Jumps in financial modelling: pitting the Black-Scholes model refinement programme against the Mandelbrot programme[La modélisation des discontinuités boursières : le programme de Mandelbrot et le programme pragmatique]

This paper gives an overview of the financial modelling of discontinuities in the behaviour of stock market prices. I adopt an epistemological perspective to present to the two main competitors for this stake: Mandelbrot's programme and the non-stable Lévy processes based approach. I explain this contest using the De Bruin's notions of refinement programme, over-mathematisation and model-tinkering: I argue that the non-stable Lévy based approach of discontinuities can be viewed as a " Black-Scholes model refinement programme " (BSMRP) in the De Bruin's sense, launched against the radical view of Mandelbrot. I use Sato's classification to contrast the two competitors. Next I present the two strands of research from an historical perspective between 1960 and 2000. Mandelbrot's initial model based on alpha-stable motions initiated huge controversies in the finance field and failed to fully describe the observed behaviour of returns due to the stronger fractal hypothesis. The mixed jump-diffusion non fractal processes began in the 1970s, followed after two decades by infinite activity processes in the 1990s. At the end, the time-change representation of the 2000s seems to unify the two competitors.