Research On Minimax Price And Choquet Price Of Several Exotic Options In Incomplete Market

With the continuous development and improvement of the financial market,exotic options have developed rapidly in recent years,and attracted more and more attention from investors.The development of exotic options provides in-vestors with more options for avoiding risks and hedging,and satisfies investors'more diverse investment preferences.Therefore,the research on the pricing of exotic options has important practical significance for the development of ex-otic options products.Since the payment at maturity of exotic options is often path-dependent,the pricing problem is more difficult than the pricing problem of standard options.According to the risk-neutral pricing theory,scholars have deduced the pricing formulas of many exotic options under the assumption of a complete market.However,because the factors that affect asset prices in the real market are very complex,the rate of return and volatility of assets often have double uncertainties-randomness and ambiguity.Such a market does not sat-isfy the completeness assumption,so the risk-neutral pricing theory is no longer applicable.Many scholars have put forward many research methods for option pricing under such an incomplete market models,such as maximum and minimum prices and Choquet prices.However,because these two prices are in the form of nonlinear expectations,it is difficult to calculate their values,which greatly limits their application in practice.Chen and Kulperger[17]used the theory of backward stochastic differential equations to give a method to calculate a class of maximum and minimum expectations and Choquet expectations,and applied it to the pricing of single-asset European options in incomplete markets.The main purpose of this paper is to generalize Chen,Kulperger and Wei[26]'s and Chen and Kulperger[17]'s results and apply them to calculate the maximum and minimum prices and Choquet prices of multi-asset European options and some exotic options.This paper mainly includes six parts.The first part is the introduction of the full text,which briefly introduces the development process of option.the research status of option pricing theory and the application of Choquet expectation and g-expectation in the economic and financial fields.Finally,the research ideas and structure of the paper a re given.The second part expounds the hypothesis of the complete market model and gives the risk-neutral pricing theory in the complete market model.Then it introduces the definition of several commonly used multi-asset options.Finally,the definition of several exotic options is introduced,and the pricing formula of multi-asset exotic options is given by using the risk-neutral pricing theory.In the third part,Chen,Kulperger and Wei[26]'s results are generalized to multidimensional cases.We study the comonotonic theorem in multi-dimensional case,that is,we consider the following forward and backward stochastic differen-tial equations#12 Where the random processW,Xi,yi,zi values at Rd,Rn,Rm,Rm×d respectively.A sufficient condition for#12 to be true is given.Then,by using the above multidimensional comonotonic theorem,we give a condition such that the g-expectation of ?i(XTi)satisfies the additivity.The fourth part is mainly based on the above conclusions to study option pricing in incomplete markets.Before this,we give the assumptions of the incomplete market model in this paper.In this paper,two fuzzy coefficients klandk2 are proposed to be obtained in two time periods of[0,q]and[q,T]respec-tively.Under this assumption,the risk-neutral measure set can be obtained,as shown below.#12#12 According to the above risk-neutral measure set,the maximum,minimum and Choquet prices of derivatives can be defined respectively.Finally,by using the multi-dimensional comonotonic theorem and the additive of g-expectation,we prove that the maximnm and minimum prices of a derivative product are equiv-alent to the Choquet price,and give a method to calculate them.The fifth part studies the maximum,minimum and Choquet prices of specific options in incomplete market models.Firstly,it is proved that the maximum and minimum prices of multi-asset European options are equivalent to the Choquet price,and taking the geometric mean basket of European call options as an example,the explicit form of the maximum price and the Choquet price formula is given,and the comparison with Chen and Kulperger[17]'s pricing formula is made.Then,taking the dual currency continuous geometric average Asian call option as an example,it is proved that its maximum price,minimum price and Choquet price are equivalent,and the explicit form of its maximum price and Choquet price formula is given when ?? 0.Finally,it is proved that when the expected rate of return and volatility of the asset are constant,the maximum and minimum prices of the barrier option are equivalent to the Choquet price of the single asset,and the explicit form of the formula of the maximum price and the Choquet price is given.The sixth part summarizes and looks forward to the full text,summarizes the main conclusions of this paper,and analyzes the advantages and disadvantages of the model in this paper.