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A Course in Financial Calculus by Alison Etheridge

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5 The Black–Scholes model

Summary

We now, finally, have all the tools that we need for pricing and hedging in the continuous time world of Black and Scholes. We shall begin with the most basic setting, in which our market has just two securities: a cash bond and a risky asset whose price is modelled by a geometric Brownian motion.

In §5.1 we prove the Fundamental Theorem of Asset Pricing in this framework. In line with our analysis in the discrete world, this provides an explicit formula for the price of a derivative as the discounted expected payoff under the martingale measure. Just as in the discrete setting, we shall see that there are three steps to replication. In §5.2 we put this into action for European options. For simple calls and puts, ...

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