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A Course in Financial Calculus by Alison Etheridge

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4 Stochastic calculus

Summary

Brownian motion is clearly inadequate as a market model, not least because it would predict negative stock prices. However, by considering functions of Brownian motion we can produce a wide class of potential models. The basic model underlying the Black–Scholes pricing theory, geometric Brownian motion, arises precisely in this way. It will inherit from the Brownian motion very irregular paths. In §4.1 we shall see why a stock price model with rough paths is forced upon us by arbitrage arguments. This is not in itself sufficient to justify the geometric Brownian motion model. However in §4.7 we provide a further argument that suggests that it is at least a sensible starting point. A more detailed discussion of the ...

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