基于退化抛物方程的期权漂移率反问题

Inverse Problem of Option Drift Rate Based on Degenerate Parabolic Equations

摘要: 在股票价格演化过程中，漂移率是一个重要的参数，对相应的期权定价具有显著影响。本文研究了一个反问题，即通过期权的当前市场价格来恢复漂移函数，由于我们的数学模型在无穷大时不趋于零，这与传统的波动率反问题不同，可能会给理论分析和数值计算带来显著困难。为了克服这一困难，我们应用线性化方法并引入变量替换，将原始问题转化为退化抛物型方程在有界区域上的逆问题。通过解决这个逆问题，我们能够恢复未知的漂移率并解决人工截断带来的误差。基于最优控制框架，我们将原始问题转化为一个优化问题，并证明了极小值的存在性。在推导出必要条件之后，我们还证明了极小值的唯一性和稳定性。

Abstract: In the process of stock price evolution, the drift rate is an important parameter that significantly in-fluences the pricing of corresponding options. In this paper, we investigate a reverse problem, which involves recovering the drift function from the current market prices of options. Due to the fact that our mathematical model does not tend to zero as it approaches infinity, unlike the tradi-tional volatility inverse problem, this poses considerable challenges for theoretical analysis and numerical computations. To overcome this difficulty, we employ a linearization method and intro-duce variable substitutions to transform the original problem into an inverse problem of degener-ate parabolic equations on a bounded domain. By solving this inverse problem, we are able to re-cover the unknown drift rate and address the limitations caused by artificial truncation. Based on an optimal control framework, we formulate the original problem as an optimization problem and demonstrate the existence of a minimum. After deriving the necessary conditions, we establish the uniqueness and stability of the minimum.