Pricing And Empirical Analysis Of Standard European Option Based On Non-linear Expectation

On December 23,2019,our first stock index option,the CSI 300 stock index option,was listed on the CICC,making investors in our country begin to pay close attention to the option market.Options,as important risk management tools,allow investors to hedge by purchasing options.For the society,options can enhance market’s liquidity and improve efficiency and anti-risk capabilities of market.Therefore,it is important to correctly price optionsIn the second chapter of this article,the classic B-S-M pricing theory is introduced first,and then the derivation process of B-S-M pricing formula is given Compared with the original version,which gives the pricing formula by solving partial differential equations,this paper gives a derivation based on the risk-neutral pricing theory and martingale theory.B-S-M option pricing formula,known as the"Second Wall Street Revolution",has greatly promoted the booming of options market.However,the B-S-M option pricing model is only established in a complete market situation and cannot describe the uncertainty of market.In this paper,we define a fuzzy coefficient k to describe the uncertainty of market and extended the risk-neutral measure to a family of probability through k.Then we propose an pricing model of European option based on the uncertain market environment.To deduce the specific pricing formula,we need to calculate the value of Choquet expectations,which is difficult to calculate in the ordinary case.But in the European option case,we can get it by using the relationship between three non-linear expectations:minimax expectations,Choquet expectations and g-expectations.Finally,we give a pricing formula to calculate the pricing range of European options.Compared with the classic B-S-M pricing formula,the model in this paper adds two other parameters:fuzzy coefficient k and expected return rate of the underlying asset,and these two parameters are easy to obtain pass historical data.So the model is feasible.Then we conducted an empirical analysis on the model.The classic BMS pricing formula assumes that the volatility is constant,this assumption is inconsistent with the market,and after it is proposed,it lays hidden dangers for the outbreak of the financial crisis.So we use an effective volatility model,GRACH(1,1)model,to estimates the volatility of each trading day.At first,we check the rationality of the pricing interval.We select a the appropriate k through the data of a sample option and verify the pros and cons of the pricing range on other options.Based on the empirical results,we find that Choquet upper price fits the option price better.So we select an optimal fuzzy coefficient k to see if the Choquet upper price can price option well.The empirical results show that the option model established in this paper is reasonable and can reflect the change range and trend of options well.Also compare the options pricing calculate by B-S-M model(the volatility is given by GARCH(1,1)),Choquet price fits the option price with obvious advantages.Finally,We Summarized the work and conclusions of this paper,expounded the limitations of this model.In later study,we hope that we can expand the fuzzy coefficient k from a constant to a random Process and further extend the model in this paper to the pricing of exotic options.