| One of the most widely studied problems in financial mathematics is the pricing of derivative securities, and it depends on the value of the underlying securities. Financial options are the most common examples of derivative securities. Options are used to hedging and speculating. The aim of hedging is to reduce the risk of adverse price movements in an asset by taking an offsetting position in future contract. So, to find the fair value of option, we need to solve a pricing problem.The risk-neutral pricing approach is a method whose key point is calculating risk-neutral probabilities. It is very important to get the risk-neutral probabilities function accurately. Without arbitrage, a cubic spline is used to approximate the risk-neutral probabilities function, and hence a quadratic programming problem can be formulated for the option pricing. And in order to ensure the non-negativity of a single variable polynomial in intervals, another model with the constraints of matrix inequalities was presented. In this thesis, we present a new model with a quadratic spline function to solve the pricing problem, and furthermore another new model with the constraints of matrix inequalities is proposed to ensure the non-negativity of a single variable polynomial in intervals. The tests show these new models are effective.The structure of this thesis is arranged as follows. In chapter1, the research background, the significance of selected topic and preliminary knowledge are introduced. In the second chapter, the present models are listed. In Chapter3, we present two new models to solve the option pricing problem. In the last chapter, the numerical experiments show our new model is effective. |
