The Continuous-Time Limit of Score-Driven Volatility Models

We provide general conditions under which a class of discrete-time volatility models driven by the score of the conditional density converges in distribution to a stochastic differential equation as the interval between observations goes to zero. We show that the form of the limiting diffusion depends only on the link function and on the conditional second moment of the score. Interestingly, the properties of the stochastic differential equation are strictly entangled with those of the discrete-time counterpart. Score-driven models with fat-tail densities lead to continuous-time processes with finite volatility of volatility, as opposed to fat-tail models with a GARCH update, for which the volatility of volatility is explosive. An extension of our results to models with a time-varying conditional mean and to conditional covariance models is also developed