AI 中文总结
针对一般一维扩散市场的欧式或有债权，提出基于PDE的自融资对冲策略，推导最小对冲资本的充要条件，通过数值实验验证方法的有效性与局限性。
AI 中文摘要
我们开发了一种基于PDE的方法，用于在仅由尺度函数和速度测度表征、可能无经典SDE表示且利率恒定的一般一维扩散市场中对欧式或有债权进行定价和对冲。我们推导了一个对冲方程，其解可生成自融资对冲策略，并给出了尺度、速度和利率的充分条件，在此条件下该策略能达到最小对冲资本，该最小对冲资本通过无消失风险的免费午餐（NFLVR）条件表述。我们进一步证明了NFLVR的充要条件，并通过一个辅助扩散刻画了等价局部鞅测度的类，该辅助扩散的尺度和速度特征由现实世界扩散的特征及利率决定。当NFLVR条件不成立时，该框架可能产生对应于非最小策略的多个对冲方程，其相关价格可超过最小对冲资本。我们通过涉及具有不规则特征的扩散模型的数值实验，说明了该方法的有效性和局限性。
英文摘要
We develop a PDE-based methodology for pricing and hedging European contingent claims in general one-dimensional diffusion markets characterized solely by their scale function and speed measure, possibly without a classical SDE representation, and with constant interest rate. We derive a hedging equation whose solution generates a self-financing hedging strategy and provide sufficient conditions on scale, speed, and interest rate, under which this strategy achieves the minimal hedging capital, expressed through the no free lunch with vanishing risk (NFLVR) condition. We further prove necessary and sufficient conditions for NFLVR and characterize the class of equivalent local martingale measures through an auxiliary diffusion whose scale and speed characteristics are determined by those of the real-world diffusion and by the interest rate. When the NFLVR condition fails, the framework may produce multiple hedging equations corresponding to non-minimal strategies, whose associated prices can exceed the minimal hedging capital. We illustrate both the effectiveness and limitations of the approach through numerical experiments involving diffusion models with irregular features.
