Some Numerical Methods Of Backward Stochastic Differential Equations And Their Financial Applications

In recent years, the theory of Backward Stochastic Differential Equations (BSDEs) has been developed rapidly and is applied to finance extensively. At the same time, the research on numerical methods of BSDEs lags behind. Since the solutions of BSDEs can rarely be solved analytically, the numerical methods for BSDEs are indispensable and significant in theory and applications.This master's thesis proposes some new numerical methods for BSDEs with path-dependent terminal values, via discretization of the filtration. Furthermore, the difference and relationship among these numerical methods are investigated. By solving some financial models, the underlying financial meanings of these numerical methods are revealed and the relationship between the numerical methods and some known stochastic simulation methods in finance is exposed.Based on the idea of Ma et al (in "Ma J, Protter P., Martin J S,and Torres S. Numerical Method for Backward Stochastic Differential Equations [J]. Ann. Appl. Probab. 2002, 12: 302-316."), which introduces a numerical method making use of the left-node discrete version of BSDE, this thesis proposes two numerical methods by using different discrete versions with the right-node and the left-and-right-nodes, respectively. After discussing the error sources of these methods, we claim that the right-node discrete version may avoid the errors of solving implicit difference equation approximately. Meanwhile, we utilize a more general random walk for discretization of Brownian motion to fulfill the requirement of financial applications.In light of its successful applications in option pricing, the trinomial model is employed in the thesis for approximation of Brownian motion to accelerate the convergence rates of the above numerical methods of BSDEs. In some difference schemes, the trinomial approximation will produce the same consequences as the binomial approximation, by using merely half nodes of the latter. The result implies that the trinomial approximation may be faster than the binomial approximation. Then, we introduce some more general numerical methods by use of polynomial approximations.Furthermore, the proposed numerical methods are applied to discuss financial models. By pricing contingent claims in completemarket, the numerical methods are compared with no-arbitrage equilibrium analysis. In this case, the different discrete versions of these numerical methods imply different discount factors. By applying these methods to price Europe option, we demonstrate the equality of the numerical methods and the conventional binomial or trinomial method for option pricing. Finally, the proposed methods and conclusions are illustrated by numerical examples. In addition, these methods are applied to price some path-dependent options, such as barrier options.