Lévy-Ito models in finance

Abstract

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.

New Approximations to Bond Prices in the Cox–Ingersoll–Ross Convergence Model with Dynamic Correlation

“…In addition, Wiener processes may not be sufficient to capture the stochastic character of the interest rates. A popular class of models uses Lévy processes to model jumps; we refer the reader to [21][22][23] for examples of interest rate models of this form.…”

Minimal Kullback–Leibler Divergence for Constrained Lévy–Itô Processes

“…. 4 For a univariate random variable Y and a probability measure \BbbQ , the \BbbQ -VaR at level \beta \in (0, 1) Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.…”

“…In this space we introduce families of so-called L\' evy--It\ô processes. For aspects of the theory of such processes, see [1,25,4]. Here, we consider an m-dimensional \BbbP -Brownian motion \bfitW = (W 1 , .…”