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A Course in Financial Calculus by Alison Etheridge

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3 Brownian motion

Summary

Our discrete models are only a crude approximation to the way in which stock markets actually move. A better model would be one in which stock prices can change at any instant. As early as 1900 Bachelier, in his thesis ‘La théorie de la speculation’, proposed Brownian motion as a model of the fluctuations of stock prices. Even today it is the building block from which we construct the basic reference model for a continuous time market. Before we can proceed further we must leave finance to define and construct Brownian motion.

Our first approach will be to continue the heuristic of §2.6 by considering Brownian motion as an ‘infinitesimal’ random walk in which smaller and smaller steps are taken at ever more frequent ...

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