Time-delayed stochastic volatility model

We propose a multivariate stochastic volatility model with time-delayed interactions, and study its emergent dynamics. The proposed model takes the form of an agent-based model with the Cucker–Smale mechanism, time-delayed interactions and regime-switching. It exhibits a collective behavior “flocking” emerging from the all-to-all coupling with time-delayed interactions induced by the finite propagation speed of communications. We assume that the realized volatility path switches randomly between two regimes. In this setting, we provide theoretical and numerical solutions of the proposed model and show that our proposed theoretical framework is sufficient for volatility’s exponential convergence toward a constant asymptotic value. The longer time-delay makes a volatility converge faster with a lower variance, and we also fit system parameters in the model with the daily observations on stock return and volatility to show model’s high prediction power in both in and out of sample tests.