Research On Asymptotics And Computations Of American Maximum Option

This paper is aimed at investigating the problem of pricing American maximum options.Both analytical methods and numerical methods are used to price American maximum options.Options were born in the tulip trading in Netherands in the 17th century.Over-the-counter options trading began in American and European markets in the late 18th century.In 1973,the Chicago Board Options Exchange（COBE）was established and the standardized options exchange began.In 2015,the Shanghai 50ETF option was listed on the Shanghai Stock Ex-change,becoming the first exchange traded option in China.During the rapid development of financial derivatives market,option pricing has always been a classic problem in the filed of financial mathematics.Ameican economists Black and Scholes^[1]gave the European option pricing model in 1973.Merton^[2]improved the model for the situation of dividend paying.Since then,many economists,financial theorists,and mathematicians have made in-depth re-search on option pricing based on their work.Now,the research on this problem covers Euro-pean,American,vanilla,and exotic options,and adopts a variety of analytical and numerical methods.Ameican maximum option is a kind of option based on one risk asset and a minimum fixed income.The option gives the right to take the maximum value between them.Different from simple options with one optimal exercise boundary,this kind of option has two optimal exercise boundaries.The particularity of American maximum option makes it difficult to price it.American maximum option is closely related to the real financial market.Its payment func-tion is quite similar to convertible bonds and break-even investment.The research on pricing Ameican maximum option is of great significance for the theoretical research of pricing com-plex option on multi assets and the practical application of determining the option price in the financial market.Firstly,according to the B-S model and option payment function,this paper gives a pricing model of Ameican maximum option.Then,the analytical method is used.The expression of the option value is obtained by solving the partial differential equation through Green’s theory.It（?） calculus^[3,4]is another way which is used to obtain the expression.The inte-gral equation containing the optimal exercise boundary is constructed by using the smoothness matching condition.Under the conditions of continuous dividend paying and closing to the expiration date,the integral equation is solved by an approximate method to obtain the asymp-totic forms of the two boundaries with respect to time-to-maturity.Furthermore,the matched asymptotic expansion for the rescaled value function of American maximum option is derived.Next,the numerical methods are used.Finite difference method and least-squares Monte Carlo simulation method are given to price American maximum option.Both methods are used in a numerical example to get the graphs of the option price and the optimal exercise boundaries.By comparing the results from two methods,the validity of finite difference method is verified.