发表机构
The University of Osaka（大阪大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文推导了短期到期障碍期权价格的渐近展开，首项为Black-Scholes价格，修正项由波动率波动阶数决定，数值实验验证其改进效果。
AI 中文摘要
我们推导了连续随机波动率下向上敲出看跌障碍期权价格在短期到期时的展开式，其中行权价和障碍在扩散尺度上接近现货价格。假设归一化期末收益、相对波动率波动和运行最大值联合弱收敛，并满足一致可积性，我们证明首项是拟合远期方差曲线的时间非齐次Black-Scholes障碍价格。第一个依赖于模型的修正项阶数为$\theta^{H+1/2}$，其中$\theta^H$，$H \in (0,1/2]$，是相对波动率波动的阶数，并通过杀死布朗转移密度显式表示。对于正则波动率模型（$H=1/2$），系数由短期平值隐含波动率偏斜决定。对于粗糙波动率模型（$H<1/2$），系数简化为一维积分。数值实验表明，该修正显著改善了Black-Scholes近似。
英文摘要
We derive a short-maturity expansion for up-and-out put barrier option prices under continuous stochastic volatility when the strike and the barrier approach the spot at the diffusive scale. Assuming joint weak convergence of the normalized terminal return, the relative volatility fluctuation, and the running maximum, together with uniform integrability, we show that the leading term is the time-inhomogeneous Black-Scholes barrier price fitted to the forward variance curve. The first model-dependent correction is of order $θ^{H+1/2}$, where $θ^H$, $H \in (0,1/2]$, is the order of the relative volatility fluctuation, and is represented explicitly through killed Brownian transition densities. For regular volatility models, where $H= 1/2$, the coefficient is determined by the short-maturity at-the-money implied-volatility skew. For rough volatility models with $H < 1/2$, the coefficient reduces to a one-dimensional integral. Numerical experiments show that the correction materially improves the Black-Scholes approximation.
