Extreme Events in Finance

Chapter 8Lévy Processes and Extreme Value Theory

Olivier Le Courtois¹ and Christian Walter²

¹EM Lyon Business School, Ecully Cedex, France

²Fondation Maison des Sciences de l'Homme, Paris, France

8.1 Introduction

There are fundamentally two different ways of viewing the uncertainty of financial asset prices in continuous time. The first assumes the principle of continuity, the second does not. According to the first view, following Bachelier (1900) legacy, price movements are modeled by continuous diffusion processes, such as, for instance, Brownian motion. According to the other view, following Mandelbrot's (1963) legacy, price movements are modeled by discontinuous processes, such as, for instance, Lévy processes. In this chapter, we develop on the relationships connecting the Lévy processes and extreme value theory (EVT).

We begin by defining the modeling alternative and the challenges contemporary finance has to tackle. Next, we present the link with EVT. A convenient way of thinking the modeling alternative for today's financial stakes is to come back to the history of science to exhibit the roots of the puzzle. The story begins in the eighteenth century: Leibniz and Newton, the inventors of differential calculus, stated that “Natura non facit saltus” (nature does not make jumps). In other words, in physics, the so-called principle of continuity states that change is continuous rather than discrete. This same principle underpinned the thoughts of Linné on the classification ...