Shot-Noise Processes in Finance

TL;DR

This paper explores the mathematical properties and financial applications of shot-noise processes, emphasizing their flexibility, tractability, and conditions for arbitrage-free modeling in finance.

Contribution

It introduces a general formulation of shot-noise processes, analyzes their Markovianity, and derives conditions for arbitrage-free measure changes in financial modeling.

Findings

Markovianity is equivalent to exponential decay of the noise function

Explicit Fourier transforms for the class of shot-noise processes

Derived drift conditions for absence of arbitrage in financial models

Abstract

Shot-Noise processes constitute a useful tool in various areas, in particular in finance. They allow to model abrupt changes in a more flexible way than processes with jumps and hence are an ideal tool for modelling stock prices, credit portfolio risk, systemic risk, or electricity markets. Here we consider a general formulation of shot-noise processes, in particular time-inhomogeneous shot-noise processes. This flexible class allows to obtain the Fourier transforms in explicit form and is highly tractable. We prove that Markovianity is equivalent to exponential decay of the noise function. Moreover, we study the relation to semimartingales and equivalent measure changes which are essential for the financial application. In particular we derive a drift condition which guarantees absence of arbitrage. Examples include the minimal martingale measure and the Esscher measure.