Recent years have witnessed great interest in financial models based on Lévy processes as possible alternatives to the traditional Black–Scholes model of financial markets, which is based on the geometric Brownian motion. The main criticism of this model concerns the fact that in the Black–Scholes world, the distribution of the log returns is normal irrespective of the spacing of the data. For widely spaced data, such as security prices sampled on a monthly time grid, the empirical distribution of the returns is close to normal. However, the daily return distributions typically are far from being normal, since they have more mass at the origin and in the tails. The discrepancy is even more visible if one goes down to an intra-day time grid. Models based on Lévy processes, however, can account for such departures from normality and they also have the ability to reproduce other important stylized features of financial time series, for instance, abrupt changes in asset prices because of financial cataclysms.
As any stochastic model, a financial model based on a Lévy process depends on various parameters (finite or possibly infinite-dimensional). Estimation of these parameters, or, in financial terminology, calibration of the model to the available data, is of critical importance to successful application of these models in practice. This is a new, challenging and rapidly growing area of statistical research. To assess recent progress achieved in inference methods for Lévy processes, to identify problems of interest, to outline future research directions and to contribute to the development of this new area, the three of us came up with the idea of organizing a workshop that would emphasize more the statistical challenges of financial modelling with Lévy processes, rather than probabilistic aspects associated with the theory of Lévy processes (path properties, fluctuation theory). EURANDOM, the Eindhoven-based European Research Institute in Stochastics and Stochastic Operations Research, kindly agreed to host the workshop, which subsequently took place on 15–17 July, 2009, and attracted some 60 participants. At the same time, the editorial board of Statistica Neerlandica supported our idea to publish contributions by invited speakers as a special issue of Statistica Neerlandica.
The six papers constituting this issue give a good impression of modern research in statistical inference for Lévy processes, as well as financial modelling with them. In particular,
- 1Antonis Papapantoleon reviews the construction and properties of some popular approaches using Lévy processes to modelling London Interbank Offered Rate (LIBOR) interest rates. He discusses different frameworks: classical LIBOR market models, forward price models and Markov-functional models, with finally some special attention to recently developed affine models.
- 2José Manuel Corcuera and Gergely Farkas consider the asymptotic behaviour of the power variation of processes that are integral transforms of α-stable processes with index of stability α between 0 and 2. They establish stable convergence of the corresponding fluctuations. These results provide tools for statistical inference from discrete observations. Their main result is a central limit theorem for the pth power variation.
- 3Non-parametric estimation of the Lévy density for pure jump Lévy processes is the topic of a paper by Fabienne Comte and Valentine Genon-Catalot. Under various sampling schemes, non-adaptive and adaptive estimators of the Lévy density are proposed and the corresponding risk bounds are derived.
- 4Johanna Kappus and Markus Reiß construct an estimator of the Lévy–Khinchine characteristic triplet corresponding to the given Lévy process and derive optimal rates of convergence under a general sampling scheme which encompasses the usual low- and high-frequency data assumptions and covers also the mid-frequency regime.
- 5Mark Podolskij and Mathias Vetter survey limit theorems for certain functionals of semi-martingales that are observed at high frequency. They explain the main ideas of the theory using the concept of stable convergence, and they demonstrate some laws of large numbers and stable central limit theorems.
- 6Starting from multivariate elliptic distributions, Nick Bingham, John Fry and Rüdiger Kiesel introduce and study multivariate elliptic processes. They discuss Lévy processes, diffusions, discrete versus continuous time, jumps versus diffusions and semi-martingales. They also analyze some data to illustrate the theory.
We expect that the topics touched upon in this issue will continue to play an important role.
Concluding this introduction, we would like to thank the authors who contributed to this issue, as well as Statistica Neerlandica for hosting the proceedings. It is our pleasant duty to acknowledge the contribution of EURANDOM to the organization of the workshop and to thank the sponsors: the European Science Foundation (through its programme AMaMeF), the Netherlands Organization for Scientific Research (NWO), the Mathematical Research Institute (MRI) and the Thomas Stieltjes Institute for Mathematics.