Relaxed Model By Quadratic Polynomial For Fitting Risk-neutral Probabilities

In Finance,an option is a contract which gives the buyer the right,but not the obligation,to buy or sell an underlying asset or instrument at a specified strike price on a specified date.It's normally used to hedging and speculating.Finding the fair value of option and solve the pricing problem are highly focused among academicians.Risk-Neutral valuation is an important method in asset pricing field.In an arbitrage-free complete market,arbitrage prices of contingent claims are their discounted expected values under the risk-neutral measure.Without arbitrage,a cubic spline is used to approximate the risk-neutral probabilities function in the pricing model for European options in Gegard's book,but in order to ensure the non-negativity of a single variable polynomial in intervals,a constraint of matrix inequalities was added which significantly increase the difficulty of this model.In this paper,we explore a new pricing model based on risk-neutral pricing approach and nonarbitrage pricing principle,the paper will transform the problem into an optimization problem.Compared with the fitting function of cubic spline model,we choose to apply a new formed quadratic function to the spline function without the constraint of matrix inequalities.The paper is organized as follows:Chapter 1,we introduce the development of options and the present situation of the derivative market in our nation.Furthermore,the chapter includes the main purpose and the main structure of the paper.In Chapter 2,basic theory of CRR is introduced.We also give a brief introduction of option pricing theory.In Chapter 3,the traditional cubic spline model and a quadratic spline model are introduced.The mathematicians gave us an idea to use the spline to approximate the risk-neutral probabilities function followed by a similar model with lower power and simplify the model.In Chapter 4,we present a new relaxed quadratic spline model.We simplify the last two models and then do some numerical experiments in the last chapter to test the applicability of our model.The numerical results also show that the optimization results of the model are close to the former one but can be pithier.Finally,we discuss the conclusion and prospection of this paper.