We propose a local likelihood estimation for the log-transformed ARCH(1) model in the financial field. Our nonparametric estimator is constructed within the likelihood framework for non-Gaussian observations: it is different from standard kernel regression smoothing, where the innovations are...

We consider implied volatility, time-dependent volatility, local volatility and stochastic volatility. We derive relationships between the different concepts. The relationships are of an exact analytical type if this is possible, else we use expansions to obtain approximate expressions. We close...

We recall some fundamentals on Levy processes. Then the Gamma distribution, the Variance Gamma process and option pricing for this process are considered in detail. To implement the Variance Gamma model for option pricing, we use the fast Fourier transform, time change and discuss error bounds