Recent progresses in option pricing using ARCH processes for the underlying are summarized. The stylized facts are multiscale heteroscedasticity, fat-tailed distributions, time reversal asymmetry, and leverage. The process equations are based on a finite time increment, relative returns,...

The influence of the past price behaviour on the realized volatility is investigated, showing that trending (driftless) prices lead to increased (decreased) realized volatility. This 'volatility induced by trend' constitutes a new stylized fact. The past price behaviour is measured by the...