On the Continuous Limit of Weak GARCH

TL;DR

This paper proves that the weak GARCH model converges to a unique stochastic volatility process characterized by kurtosis, improving the modeling of implied volatility surfaces and generalizing Nelson's limit.

Contribution

It establishes the continuous limit of weak GARCH as a geometric mean-reverting process, independent of parameter convergence assumptions, and relates it to kurtosis of returns.

Findings

Limit is a geometric mean-reverting stochastic volatility process.

Limit coincides with Nelson's when returns are normal.

Provides an improved fit to implied volatility surfaces.

Abstract

We prove that the symmetric weak GARCH limit is a geometric mean-reverting stochastic volatility process with diffusion determined by kurtosis of physical log returns; this provides an improved fit to implied volatility surfaces. When log returns are normal the limit coincides with Nelson's limit. The limit is unique, unlike strong GARCH limits, because assumptions about convergence of model parameters is unnecessary -- parameter convergence is uniquely determined by time-aggregation of the weak GARCH process.