Small-time central limit theorems for stochastic Volterra integral equations and their Markovian lifts

Abstract

We study small-time central limit theorems for stochastic Volterra integral equations with Hölder continuous coefficients and general locally square integrable Volterra kernels. We prove the convergence of the finite-dimensional distributions, a functional CLT, and limit theorems for smooth transformations of the process, covering a large class of Volterra kernels including rough models based on Riemann-Liouville kernels with short- or long-range dependencies. To illustrate our results, we derive asymptotic pricing formulae for digital calls on the realized variance in three different regimes. The latter provides a robust and largely model-independent pricing method for small maturities in rough volatility models. Finally, for the case of completely monotone kernels, we introduce a flexible framework of Hilbert space-valued Markovian lifts and derive analogous limit theorems for such lifts. The latter provides new small-time limit theorems for stochastic Volterra processes obtained by transformation of the underlying Volterra kernels.