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

TL;DR

This paper establishes small-time central limit theorems for stochastic Volterra integral equations with various kernels, including rough models, and applies these results to derive asymptotic pricing formulas for digital options in rough volatility models.

Contribution

It introduces new CLTs for stochastic Volterra equations with broad kernels and develops Markovian lifts for monotone kernels, extending the theoretical understanding of rough volatility models.

Findings

Proved convergence of finite-dimensional distributions for Volterra processes.

Derived asymptotic pricing formulas for digital options on realized variance.

Established CLTs for Markovian lifts of Volterra equations.

Abstract

We study small-time central limit theorems for stochastic Volterra integral equations with H\"older 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, which covers a large class of Volterra kernels that includes rough models based on Riemann-Liouville kernels with short- and 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 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.