Actuarial Finance

10 Two-period binomial tree model

For more flexibility and accuracy, we can add an intermediate time step, or equivalently another time period, to the one-period binomial tree model: the result is the two-period binomial tree model.

In the next chapter, we will focus on the general multi-period binomial tree model. In this chapter, we will take the time to provide a thorough treatment of its two-period version. Doing so will allow us to:

1. better understand the construction of the general binomial tree and the dynamics of the underlying asset over time;
2. focus on the replication procedure over more than one period, which is important for risk management purposes;
3. analyze the complexity of pricing path-dependent derivatives and variable annuities;
4. consider more complex situations, such as dollar dividends and stochastic interest rates.

Consequently, the main objective of this chapter is to replicate and price options and other derivatives in a two-period binomial tree model. The specific objectives are to:

• recognize how to build a two-period binomial tree using three one-period binomial trees;
• understand the difference between recombining and non-recombining trees;
• build the dynamic replicating strategy to price an option;
• apply the risk-neutral pricing principle in two periods;
• determine how to price options in more complex situations: path-dependent options, dollar dividends, variable annuities, stochastic interest rates.

10.1 Model

In a two-period model there ...