Application On Options And Futures For Backward Stochastic Differential Equations And Monte-Carlo Methods

In recent years, more and more people are concerned about the derivatives market, for the improvement of the financial markets and the sharp fluctuations in the spot market. There are two kinds of basic research in derivatives market, they are is pricing and hedging. In order to satisfy the different requirements for the customers and to avoid the marker risk, there are many non-standardized derivatives in Over-the-Counter market in addition to standard options in exchange market. Many financial issues concern to the pricing of these exotic derivatives, but the calculation is difficult because many of those products are path-dependent, therefore the pricing method is more complex than the traditional European options.This paper studies two different pricing methods for standard options and exotic options. One approach is Backward Stochastic Differential Equations methods, the other one is the Monte-Carlo method. It calculates Standard European options and two kinds of exotic options with both of the methods, and then compares the results for these two methods. The reason it calculates the traditional European options is that most of exotic options are the innovation from the tradition ones, therefore the exotic options and the standard ones have a great relationship. The main content is as follows:Chapter 1 is the introduction of this article. It reviews the conception of the option briefly and several methods of option pricing at present. Furthermore, is summarizes some history documents which is related to option pricing. In the end, the main contents and the idea of this article is given.Chapter 2 is the introduction for basic mathematical models. It describes in detail the no-arbitrage pricing theory which is the most commonly used in models. Both of the two method discussed are base on no-arbitrage theory. It introduces some exotic options as examples used in the article later.Chapter 3 provides a higher accuracy numerical method of backward stochastic differential equations, the numerical method is based on the fine nature for relationship for the backward stochastic differential equations and partial differential equations. The algorithm also takes the character of Markov for the solution of backward stochastic differential equations. It gives a few examples to confirm the high-precision for this kind of algorithm.Chapter 4 introduces Monte-Carlo method and the principle of this method in calculating standard European options, Asian options and binary options. It provides the difference between the backward stochastic differential equations and Monte-Carlo methods. The backward stochastic differential equations provides the higher accuracy and it takes much shorter time than Monte-Carlo method. But Monte-Carlo method is more simple and convenient. Compared with backward stochastic differential equations, Monte-Carlo method is more flexible, it can not only solve the pricing problem, also can borrow the idea of pricing to calculate risk-free arbitrage problem, which is an innovation in this article.Chapter 5 elaborates exotic options is playing an very important role in financial market, and summarizes this article.