10. Lévy Processes

  1. Jacques Janssen,
  2. Oronzio Manca and
  3. Raimondo Manca
  1. Jacques Janssen,
  2. Oronzio Manca and
  3. Raimondo Manca

Published Online: 4 APR 2013

DOI: 10.1002/9781118578339.ch10

Applied Diffusion Processes from Engineering to Finance

Applied Diffusion Processes from Engineering to Finance

How to Cite

Janssen, J., Manca, O. and Manca, R. (2013) Lévy Processes, in Applied Diffusion Processes from Engineering to Finance, John Wiley & Sons, Inc., Hoboken, NJ, USA. doi: 10.1002/9781118578339.ch10

Publication History

  1. Published Online: 4 APR 2013
  2. Published Print: 4 MAR 2013

ISBN Information

Print ISBN: 9781848212497

Online ISBN: 9781118578339

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Keywords:

  • Brownian-Poisson model;
  • finance;
  • Lévy processes;
  • Lévy–Khintchine formula;
  • variance gamma (VG) process

Summary

The main fact of Lévy processes is to drop the normality assumption in the definition of a standard Brownian motion (SBM) and keep its other assumptions, with another slight assumption replacing the normality. This chapter gives some examples of Lévy processes and presents the Lévy–Khintchine formula. Variance gamma (VG) process and Brownian-Poisson model are considered in detail. If the approach of option pricing using Lévy processes really extends the classical Black and Scholes approach, the fact that, except with the Merton model, we lose the unicity of the risk neutral measure leads to other problems, which must still be studied, and simulation is the only way to use such models from a practical point of view. Lévy processes still have independent increments, an assumption that is also criticized from the viewpoint of realistic stochastic models in finance.