Some Properties In The Pricing Of The Options

The purpose of this thesis is to use the method of the BSDE to study some properties in the pricing of the options. In the base of the classical options pricing theory,some economists have already obtained many properties in the pricing of the options from the expectation and equivalent martingale theories(see[7][8][10][11][12][13] [14][15]).Here,we use the Backward Stochastic Differential Equa-tion(BSDE) to get the pricing of the options and to study the properties of the options pricing.First,we begin with the complete condition. The options pricing satisfies a linear BSDE under this condition.We use the hypothesis of smooth quality of the equation in [3] to study the derivative of the solution to the equation in the initial time(i.e the options pricing) with regard to the stock initial value.We find that this derivative is dominated by the derivative of the terminal value function and the options pricing inherits the convexity of the terminal value function.From the bound of this derivative,we make use of the Cauchy problem and Feynman-Kac formula to induce other relevant properties. We make a contrast with the properties of the options pricing the economists have obtained. We discover it is easier to use the BSDE to make the financial meanings. Then,we turn to the uncomplete condition and find some properties are also satisfied.It is notable to see that under this uncomplete condition the corresponding options pricing satisfies the non-linear BSDE. The method we use is to make the equation linear and to use the same method to the complete condition.In the thesis,we give the example to each condition. Thispaper is organized as follows:Chapter 1 mainly introduces the development of the pricing of the options,the development of the BSDEs and the content of this thesis.Chapter 2 gives the prepared knowledge,include the necessary knowledge of the Stochastic Differential Equation,some useful knowledge of the Backward Stochastic Differential Equation,some notations and conditions,and some propositions and lemmas.Chapter 3 under the complete condition,shows the pricing of the options from the financial angle,gives the BSDE model,analyzes some options pricing properties according to the BSDE.At last,we give the example of the European Stocks Put Options.Chapter 4 discusses the properties of the pricing of the options under the uncomplete condition.