The Application Of Girsanov Transformation In Backward Stochastic Differential Equations And Asian Options Pricing

The Girsanov theorem, which is fundamental in the general theory of stochastic analysis. It is also very important in many applications, for exam-ple in mathematics finance. Basically the Girsanov theorem describes how the dynamics of stochastic processes change when the original measure is changed to an equivalent probability measure. Specially, with proper choosing the pa-rameter, by Girsanov transform, we can change an Ito process into Brownian motion under new probability measure. The Girsanov theorem can be used to solve the problem of pricing options and other financial derivative products.Option, as one of the most basic of financial derivatives, is an important product in financial market. The pricing problem of options has been studied extensively. Option pricing model depends on the native model of the evo-lution of underlying asset prices. Under continuous-time model, we can use stochastic differential equation to describe the price of underlying asset, then the wealth process of option holder can be described as a backward stochastic differential equation. So the price of an option can be calculated by a backward stochastic differential equation. Then we can build the model of the option pricing and derive the option pricing formula, by the theories of backward s-tochastic differential equations. But for many backward stochastic differential equations, it is difficult to find the explicit solution. We can only solve the backward stochastic differential equations by numerical methods.In this paper, we study a class of backward stochastic differential equa-tions with some special terminal conditions, and solve it by numerical methods. We study the1-dimensional backward stochastic differential equations as fol-lows First we give the numerical method to solving the standard backward stochas-tic differential equations, with the terminal condition yT=ξ=Φ((Bt)0≤t≤T), where{(Bt)0≤t≤T} is a1-dimension standard Brownian motion. We define the scaled random walk to approximate the Brownian motion. Then we get the discrete backward equations and the discrete terminal condition. We de-part from the discrete terminal condition, iterate forward, and calculate the possible values of variables at each discrete time in turns. Then we can get the solution of the discrete backward equation at the initial time t=0. When certain conditions are assumed, the solution of the discrete backward equation converges to the solution of backward stochastic differential equations.In this paper, we study the backward stochastic differential equations with the terminal condition like yT=ξ=Φ((xt)0≤t≤T), where xt is a diffusion process, as solution of a SDE. If there is no explicit solution for this SDE, we can’t solve the equation with the numerical methods discussed before. We can use another more useful and efficient method. We can transform the Ito process to a standard Brownian motion under new probability, by Girsanov transform, get a equivalent backward stochastic differential equation under new probability measure with a simple terminal condition. We analyze three special forms of terminal conditions, with Girsanov transform, change discrete backward equation and discrete terminal condition, introduce the iterative for-mula of explicit scheme and of the implicit scheme, and prove the convergence. We supplied some examples, including a example of European option pricing by backward stochastic differential equation.Asian option is a path-dependent options contracts, dependent on the av-erage of the price of underlying asset before the maturity time. Asian option pricing model can be described by the backward stochastic differential equa-tions with special terminal conditions. We analyzed the pricing problem of the European arithmetic average Asian options with fixed strike price. First we simplified the model by Girsanov transformation. Then we listed the iterative formula of explicit scheme and of the implicit scheme. Finally, We analyzed the solution process. Finally, we carried out summary of this thesis and give some possible future researches.