基于方差模糊与Gamma分布的期权定价研究

Option Pricing Based on Variance Ambiguity and Gamma Distribution

摘要: 经典期权定价模型依赖于波动率已知的先验假定，难以刻画深度不确定性下的资产价格行为。本文将二阶方差模糊性引入离散时间一般均衡框架，考察其对期权定价的影响。模型设定宏观冲击服从条件正态分布，投资者对客观方差的主观信念则遵循Gamma分布。结合平滑模糊理论与广义递归效用框架，本文通过构造二阶效用函数，在不完备市场设定下推导出唯一的主观等效测度，并基于跨期欧拉方程给出了欧式看涨期权的半闭式解析解。比较静态分析表明，由于期权收益结构具有凸性，客观方差膨胀与主观模糊厌恶增加均会推高期权理论溢价。本文突破了传统定价框架中单一风险维度的局限，为解释期权市场中的方差风险溢价现象提供了微观理论基础。

Abstract: Classical option pricing models rely on the prior assumption of known volatility, making it difficult to capture asset price dynamics under deep uncertainty. This paper introduces second-order variance ambiguity into a discrete-time general equilibrium framework to investigate its impact on option pricing. The model assumes that macroeconomic shocks follow a conditional normal distribution, while investors hold subjective prior beliefs about the objective variance that conform to Gamma distribution. By integrating smooth ambiguity preferences with recursive utility, this paper derives a unique subjective equivalent measure within an incomplete market setting, and provides a semi-closed-form analytical solution for European call options based on the intertemporal Euler equation. Comparative static analysis reveals that, due to the convexity of the option payoff structure, both the expansion of objective variance and an increase in subjective ambiguity aversion will drive up the theoretical option premium. This paper moves beyond the single risk dimension of conventional pricing frameworks and provides a micro-theoretical foundation for explaining the variance risk premium observed in option markets.