Estimating the truncation error in the case of solving one dimensional Black-Scholes equation

Abstract

In the early 1970s, Fischer Black and Myron Scholes made a breakthrough by deriving a differential equation that must be satisfied by the price of any derivative security dependent on a non-dividend-paying stock. They used the equation to obtain the values for European call and put options on the stock. Options are now traded on many different exchanges throughout the world and are very popular instruments for both speculating and risk management. There are several approaches to option pricing but however we only consider Partial Differential Equations(PDE) approach, where options are expressed as solutions to certain partial differential equations. These equations are specified over an infinite(unbounded) region and usually cannot be solved exactly. Most numerical methods for solving partial differential equations require the region to be finite, so before applying numerical methods the problem is changed from infinite to finite region. The aim of our thesis is to study the error caused by this change, will do that by estimating the error at the boundaries and use these estimates to get pointwise error inside the domain, followed by numerical verification. The structure of the thesis is as follows: Chapter one provides a brief introduction of option pricing and includes neccesary results. In chapter two we give a defination of maximum principle for backward parabolic equations and prove some lemmas based on this principle which will be useful throughout this thesis. We further outline ways of getting estimates with the aid of the results we got in our lemmas. In chapter three we will obtain estimates at the truncation boundaries for both call and put option. In chapter four we use the estimates of the previous chapter to find the estimates inside the region. In chapter five we demonstrate the process of using our estimates in the case of pricing concrete put and call options and show the validity of the estimates by finding numerically the values of the solution of this truncated problem.