weak, due probably to mixed sources of error such as initial and boundary layers and the discontinuity in the derivative of the solution. |
| Keywords: | Options singularities finite difference methods monotone robust method |
| 本文献已被 CNKI 维普 万方数据 等数据库收录！ | |