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Financial Calculus by Martin Baxter, Andrew Rennie

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Chapter 3

Continuous processes

 

 

 

Stock prices are not trees. The discrete trees of the previous chapter are only an approximation to the way that prices actually move. In practice, a price can change at any instant, rather than just at some fixed tick-times when a portfolio can be calmly rebalanced. The binary choice of a single jump ‘up’ or ‘down’ only becomes subtle as the ticks get closer and closer, giving the tree more and ever-shorter branches. But such trees grow too complex and we stop being able to see the wood.

We shall have to start from scratch in the continuous world. The discrete models will guide us – the intuitions gained there will be more than useful – but limiting arguments based on letting δt tend to zero are too dangerous ...

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