A central limit theorem and its application to the limiting distribution of volatility target index
arXiv:2503.16878
The paper proves a strong law of large numbers and a central limit theorem for a volatility‑target index as the discretisation step goes to zero, derives the exact limiting distribution and its volatility properties, and provides a formula converting rho‑1 sensitivity to vega for derivatives on the index.
Abstract
We study the limiting distribution of a volatility target index as the discretisation time step converges to zero. Two limit theorems (a strong law of large numbers and a central limit theorem) are established, and as an application, the exact limiting distribution is derived. We demonstrate that the volatility of the limiting distribution is consistently larger than the target volatility, and converges to the target volatility as the observation-window parameter in the definition of the realised variance converges to 1. Besides the exact formula for the drift and the volatility of the limiting distribution, their upper and lower bounds are derived. As a corollary of the exact limiting distribution, we obtain a vega conversion formula which converts the rho1 sensitivity of a financial derivative on the limiting diffusion to the vega sensitivity of the same financial derivative on the underlying of the volatility target index.