发表机构
Nomura Securities（野村证券）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明波动率目标指数的结构极限在广义随机波动率下仍成立，并推导离散再平衡的期望二次变异修正公式，给出收敛条件，数值验证有效。
AI 中文摘要
波动率目标指数的精确极限分布最近已在风险资产服从对数正态假设的情况下建立。本文证明，在更广泛的随机波动率过程类别下，该结构极限仍然成立。此外，我们表明，当考虑波动率目标指数与驱动布朗运动的联合分布的极限时，必须引入一个额外的独立布朗运动。除结构极限定理外，我们还在标的资产波动率为时间的确定性函数的假设下，推导了离散再平衡波动率目标指数期望二次变异的修正公式。作为离散修正公式的一个推论，表明当再平衡时间步长 $\Delta t\to 0$ 与观测窗口参数 $\lambda \to 1$ 沿同一路径同时趋于极限时，期望二次变异收敛于目标方差当且仅当 $(1-\lambda)^{-1}\Delta t \to 0$ 沿同一路径成立。数值结果以局部波动率模型和Heston随机波动率模型为例，支持了结构极限及离散修正公式的有效性。
英文摘要
The exact limiting distribution of a volatility target index has recently been established under a log-normal assumption on the risky asset. In this paper, we show that the structural limit remains valid under a broad class of stochastic volatility processes. In addition, we show that, when the limit of the joint distribution of the volatility target index and the driving Brownian motions is considered, an extra independent Brownian motion must be involved. Besides the structural limit theorem, we derive a correction formula for the expected quadratic variation of the discrete rebalancing volatility target index, under the assumption that the volatility of the underlying risky asset is a deterministic function of time. As a consequence of the discrete correction formula, it is shown that, as the rebalancing time step $Δt\to 0$ and the observation window parameter $λ\to 1$ simultaneously along a path, the expected quadratic variation converges to the target variance if and only if $(1-λ)^{-1}Δt \to 0$ along the same path. Numerical results are provided to support the structural limit, using the local volatility model and the Heston stochastic volatility model as examples, as well as the effectiveness of the discrete correction formula.
