On the exact limiting distribution of a volatility target index

Assuming a log-normal distribution for the underlying risky asset, we study the limiting distribution of a volatility target index as the rebalancing time step approaches 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 study the dependence of the convergence speed on the observation-window parameter $\lambda$.

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 rho 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.