We address the problem of defining and calculating forward volatility implied by option prices when the underlying asset is driven by a stochastic volatility process.We examine alternative notions of forward implied volatility and the information required to extract these measures from the...

We derive an explicit representation of the transitions of the Heston stochastic volatility model and use it for fast and accurate simulation of the model. Of particular interest is the integral of the variance process over an interval, conditional on the level of the variance at the endpoints....

The number of crossings of the implied volatility function with a fixed level is bounded above by the number of crossings of the risk-neutral density with the density of a log-normal distribution with the same mean as the forward price. It is bounded below by the number of convex payoffs priced...