We present an overview of the broad class of financial models in which the prices of assets are Lévy-Ito processes driven by an n-dimensional Brownian motion and an independent Poisson random measure. The Poisson random measure is associated with an n-dimensional Lévy process. Each model consists of a pricing kernel, a money market account, and one or more risky assets. We show how the excess rate of return above the interest rate can be calculated for risky assets in such models, thus showing the relationship between risk and return when asset prices have jumps. The framework is applied to a variety of asset classes, allowing one to construct new models as well as interesting generalizations of familiar models.
The authors thank J. Armstrong, D. Brody, Á. Cartea, E. Eberlein, M. Grasselli, T. Hurd, A. Lokka, A. Macrina, D. Meier, B. Meister, K. Owari, G. Peskir, M. Pistorius, A. Rafailidis, M. Schweizer, T. Tsujimoto and R. Zimmer for helpful comments, along with participants at the 2019 Research in Options Conference, IMPA, Rio de Janeiro, and the University of Manchester financial mathematics seminar. This work has also benefitted from comments made by anonymous reviewers. We gratefully acknowledge support from (a) Timelineapp Limited, Basildon [GB], (b) the Fields Institute for Research in Mathematical Sciences, the Aspen Center for Physics, and the Simons Foundation [LPH], (c) the Natural Sciences and Engineering Research Council of Canada, grants RGPIN-2018-05705 and RGPAS-2018-522715 [SJ], and (d) Consejo Nacional de Ciencia y Tecnología (CONACyT), Mexico, LMAX Exchange, London, and Oriel College, Oxford [LSB].
"Lévy-Ito models in finance." Probab. Surveys 18 132 - 178, 2021. https://doi.org/10.1214/21-PS1