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The Brownian Motion in Finance: An Epistemological Puzzle

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Abstract

While in medicine, comparison of the data supplied by a clinical syndrome with the data supplied by the biological system is used to arrive at the most accurate diagnosis, the same cannot be said of financial economics: the accumulation of statistical results that contradict the Brownian hypothesis used in risk modelling, combined with serious empirical problems in the practical implementation of the Black-Scholes-Merton model, the benchmark theory of mathematical finance founded on the Brownian hypothesis, has failed to change the Brownian representation, which has endured for more than fifty years despite the extent of its invalidation by experience. Without any statistical foundations, one mathematical representation (Brownian motion) has become the established approach, acting in the minds of practitioners as a “prenotion” in the sense the word is used by Durkheim (The rules of sociological method: and selected texts on sociology and its method, Free Press, New York, 1894), i.e. a “schematic, summary representation” which has produced a kind of spontaneous epistemology. The question arises of the persistence of this mathematical (Brownian) representation, which has been the basis for every financial risk modelling approach: how can its long life be explained? How was this spontaneous epistemology formed, and why did it prove to be so persistent? To address this question and offer an answer, I will test the various dynamics of scientific knowledge used with reference to financial modelling. All these dynamics are specific ways of describing the relationship between knowledge of a phenomenon (here, the representation of a stock market dynamic) and the phenomenon itself (here, stock price fluctuations). Observing that it is impossible for the positivist approach to solve the financial puzzle, I turn to the three principal postpositivist dynamics, developed by Kuhn, Lakatos and Quine. I shall try to make to speak these representations of science for financial research, in order to reflect on the dominance of the Brownian representation in finance. We shall see that none of the epistemologies examined can explain why the Brownian representation continues to be used in mathematical finance research. I shall then propose an alternative hypothesis, concerning a significant pervading mental model that has irrigated both academics and practitioners in the financial sector: the “principle of continuity” introduced into economics by Marshall in (Principles of economics, Macmillan, London, 1890). I consider that this principle of continuity has become a “persistent idea” in the form of a viral approach to representations, and I give this persistent idea the metaphorical name of the “Brownian virus”. Then, to explain the spread of this Brownian virus through the financial sector, the contamination of financial practices by this mental representation founded on the principle of continuity, I introduce the concept of the “financial Logos”, a discourse that structures practices and organisations, calculations, prudential regulations and accounting standards, leading to a general financialisation of society from the 1980s onwards.

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Notes

  1. French: “continue à droite, limite à gauche”, i.e. right continuous with left limits.

  2. Lévy processes, labelled after the French mathematician Paul Lévy, are continuous-time stochastic processes with independent and identically distributed (IID) increments. With the exception of Brownian motion with drift, they consist entirely of jumps. See for example Bertoin (1996) and Sato (1999).

  3. “Standard” means W0 = 1, the increments are independent and for 0 < s <t, the increment W(t)−W(s) is normally distributed with mean zero and variance t−s.

  4. A replicating portfolio for a given asset is a portfolio of assets with the same financial properties (e.g. cash flows).

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  1. Kedge Business School, Marseille, France

    Christian Walter

  2. Chaire Ethique et Finance, Collège d’Études Mondiales, FMSH, ISJPS (UMR 8103), Univ. Paris 1-Panthéon-Sorbonne, Paris, France

    Christian Walter

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Walter, C. The Brownian Motion in Finance: An Epistemological Puzzle. Topoi 40, 1–17 (2021). https://doi.org/10.1007/s11245-019-09660-7

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