Backward Stochastic Differential Equation And Malliavin Derivative Applied In Finance

Linear Backward Stochastic Differential Equation was first introduced by Bismut[1]in 1973 when he studied the maximum principle of Stochastic optimal control.Ever since that Pardou and Peng[2]proved the existence and the uniqueness of general Backward Stochastic Differential Equations. In 1992,two well-known economists,Duffle and Epstein also succeeded in introducing a special type of backward stochastic differential equations to characterize the recursive utility function in finance area.Backward stochastic differential equation(hereafter referred to as BSDE )now has become a rapidly-developing branch of Stochastic Theory.Backward stochastic differential equation provides ways to make nowaday decisions pertaining to certain(or uncertain)goals in the future,which happen to have the same view with the idea of futures options in the financial markets.In 1997, ElKaroui,Quenez and Peng[4]found BSDE's applications in the pricing of derivative securities,and it provided a framework to describe financial mathematical problems.Although had been studied for many years,the Malliavin derivative was not found to have many applications,until the discovery of the formula of integration by parts.For example,in the pricing of Asian options,the expression of Greeks are not easily got,but these parameter are frequently used in financial practices;the Malliavin derivative provide a good approach,however.This paper includes the following four chapters:The first chapter introduce the form of BSDE.BSDE can be naturally introduced from some baltic assumptions of the financal market,and this fully illustrated the mightiness of BSDE as a powerful tool to describe financial issues.To facilitate the application,we used the form of existence and uniqueness of solution given by N.El Karoui,S.Peng and M.C.Quenez in cite EPQ1997.Chapterâ…¡introduces the definition of the Malliavin derivative,as well as a number of its very important properties.As for the formula of integration by parts,we give a useful form since it plays a key role in the calculation of Greeks later in the paper.Chapterâ…¢illustrates a paper by N.El Karoui,S.Peng and MC Quenez in cite(EPQ1997) which deducted a new BSDE satisfied by the Malliavin derivative of the solution of an original BSDE and pointed out that the Malliavin derivative of Y_t is an amendment of Z_t's.Hence,we can use the known results of pricing process(also known as the wealth process) to get the investment strategy by calculating its Malliavin derivative.Taking European call option as an example,we can see that this theorem is very powerful.In the process of figuring the problem out,in order to better tackle the problem, we used Girsanov transform,and the expanded form of Clark-Ocone formula, as well as the diffusion process of Markov etc,then finally we reach the expressions of portofolio when fluctuation rate and volatility are both time dependent.Black-Scholes formula can be considered as a special case of this expression when the parameter is a constant.The fourth chapter talks about the our practical application.In the financial practice,people are very concerned about the Greeks parameters. In this part,we describe the principle of duality and integration formula of Malliavin derivative by means of BSDE,and finally come to the Delta, Vega,Gamma cxprcssions under the condition that both fluctuation rate and Volatility are time dependent.A similar approach will be transfer to the Asian option.