Multivariate Stochastic Volatility under the Assumption of Stochastic Volatility of Volatility

Abstract

Multivariate stochastic volatility (MSV) models typically allow volatility to evolve stochastically but do not explicitly account for stochastic volatility of volatility. The main aim of this paper is to develop nonparametric MSV models under the assumption of stochastic volatility of volatility. This study also aims to be the first to examine the dynamics within and across stock prices, volatilities, and volatilities of volatilities. The models developed are simpler and more closely aligned to the continuous-time models found in financial theory than what is currently employed. Statistical inference procedures to test for the stochasticity of the estimated co-volatilities are also developed. The models are applied to a sample of six stocks listed on the S&P 100 index. The results provide strong evidence that not only volatility and volatility of volatility are stochastic, but that significant stochastic relationships also exist within and across stock prices, volatilities, and volatilities of volatilities. The findings further suggest a hierarchical dependence structure, whereby stock price co-movements influence volatility co-movements, which in turn shape higher-order volatility dynamics. These results contribute to the growing stochastic volatility literature and provide useful insights for multivariate risk modelling, portfolio management, and hedging under higher-order uncertainty.