In this chapter, we will discuss the binomial model. Although the use of binomial tree has been superseded by other numerical techniques in recent years, they remain a good way to understand certain option structures. Regardless of whether the binomial model is still used in practice to price options and other derivatives, it is still very worthwhile to understand the approach.^{1}

The binomial model is useful in that it demonstrates clearly, in discrete time, a concept that we will introduce in Chapter 5, i.e., option replication and risk-neutral valuation. In particular, the binomial model provides a link between risk-neutral probability and payoff replication.^{2} A key message is that there will only be one price generated for a given option specification since any other price would offer arbitrage opportunities. This unique price is independent of the risk aversion of market participants and the distribution of future price movements of the underlying asset.

We will see how the binomial model can be used to price both European and American call and put options, with examples that will reference a variety of different underlying asset classes. In these examples, we will generally assume that we are working with mid prices but this is just for simplicity and, in reality, one will almost always have to deal with a bid or an offer price. We should note that prices for vanilla European calls and puts can be calculated as a closed-form solution from, say, the ...

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