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Pricing the Future: Finance, Physics, and the 300-year Journey to the Black-Scholes Equation by George Szpiro

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ELEVEN
Accounting for Randomness
I HAVE ALREADY ALLUDED TO THE MATHEMATICAL TECHNIQUES of calculus, that is, of differentiation and integration as formalized by Isaac Newton and Gottfried Wilhelm Leibniz. Bachelier was the first mathematician to bring calculus to bear on problems involving probability, Kolmogorov the most recent. But there was a problem. Simple calculus deals with smooth mathematical functions that are, for that reason, aptly called differentiable. Brownian motion, however, the linchpin for the understanding of financial markets, is jerky and jiggly and nowhere near smooth. This is why it is said to be “nondifferentiable.” In order to apply techniques from regular calculus to jerky processes, a new approach was required—stochastic ...

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