不连续收益的无偏蒙特卡洛 Greeks

中文摘要 AI 辅助

路径wise 微分在收益不连续处失效，对障碍期权、自动赎回期权和数字期权产生零或偏倚的敏感性。行业变通方法——平滑指示函数——引入偏差并要求按产品校准。我们推导出一个修正公式，无需平滑即可恢复无偏 Greeks。对于分段光滑且在曲面 {g_i = 0} 上具有不连续性的收益 F(Z,θ)，我们证明敏感性可分解为路径wise 项（由标准 AAD 计算）加上边界修正之和，每项涉及收益跳跃、边界处的高斯密度以及边界对参数的敏感性。修正通过在正态随机空间中进行牛顿求根计算，跳跃通过定价内核的两次前向重放评估。实现使用 AADC（pip install aadc），其磁带重放和自动不连续性跟踪使方法完全自动化——量化分析师编写标准定价代码，修正驱动器识别并处理所有不连续性。我们证明了任意光滑函数和指示函数组合（不仅仅是外积）的公式，涵盖具有递归存活/死亡逻辑的真实自动赎回收益结构。在 QuantLib 模型（GBM、Heston、Hull-White）上的基准测试显示，所有 Greeks 在解析或凹凸重估参考值的 0.1–4% 范围内。

英文摘要

Pathwise differentiation of Monte Carlo estimators fails at payoff discontinuities, producing zero or biased sensitivities for barriers, autocallables, and digital options. The industry workaround --- smoothing the indicator functions --- introduces bias and requires per-product calibration. We derive a correction formula that restores unbiased Greeks without smoothing. For a payoff $F(Z,θ)$ that is piecewise smooth with discontinuities on surfaces $\{g_i = 0\}$, we show that the sensitivity decomposes into a pathwise term (computed by standard AAD) plus a sum of boundary corrections, each involving the payoff jump, the Gaussian density at the boundary, and the sensitivity of the boundary to the parameter. The correction is computed by Newton root-finding in the normal-random space, with the jump evaluated by two forward replays of the pricing kernel. The implementation uses AADC (\texttt{pip install aadc}), whose tape replay and automatic discontinuity tracking make the method fully automatic --- the quant writes standard pricing code, and the correction driver identifies and handles all discontinuities. We prove the formula for arbitrary compositions of smooth functions and indicator functions (not just outer products), covering real autocallable payoff structures with recursive alive/dead logic. Benchmarks on QuantLib models (GBM, Heston, Hull-White) show all Greeks within 0.1--4\% of analytic or bump-and-revalue references.

发表机构

• CIDMA, University of Aveiro（阿维罗大学CIDMA）

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