Numerical Solution For Pricing Options And Its Asymptotic Limit

In financial Mathematic and Financial Engineering, the theory of options pricing is the core of our study fields. As the financial instruments and financial derivatives created, the theory of options pricing is of its heart importance. The theory of options pricing of Black-Scholes has been treated as classical financial analytic technique. Under the base of the theory, except the European vanilla options and American vanilla options, some kinds of exotic options are considered, Black-Scholes equation gains much extension.However, since the financial instruments continuously created, closed form solutions of Black-Scholes equation and its extension equations are impossibly derived on many occasions, especially some exotic options, for example, barrier options.In this article, option valuation is resorted to numerical procedures. We make use of binomial model and its extension model to estimate barrier options valuation. Main contents of this thesis are listed as followed:The first chapter introduces general characteristic of financial derivatives, fundamental concepts and some basic assumptions in financial option theory. Then a kind of exotic option__barrier option concept is presented.In chapter 2, the basic ideas of binomial pricing model are concerned, we discuss binomial pricing model of discrete up-and-out calls and show the formula, it extends binomial pricing model of European option. It is then followed by the discussion of the structure and thought of trinomial pricing model, the formula of barrier option is derived. At last, we provide mixed trinomial pricing model.The asympostic limits of the binomial formula are considered in chapter 3.And we proved the equation of option value of binomial and trinomial scheme satisfying is one order approximation.