The editors of this special issue and several of the contributing authors have known Peter for a long time. We thought that the special issue will be enriched by adding a few personal notes and recollections about our interactions with Peter.

Volatility derivatives are a class of derivative securities where the payoff explicitly depends on some measure of the volatility of an underlying asset. Prominent examples of these derivatives include variance swaps and VIX futures and options. We provide an overview of the current market for...

Models which hypothesize that returns are pure jump processes with independent increments have been shown to be capable of capturing the observed variation of market prices of vanilla stock options across strike and maturity. In this paper, these models are employed to derive in closed form the...

Three processes reflecting persistence of volatility are formulated by evaluating three Levy processes at a time change given by the integral of a square root process. A positive stock price process is then obtained by exponentiating and mean correcting these processes, or alternatively by...

We investigate the importance of diffusion and jumps in a new model for asset returns. In contrast to standard models, we allow for jump components displaying finite or infinite activity and variation. Empirical investigations of time series indicate that index dynamics are devoid of a diffusion...

In finance, optionality is a possible property of a financial contract giving the owner a choice between two or more assets. For example, a convertible bond has optionality because its owner must choose between having a bond or having some shares of stock. In mathematics, a binary operation acts...