The dynamic dependencies in financial market volatility are generally well described by a long-memory fractionally integrated process. At the same time, the volatility risk premium, defined as the difference between the ex-post realized volatility and the market’s ex-ante expectation thereof,...

Traders in the nineteenth century appear to have priced options the same way that twenty-first-century traders price options. Empirical regularities relating implied volatility to realized volatility, stock prices, and other implied volatilities (including the volatility skew) are qualitatively...