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传统的期权定价理论总是建立在标的资产价格分布的严格假设下,而没有考虑分布的不确定性。本文对标的资产价格分布的严格假设进行放松,分别在仅知到期日标的资产价格的前二阶矩及前三阶矩,而不知道其具体分布的条件下,对期权进行定价。由于信息不充分及分布不确定,推导出的期权价格为一个区间。我们针对有限信息条件下求解期权价格上下界的问题,建立数学规划模型,并将其转化为对偶规划问题进行求解。对此上下界和Black-Scholes价格进行对比分析后发现,Black-Scholes价格介于此上下界之间,相对于采用前二阶矩推导的上下界,采用前三阶矩信息推导的上下界更窄。在使用香港恒生指数权证数据进行的时序分析及横截面分析中发现,市场价格确实介于上下界之间,上下界区间随波动率及剩余存续期的减小而缩小。采用本文的定价方法,不需要对资产价格分布进行严格假设,故可提高定价模型的稳健性,有助于投资者结合期权价格上下界及自己的主观判断进行投资决策。
Traditional options pricing theory is always based on the strict assumptions of the underlying asset price distribution, without considering the uncertainty of the distribution. In this paper, we relax the strict assumptions of the underlying asset price distribution and price the options under the condition that only the first two moments and the first three moments of the asset price on the maturity date are known, respectively, without knowing the specific distribution. Due to insufficient information and uncertain distribution, the derived option price is a range. In order to solve the upper and lower bounds of option prices under the condition of limited information, we establish mathematical programming model and solve it as dual programming problem. By comparing the upper and lower bounds with the Black-Scholes price, we find that the price of Black-Scholes is between the upper and lower bounds. Compared with the upper and lower bounds derived from the first two moments, the upper and lower bounds derived from the first three moments are more narrow. Using the time series analysis and cross-sectional analysis of the warrants data of Hong Kong Hang Seng Index, it was found that the market price is indeed between the upper and lower bounds, and the upper and lower bounds of the market narrowed as the volatility and remaining duration decreased. The pricing method of this article does not need strict assumptions about the distribution of asset prices, so it can improve the robustness of the pricing model and help investors to make investment decisions based on the upper and lower options prices and their own subjective judgments.