# Chapter 3 The Binomial Model

• Published 2017

#### Abstract

In Chapter 1 the linear derivatives were considered. They were priced with static replication and payo tables. For the non-linear derivatives in Chapter 2 this will not work since the payo function is non-linear and then not possible to replicate with a linear product, using a buy-and-hold strategy like the payo table. The non-linear derivatives have much more sophisticated payo functions and the price will depend on the probability distribution of the underlying asset. That is, we need a stochastic model for the underlying asset in order to price the derivatives in Chapter 2. This chapter considers one of the most simple and yet non-trivial stochastic models, namely the binomial model. The binomial model has previously been used in practice but is nowadays rarely used, it have been exchanged with more general nite di erence methods. However, the binomial model turn out to be an excellent pedagogical example on how the arbitrage theory applies in a stochastic model. Along with the presentation of the binomial model we will point out how the arbitrage theory applies to a stochastic model. There are some general facts that can be stated already in this simple model. Furthermore, we will only consider European type derivatives. From a practical point of view this is strange since the bene t of the binomial model is for path dependent options, in particular American and Bermudan options. However in this presentation we have decided to put practice aside in favour for teaching arbitrage pricing. To set the scene the chapter begins with a section where the European call option is considered in the payo table. It is a ridiculous example but it clearly shows why the buy-and-hold strategy fails for the non-linear derivatives and why a stochastic model is needed. The example is good to keep in mind throughout the book since it is easy to drown in the models and forget why they are needed in the rst place. After this introduction the binomial model is de ned. This is done in several steps. The binomial model is a discrete model in both time and space (that is price of the underlying) and

### Cite this paper

@inproceedings{2017Chapter3T, title={Chapter 3 The Binomial Model}, author={}, year={2017} }