Research On The Maximum And Minimum Prices Of American Options In Incomplete Market

With the continuous opening of the domestic financial market,the emergence of a large category of financial native derivatives has added a lot of vitality to the domestic financial market.As an important financial derivative tool in the financial market,option contract has a wide variety of types.Because of their flexible trading methods,they provide investment traders with many strategies for hedging and risk aversion.Therefore,studying the pricing of options has become one of the key research directions in the financial field.American option,as a highly flexible option contract,its pricing is often more complicated than ordinary traditional European option pricing because of the holder’s arbitrariness to exercise before the deadline.For American call options that do not pay dividends in the traditional complete market,their value is not as good as selling directly due to their early exercise,which makes their value consistent with European call options.For American put options,its value is determined by a set of linear complementary conditions,which need to be determined numerically using finite difference format or Monte Carlo method.However,the financial market is often not a complete market with ideal assumptions.The rate of return and volatility of the underlying asset is usually difficult to determine,with randomness and ambiguity.Therefore,in such a market,the risk-neutral measure is not unique,and a series of pricing formulas under the assumption of a complete market will also fail in the real market.For the study of incomplete markets,many scholars have proposed maximum and minimum prices and Choquet upper and lower prices to calculate the upper and lower bounds of option prices under given conditions.However,since these prices are in the form of non-linear expectations,how to calculate them has become a difficult problem.Chen and Kulperger[24]combined the backward stochastic differential equation theory created by the famous financial mathematician Peng Shige[18],and gave a relationship between the maximum and minimum expectations and Choquet’s upper and lower prices,and used this to solve the pricing of European options in an incomplete market and gave the calculation formula.For the pricing of American put options in an incomplete market,their value depends on a special kind of reflected backward stochastic differential equation(RBSDE).And this type of RBSDE inherits BSDE in some key properties,so the use of RBSDE properties to establish the calculation of the maximum and minimum expectations of American put options constitutes the main content of this article.The main work of this paper is to generalize the co-monotonic theory under BSDE[27],establish the co-monotonic theory under RBSDE and the nonlinear Feynman-Kac formula for RBSDE under given strong conditions,and use the RBSDE model to describe the maximum and minimum prices of American put options and the corresponding obstacle problem PDE satisfied.This article first introduces the background of current options research,and at the same time briefly introduces the current research status of American put options and the financial application of g expectation and RBSDE.The second chapter introduces various methods of option pricing in the complete market,including risk-neutral pricing,backward stochastic differential equation pricing,reflected backward stochastic differential equation pricing,and PDE pricing derived from hedging portfolios.In third chapter,according to the co-monotonic theorem of backward stochastic differential equations given by Chen,Kulperger and Wei[27],the co-monotonic theorem on reflected backward stochastic differential equations is obtained.At the same time,a nonlinear Feynman-Kac formula based on RBSDE is established under strong conditions.The fourth chapter is analogous to Chen and Kulperger’s theory of incomplete markets[24],and gives the maximum and minimum prices in the incomplete market of American put options.At the same time,according to the martingale representation theorem for maximum and minimum prices[24],the RBSDE model is used to prove that under a given market risk price,the corresponding solution is the highest price and the lowest price defined by the American put option,which satisfies the corresponding obstacle problem PDE,Finally,we use the American put bonus model to illustrate that this is a form of our American put option,resulting in a simplified expression for the maximum price of the American put option.The fifth part summarizes the main points of the full text,and points out the advantages and disadvantages of the model in this paper and the prospect of future research.