Financial Signal Processing and Machine Learning

Chapter 9Correlated Poisson Processes and Their Applications in Financial Modeling

Alexander Kreinin

Risk Analytics, IBM, Canada

9.1 Introduction

Multivariate risk factor models set the foundation of financial risk measurement. The modern financial theory often recommends jump-diffusion models to describe dynamics of the individual risk factors, such as interest rates, foreign exchange rates, stock indices, and volatility surfaces (Kou, 2002; Lipton and Rennie, 2008; Merton, 1976; Musiela and Rutkowsky, 2008), One of the most popular model of jumps, the Poisson model, requires introduction of a codependence structure in the multivariate setting.

The multivariate Gaussian diffusion models are traditionally popular in financial applications (Musiela and Rutkowsky, 2008). In this class of models, the dynamics of the risk factors are described by the Gaussian stochastic processes. The calibration problem in this case can be reduced to the estimation of the drift vector and the diffusion matrix describing the covariance structure of the risk factor space. The only constraint imposed on the covariance matrix is nonnegativity of its eigenvalues.

It is very well known that calibration of the models for equity derivatives pricing can be performed satisfactorily in the class of jump-diffusion processes (Kou, 2002; Kyprianou et al. 2005; Merton, 1974). Once the jump processes are introduced, the calibration problem becomes more challenging and, technically, more demanding. In particular, ...