Actuarial Finance

9 One-period binomial tree model

In the first part of this book, whenever we discussed options and other derivatives, we always considered the initial premium as given. We briefly discussed that to find the price of an option, a mathematical model for the evolution of the underlying asset price would be needed.

In the next chapters, we will introduce our very first class of market models for the price of the underlying asset, namely the binomial tree model. It is probably the most famous discrete-time financial model to represent the evolution of a stock price, an index value, etc. We will mostly focus on pricing and hedging financial and actuarial derivatives written on a risky asset whose price follows such a model.

Before introducing the general multi-period binomial tree model, we will have a first (but deep) look at the very important one-period and two-period versions of this model. These two special cases will allow us to introduce all the main ideas and concepts in a technically (and financially) easy fashion.

The main objective of this chapter is to investigate the one-period binomial tree model to price options and other derivatives. The specific objectives are to:

• describe the basic assets available in a one-period binomial model;
• identify the assumptions on which a one-period binomial model is based;
• understand how to build a one-period binomial tree;
• understand how to price a derivative by replicating its payoff;
• rewrite the price of a derivative using risk-neutral ...