Structural limit of volatility target under stochastic volatility and discrete corrections

Abstract

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 $\Delta t\to 0$ and the observation window parameter $\lambda \to 1$ simultaneously along a path, the expected quadratic variation converges to the target variance if and only if $(1-\lambda)^{-1}\Delta t \to 0$ 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.