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# Application of numerical computation of American options

Author: WuQiang
Tutor: ZhangJiZhou
School: Shanghai Normal University
Course: Applied Mathematics
Keywords: American options Self-financing continuation region no arbitrary principle Ito lemma compound option slip method SOR method artificial boundary method maximum princi-ple early exercise boundary optimal stop problem Arbitrage Richardson extrapolation Portfolio variational inequality penalty problem
CLC: F224
Type: Master's thesis
Year: 2006
Quote: 2

### Abstract

 American option is so difficult to us because it can implement prematurely.but it is so close to our life that plays a very important role and can be applied in very wide range,from the American stock option which we realize option initially,to all kinds of foreign exchange option,weather option currently,and now investment choose in use of American option,furthermore,the bubble price and corporate bonds price can be worked out according to the American option principle .at the same time we can investigate corporate default probability and credit risk .nowadays the research range has go beyond its traditional category, we emphasize on how to computer the American option and find the free boundary by the numerical methods.in the last two chapters,mainly clarify the wide application of American option in the mortgage bond and insurance .besides price the finance derivation base on the American option.In the first chapter,we introduce the basic option character ,at the same time introduce the random walk model .from this we can deduce the relationship between the probability and partial differential equation,also we can deduce the Ito lemma.base upon these we get the mathematical model of option price.In the second chapter,give the American option property and establish American option’s differential equation which exists free boundary .further more we deduce the analytical price formula according to the probability method,(of course some unknown variable among the formula ),next ,we utilize the numerical method to put forward two feasible means.In the third chapter,firstly introduce what the Successive Over Relaxation meanings .secondly introduce the particularly operating steps.furthermore under this method we get the programme .lastly illustrate it by an example .In the forth chapter, we introduce the artificial boundary method ,then utilize the maximum principle to analyze the Black-Scholes equation’s character.under these conclusion we find the free boundary location by the artificial boundary .lastly we reach the upon triangle matrix also we illustrate the numerical example .In the fifth chapter, we establish the mortgage loan model according to the American op-tion, and deviate the partial different equation by no arbitrary principle .lastly we give the computer method thereby characteristic line method and spilt method separately.lastly we derive the according means.In the sixth chapter,we establish the investment couple insurance guarantee slip model by the same principle which utilized by American option ,then transforms to the variational inequality.we

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