Numerical Methods For Backward Stochastic Differential Equations, Nonlinear Expectation And Their Application In Finance:g-Pricing Mechanism And Risk Measure

The linear form of Backward Stochastic Differential Equation (BSDE) was first introduced by Bismut (1973). Pardoux & Peng (1990) proved the existence and uniqueness theorem of the solution of nonlinear BSDE when generating function satisfied Lipschitz condition. Duffie & Epstein (1992b) also proposed a type of BSDE independently to characterize the stochastic differential utility. 1991 Peng[Peng (1991)] proposed nonlinear Feymann-Kac formula by using BSDE, established a close link between BSDE and PDE. 1997, N.E1 Karoui, Peng and Quenez[El Karoui et al. (1997b)] obtained the extended Black-Scholes formula, from then on, BSDEs theory applied to financial theory gradually. After 10 years of development, BSDE is further studied and applied widely in stochastic control theory, partial differential equations, mathematical finance, control theory, economics and other fields.BSDEs in various fields of application needs to answer the most important question is : How to solve it When given the terminal conditions and the corresponding generating function.But as PDE areas, Only when the generating function for the other linear or nonlinear function of several special categories, We can only get explicit solutions similar to the Black-Scholes formula, but in most cases, We can only rely on numerical methods for solving BSDEs.In the past 10 years, many efforts have been made in the approximate methods for BSDEs. The principle of these methods can be divided into two categories, the first one transformed BSDEs into corresponding quasi-linear parabolic partial differential equations by using nonlinear Feymann-Kac formula developed by Peng, such as "Four-step scheme" developed by Ma, Protter, Yong[Ma et al. (1994)] and Douglas, Ma, Protter, Yong[Douglas, Jr. et al. (1996)], the forward-backward stochastic algorithms proposed by Deiarue and Menozzi[Delarue & Menozzi (2006)], the layer method developed by Milstein and Tretyakov[Milstein & Tretyakov (2006)]. The second category of methods developed directly from the characteristics of BSDEs. The earliest one among these methods is the random time discrete scheme developed by V.Bally[V. Bally (1997)]. Chevance proposed a dynamic programming principle to solve BSDEs. F.Coquet, V.Mackevicius, J.Memin[Coquet, Mackevicius & Memin (1998)], Ph.Briand, B.Delyon, J.Memin[Briand et al. (2001)], S.Peng, M.Xu[Peng & Xu (2003)] and J.Ma, Ph.Protter, J.S.Martin, S.Torres[Ma et al. (2002)] proposed the Binary Tree method for BSDEs from different points of view, and have proven its convergence. Recently, Bouchard and Touzi[Bouchard& Touzi (2004)] presented a kind of Monte-Carlo method for solving BSDEs by using Malliavin calculus. E.Gobet, JP.Lemor & X.Warin (2005)] proposed a regression based Monte-Carlo method for solving BSDEs. Zhao, Chen, Peng[WD.Zhao, LF.Chen & SG.Peng (2006)] developed a kind of high accurateθ-scheme for BSDEs.Peng (1997) introduced the conception of g-expectation via backward stochastic differential equation, g-expectation has a obvious advantages, it is easy to define conditional expectation by Using Peng's g-expectation. In the following study, Peng found that some good properties of g- expectations can be simply reflected by its generating function g. For the financial derivatives market, it means that we can using different g indicate Different types of participator in the market. Based on the conclusion of Coquet, Hu, Memin , Peng[Coquet et al. (2002)], Peng (2004b,d, 2005a) elaborated the theory of dynamic consistent evaluation and g-evaluation, he also proved that under some reasonable assumptions a dynamic consistent evaluation is a g-evaluation. This means that if a certification of financial market pricing mechanisms to meet these conditions. Behind the pricing mechanism must exist a g to characterize their characteristics. This resulting two issues : how to verify these conditions? How to find the hidden g behind the market pricing mechanism?For such an inverse problem, Briand, Coquet, Hu, Memin & Peng (2000) proposed the inverse comparison theorem for BSDEs and the representation theorem for generating function g. This result ensures the uniqueness of g been found and the computability of the pricing mechanism g. Yang (2006) has used this representation theorem to calculate pricing mechanism g. The resulting g coincide with the test g well. He also found that in actual financial market there is no such kinds of data and conditions to support this calculation. Therefore, this method can only be used to test and not used to calculate pricing mechanism g.Yang (2006) treated the data of option price as a trajectory of SDE. He used non-parametric statistical methods for SDEs to estimate the pricing mechanisms g and z, the Solutions of BSDEs. He chose Kernel-Regression non-parametric method and given the estimate formula. He used simulation to test his method. The results are similar as the results of SDEs case: For diffusion term (corresponding to the solution z of BSDEs ), the estimation is effective, but for drift term(corresponding to the generating function g of BSDEs ) the estimation completely submerged in the noise. The estimation could not be effective.In this paper, there is a new idea to consider this inverse problem. This method comes from the decomposition theorem of g- submartingale proposed by Peng[2003]. For given option price process Y_t, firstly finding a g~μ-evaluation to dominate the g-pricing mechanism of the actual market. Under such circumstances, Y_t is a g~μ-submartingale, g- submartingale decomposition method can be used approximate generating function g. In this thesis, some numerical simulations have been used to verify the effectiveness of our method.Financial derivative products from the day of its birth, has been accompanied by risks. In the history of financial derivative products, there has been a lot of well-known risk events, such as "Dutch tulips incident", "British Barings Bank incident" and "China Aviation Oil's case" and so on. Of course, in these incident there is more or less some of other risk factors. Here we consider the market risk only, which means the risk exposure caused by the less of preparation to the price fluctuations of the underlying products. Artzner, Delbaen, Eber &; Heath (1997, 1999) introduce the coherent risk measure. They put Currency type risk-Metric into the axiomatic system. Follmer & Schied (2002a,b,c) and Frittelli & Rosazza Gianin (2002, 2004) followed the concept of a convex risk measure. Rosazza use g- expected to bring in a class of dynamic risk measure, Jianglong presented and proved the necessary and sufficient conditions of g-expectation being a coherent risk measure or convex risk measure.In fact, a good risk measure is a strong pricing mechanism, can cover the risk of financial products caused by the uncertainty of the price. The uncertainty of expected return rate can be included in the scope of g-expectations. For the uncertainty of volatility, Peng (2005b, 2006a,b) introduced a new class of nonlinear expectation: G-expectation, which can be used for this kind of risk measurement.Full paper is divided into five chapters. It's organized as follows:Chapter 1 introduces the background knowledge and basic concepts of BSDEs and g- expectations.Chapter 2 describes and summarizes some previous works for numerical solution of BSDEs. Several representative algorithms for solving BSDEs were explained in detail. Chapter 3 presents a high accuracyθ-scheme for solving BSDEs, and describes the construction of this scheme in detail. For some representative BSDEs models, a large number of numerical results have been made.Chapter 4 ,section 1 describes the origins and development of the financial derivatives, and the scale and status. The enormous scale of the transaction of Derivatives Market shows the urgency and great significance of this kind of research. Section 2 gives a description of the inverse problem for BSDEs and related theoretical results. Section 3 summarizes the results of previous relevant, gives a calculation method of the generating function g : g- submartingale decomposition method.Chapter 5 introduces some related concepts of financial risk management and the basic concept of risk measurement theory and development.SPAN margin system which is widespread use by the large number of international exchange is explained here. We gives the definition of G-risk Measure and the roles for setting parameters of G-risk measure. Finally, an empirical comparison between SPAN system and G-risk measure system is presented by using of financial market data.