Actuarial Finance

11 Multi-period binomial tree model

The one-period and two-period binomial trees presented in Chapters 9 and 10 had the advantage of introducing important concepts and procedures, such as replication, portfolio dynamics and risk-neutral formulas, in fairly simple setups. Now, we seek to generalize these ideas to a model with more than two time steps.

The main objective of this chapter is to introduce the general multi-period binomial tree model and its own challenges. The algorithmic approach we will use throughout the chapter will allow a straightforward computer implementation of the model and it will lay the foundations for the limiting case known as the Black-Scholes-Merton model. The specific objectives are to:

• build a general binomial tree and relate the asset price observed at a given time step to the binomial distribution;
• understand the difference between simple options and path-dependent options;
• establish the dynamic replicating strategy to price an option;
• apply risk-neutral pricing in a multi-period model.

11.1 Model

Fix n ≥ 1, the number of time steps or periods. In an n-period model there are n + 1 time points: 0, 1, 2, …, n. One should think of time 0 as today while time 1, time 2, etc. are in the future. Often, time 0 will represent the issuance date of a derivative whose maturity occurs at time n.

When time is expressed without units, the timeline is as follows:

As is the case for most discrete-time financial models, it is often more convenient to ...