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...

The stationary distribution of a GARCH(1,1) process has a power law decay, under broadly applicable conditions. We study the change in the exponent of the tail decay under temporal aggregation of parameters, with the distribution of innovations held fixed. The parameter transformation we study...

We derive the exact solution of a one-dimensional Markov functional model with log-normally distributed interest rates and constant volatility in the terminal measure. The model is shown to have two distinct limiting states, corresponding to small and asymptotically large volatilities,...