中文摘要：

确定原生资产的隐含波动率无论是在理论还是实际应用上都有重要意义。本文讨论在期权平均价格己知的前提下如何重构隐含波动率的反问题，利用Green函数法将此问题化为一个“终端”控制问题，通过最佳控制解法讨论了控制泛函极小元的存在性与唯一性，并给出了极小元所满足的必要条件。

英文摘要：

Identifying the implied volatility of underlying assets is very important for both theoretical and practical applications. Discussed in this paper is an inverse problem of recovering the implied volatility when the average option price is known. The issue is converted into a terminal control problem by the Green function method, and the existence and uniqueness of the minimum of the control functional is also deduced by the optimal control method. Finally, the necessary condition for the minimum is given.