A tale of two volatilities

We show that there are two distinct ways to make volatility stochastic that are differentiated by their consequences for skewness. Most models in the literature have adopted the relatively tractable methodology of using stochastic time changes to engineer stochastic volatility. Unfortunately, this is also the one that can conflict with the relationship occasionally observed in markets between volatility and skewness. Research enhancing the tractability of the second approach to stochastic volatility based on scaling is called for.